Modeling the Dynamic of Herpes Simplex Virus II Incorporating Voluntary Laboratory Test and Medical Treatment

Abstract

This study develops a mathematical model to investigate the transmission dynamics of HSV-II within the framework of symmetry in dynamical systems. The basic reproduction number ($R_0^{HSV}<1$) of the model was determined using the next generation method (NGM). The stability of the disease-free equilibrium point was also investigated using the Routh–Hurwitz Criterion and was found to be locally asymptotically stable (LAS) when $R_0^{HSV}<1$ but not globally asymptotically stable (GAS). To help ensure that the control variables were included correctly, sensitivity analysis was performed on the fundamental reproduction number parameters. Four control variables were applied for the model: HSV-II vaccination, effective condom use, laboratory test, and treatment. The optimality system was solved using Pontryagin’s maximum principle (PMP) to establish the optimal control strategy for combating the spread of the disease. Numerical solution was obtained by using the forward-backward Runge–Kutta fourth-order approach. The most effective approach to help eradicate HSV-II disease in the system is to combine the HSV-II vaccine, effective condom use, laboratory testing, and HSV therapy (strategy D).

1. Introduction

HSV is a chronic incurable disease that causes a latent infection in its human host, resulting in recurrent outbreaks of mucocutaneous lesions and other symptoms over the course of a person’s life [1]. There are two types of HSV, HSV-I and HSV-II. HSV-I, which is usually less serious than HSV-II, is mostly linked to oral infections like cold sores and is commonly spread by direct contact with bodily fluids like saliva during kissing or sharing utensils. Through oral–genital contact, HSV-I can also result in genital infections, despite its normally milder nature. On the other hand, HSV-II is more dangerous and is primarily identified as a sexually transmitted virus. It is spread through sexual contact with an infected person’s bodily fluids, such as semen or vaginal secretions, and mainly causes genital herpes. Intimate skin-to-skin contact, including unprotected sexual contact, can expose a healthy, non-infected person to HSV-II [2]. Both types of HSV have important public health concerns.

A number of factors greatly increase the likelihood of HSV transmission [1,3]. These include other co-occurring STIs, using condoms inconsistently or non-systematically, and having several sexual partners. In addition to raising the risk of exposure, these factors also help explain why HSV is so common in certain groups. A significant trend has been noted among pregnant women within the subgroup of women who reported having three or fewer lifetime sexual partners. The percentage of people who are still seronegative for HSV is rising. This implies that these women have either not been exposed to the virus or have not produced detectable antibodies against it, even if they engage in lower-risk sexual behavior. This development raise serious concerns. The risk of primary HSV acquisition during pregnancy is higher for a greater proportion of these seronegative expecting mothers. Due to the fact that the immune system has not been exposed to the virus before, primary infections during pregnancy are more harmful and can worsen the disease and increase the risk of transmission to the foetus. The risk of serious neonatal problems, such as neonatal herpes, which can result in long-term morbidity or even death, is present with this vertical transmission. This can happen during pregnancy, delivery, or the first few days after giving birth [4]. These results highlight the necessity of focused public health initiatives to lower the risk of HSV acquisition in this susceptible population and safeguard the health of expectant mothers and newborns.

In many nations, the most prevalent cause of genital ulcers is the herpes simplex virus type II (HSV-II). Globally, adults and adolescents between the ages of 15 and 49 were predicted to have contracted HSV-II in 23.2 million new cases in 2016, with younger age groups having the highest prevalence. HSV-II is present in more than 10% of people in the Western Pacific, Southeast Asian, Eastern Mediterranean, and European regions [5]. Africa has the highest frequency of HSV-II, with about 40% of the population infected with HSV-II. Women were expected to have 14.7 million new infections and 313.5 million prevalent infections in 2016, while men were estimated to have 9.2 million new infections and 178.0 million prevalent infections. This suggests that females consistently have a higher rate of HSV-II infection than males. The higher infection prevalence in women is explained by their biological susceptibility to HSV-II infection. In 2015–2016, the National Centre for Health Statistics (NCHS) reported that approximately 12% of sexually active individuals in the United States between the ages of 15 and 49 had HSV-II [6]. Recurrence episodes can happen at any point in the infected person’s life, which facilitates HSV-II disease dynamics and transmission until the virus replicates itself once more, a process known as viral reactivation [7,8]. It is important to keep in mind that a reinfection and HSV recurrence are not the same thing. Recurrence of HSV-II is a condition in which the virus periodically reactivates from its latent state in the dorsal root ganglia and is linked to a variety of other conditions. Frequent triggers include emotional and physical stress, exhaustion, exposure to intense heat or sunlight, fever, immunosuppression, and hormonal changes like those that occur during menstruation. These factors can cause a recurrence of symptomatic lesions by weakening the body’s defences or directly stimulating viral reactivation [1,7,8]. The median number of recurrences in the first year after the original outbreak is usually two. This median increases to five by the second year, indicating a higher rate of reactivation with time ([7]). Individual recurrence rates, however, can vary greatly. Studies by [1,9] show that 16% of patients have more than five outbreaks yearly, 36% report two to five recurrences, and 47% have fewer than two recurrences annually. This variation highlights the impact of lifestyle choices, suppressive antiviral medications, and individual immunological responses. Comprehending these trends is essential for customizing treatment plans, which frequently involve antiviral drugs, stress-reduction methods, and lifestyle changes to lessen the negative effects of HSV-II recurrences on a patient’s quality of life.

Constitutional and non-constitutional symptoms are the two primary categories of HSV-II infection symptoms [8]. Constitutional symptoms, such as localised discomfort, and other symptoms that are sometimes mistaken for indications of another infection or disease, might make it difficult to properly diagnose HSV-II. On the other hand, lesions and ulcers—both of which are readily identifiable and fluctuate in size and severity—are examples of severe symptoms that are not constitutional.

In order to establish the clinical diagnosis of genital herpes, laboratory testing—also known as microbiological testing—is essential. Using a range of diagnostic methods, this procedure looks for the presence of HSV-II in the vaginal or genital area. These techniques include antigen detection assays, virus isolation and culture, and sophisticated molecular diagnostic techniques like polymerase chain reaction (PCR), which can detect HSV-II DNA with high sensitivity and specificity. For an appropriate diagnosis, vesicular lesions must be sampled within three days of commencement. It is more likely that a trustworthy result will be produced during this time since the virus is actively multiplying and the concentration of viral particles is at its peak. After this period, when the lesions start to heal and virus shedding slows down, the sensitivity of these tests may diminish. Early detection of an infection suggests that the virus has already taken hold of the host and that tissue damage linked to viral replication may have started [10]. The clinical setting frequently influences the selection of diagnostic method. Because of their specificity, viral isolation and culture continue to be the gold standard; nonetheless, they are time-consuming and need specialised laboratory facilities. On the other hand, molecular diagnostics such as PCR have gained popularity due to their quick response time, capacity to identify low viral levels, and suitability even in situations when sample quality may be below ideal. Such accurate diagnostic techniques must be incorporated into clinical practice in order to manage symptoms, inform treatment choices, and lower the chance of additional transmission.

The two techniques used to diagnose HSV-II are the standard diagnostic test and the microbiological lab test. The phrase “routine diagnostic test” describes a clinical diagnostic evaluation of common symptoms including painful anogenital ulceration, vaginal or urethral discharge, superficial dyspareunia, and external dysuria [4]. The body’s overall reaction to a viral infection is reflected in these localized symptoms, which are frequently accompanied by systemic symptoms including fever, flu-like symptoms, and general malaise.

There is considerable variation in the anogenital ulcers brought on by HSV-II, from small, isolated lesions to larger ones. Many ulcerations may vary in severity. These variations frequently rely on the presence of concurrent diseases or infections, the person’s immunological response, and the stage of the infection (primary versus recurring). The symptoms of primary infections are typically more severe, exhibiting prominent systemic signs, extensive ulceration, and considerable discomfort. On the other hand, recurrent episodes can still be extremely distressing, but they are typically milder and more localized.

Even though standard diagnostic techniques are very helpful for preliminary assessment, they have drawbacks, especially when it comes to differentiating HSV-II from other anogenital ulceration causes including syphilis, chancroid, or autoimmune diseases. Therefore, in order to establish the existence of HSV-II and provide suitable therapeutic options, microbiological laboratory tests are frequently added to clinical diagnosis. Particularly in uncommon or subclinical instances, these tests offer a more conclusive diagnosis, guaranteeing proper treatment and lowering the possibility of consequences or transmission.

The antiviral drug acyclovir is one of the most popular and successful ways to treat and stop the spread of HSV-II. Acyclovir functions by selectively targeting infected cells, where the viral enzyme thymidine kinase phosphorylates, activating it. While successfully preventing viral replication in HSV-infected cells, this selective activation guarantees that acyclovir is harmless for healthy cells. It is appropriate for use in both healthy people and those with weakened immune systems because of its effectiveness in treating a variety of HSV-II infections, from mild instances to severe symptoms [11].

Whether the infection is a primary episode or a recurrence determines the course of acyclovir treatment. Oral administration of 200 mg or 400 mg three times a day for seven to ten days is the recommended dosage for early HSV-II infections, which are characterised by more severe symptoms and systemic involvement. Patients benefit greatly from this regimen, which shortens the length of symptoms by roughly one week and speeds up lesion healing by roughly six days [12].

The dosing approach is different for recurring episodes of HSV-II, which are usually less severe but nonetheless problematic. Acyclovir can be taken orally in a dose of 800 mg or 200 mg five times a day. These protocols are intended to reduce the intensity of symptoms and shorten the duration of recurring outbreaks. To maximize effectiveness, treatment must be started as soon as possible, ideally at the first indication of symptoms or prodromal feelings. There is currently no effective vaccine to prevent HSV-II, despite the efficacy and widespread use of antiviral medications like acyclovir. The lack of a vaccine emphasizes how difficult it is still to stop early infections and manage the virus’s wider effects on public health. To combat this, several studies are being conducted to create preventive vaccinations and alternate treatment approaches.

Numerous mathematical models have been developed to study the transmission dynamics, treatment strategies, and potential impacts of vaccination programs on herpes simplex virus (HSV) infections. These models provide valuable insights into how various biological, social, and policy factors influence the spread and control of HSV infections. One of the earliest studies, carried out by [13], assessed the kinetics of HSV transmission in both humans and animals, providing the groundwork for comprehending how various hosts interact to propagate the virus. Several studies have been conducted expanding on [13] and examining how vaccination campaigns have affected HSV-II prevalence. These studies demonstrated the substantial potential of vaccination in lowering the disease’s burden by estimating that a vaccination program might stop 11 million HSV-II infections in the US within 10 years of its start.

A qualitative examination of the dynamics of HSV-II with and without vaccine treatments was presented in 2009 by [14]. Their research provided a better knowledge of how vaccinations can change the prevalence and recurrence patterns of diseases. The modeling attempts also included HSV-I. In [15], the authors developed an epidemic model that focused on HSV-I and included vaccine-therapy reproductive numbers. By contrasting various immunization and treatment approaches, they offered valuable information for improving interventions to successfully contain epidemics. In [16], the dynamics of HSV-II transmission among Zimbabwean inmates in a more localized setting was investigated. This study showed how customised treatments could slow the spread of disease in high-risk groups and emphasized the importance of focused counselling and education programs as important control measures within this particular community. Other control stategies applied to different dynamics, as in [17], were also examined to gain more insight into methodologies around control strategies.

Furthermore, ref. [18] examined how behavioral factors like sexting and peer pressure affect the spread of HSV-II in teenagers. Their mathematical model emphasized how crucial it is to take sociocultural factors into account when creating successful prevention initiatives, particularly for populations of vulnerable young people.

These investigations collectively demonstrate the value of mathematical modeling in tackling the intricate dynamics of HSV infections. They offer essential frameworks for assessing intervention tactics, refining immunization campaigns, and comprehending the ways in which behavioral and social factors contribute to the persistence and spread of the disease.

This study expands on the work of the HSV-II vaccination model of [5] by analyzing HSV-II dynamics while taking laboratory tests into account. The purpose of the expanded model is to assess the ways in which vaccination and other control strategies work in conjunction with diagnostic tests to affect the prevalence and transmission of HSV-II in a community. The interaction between infected individuals, those undergoing testing, and those receiving treatment or immunization is captured by the model using a system of differential equations.

The study aims to gain a better understanding of laboratory testing’s involvement in early detection and management of HSV-II infections. In order to lessen the chance of additional transmission, laboratory testing can help with prompt measures such as partner notification, patient counselling, and targeted antiviral therapy. Additionally, the model assesses the ability of testing to detect asymptomatic carriers, who frequently do not realize they are infected but are yet capable of spreading the virus to others.

The study also investigates the best control methods for preventing HSV-II transmission. To have the greatest possible impact on halting the disease’s spread, these tactics entail allocating resources among testing, treatment, and immunization initiatives. A dynamic examination of the ways in which diverse factors, including vaccination rates, treatment effectiveness, and testing coverage, impact the disease’s growth and management over time is made possible by the application of differential equations.

Through the extension of the approach used in [5], this study offers a more thorough comprehension of the dynamics of HSV-II transmission and emphasizes the significance of incorporating laboratory testing into public health plans for disease management. Policies intended to reduce the prevalence of HSV-II and lessen its effects on impacted groups may be influenced by the findings.

There are seven sections in this paper. Background data and an overview of pertinent research on the HSV-II virus are given in Section 1. The extended HSV-II model is presented in Section 2, together with information on its parameters, initial research, and crucial mathematical evaluations. In Section 3, the basic reproduction number and the stability analysis of the model’s equilibrium states are presented. The reproduction number’s sensitivity analysis is examined in Section 4. The optimal control problem, including the formulation and solution of the optimality system, is presented in Section 5. Section 6 presents numerical simulations, graphical interpretations, and a brief discussion. Lastly, a conclusion is presented in Section 7.

2. Herpes Simplex Virus-II Model

2.1. HSV-II Model Description

The HSV-II model describes the transmission dynamics of the herpes simplex virus type II epidemic with vaccination, screening, and treatment compartments. The model specifies the rate of change over time of individuals in eleven (11) state variables: unvaccinated susceptible, $S\left(t\right)$; vaccinated susceptible, $V\left(t\right)$; unvaccinated infected individuals who are infectious (either asymptomatic or symptomatic), $H_u\left(t\right)$; vaccinated but infected individuals who are infectious (either asymptomatic or symptomatic), $H_v\left(t\right)$; unvaccinated infected individuals who are quiescent, $Q_u\left(t\right)$; vaccinated but infected individuals who are quiescent $Q_v\left(t\right)$; unvaccinated exposed individuals, $E_u\left(t\right)$; the vaccinated exposed population, $E_v\left(t\right)$; aware infectious individuals (either asymptomatic or symptomatic), $H_A\left(t\right)$; quiescent aware infectious individuals, $Q_A\left(t\right)$; and the treated class $T_h\left(t\right)$. A fraction, $p \in$, of the newly recruited sexually active (adolescent) individuals have received vaccines at the rate $\mathrm{\Pi }$; in this instance, p denotes the percentage of vaccinated individuals to whom the vaccine is administered. At a rate $\zeta$, susceptible people receive vaccinations, and it is predicted that the vaccine will decline at a rate $\varpi$. The unvaccinated susceptible population is made up of people who are not covered by the vaccination program or who have received vaccinations but for whom the shots do not take effect (that is, do not trigger an immunological response).

Consequently, susceptible individuals who enter the sexually active pool are deemed to have been “successfully vaccinated,” contingent upon the vaccination coverage rate and the proportion of vaccinated persons in whom the vaccine takes ∈. As long as the vaccination is effective, these people are immune to infection; but, because the vaccine offers only partial protection, some of these people will contract the disease. $\mathrm{\Psi }$ indicates the level of immunological protection; if $\mathrm{\Psi }=1$, the vaccination offers 100% protection against disease. Three therapeutic benefits are available to those who have received vaccinations but still become infected (because $\mathrm{\Psi }$ is less than 1):

• The average length of their viral shedding episodes is shorter;
• They have fewer viral shedding episodes;
• Compared to those who are not vaccinated, they are less likely to spread infection.

Thus, vaccinated infected individuals have reduced shedding. Individuals in both the vaccinated and unvaccinated quiescent state are assumed to be infectious but at a reduced rate compared to those populations in the vaccinated infected $H_u\left(t\right)$ and unvaccinated infected $H_v\left(t\right)$ class, respectively.

Further, the aware infectious class is made up of HSV-II infected individuals who, after being diagnosed or becoming aware of their infection, alter their behavior by partially adhering to control measures (such as limiting sexual contacts or receiving treatment), hence maintaining a decreased but non-zero transmission potential. Those whose awareness results in successful withdrawal from transmission, such as abstinence or rigorous adherence to suppressive therapy, are included in the quiescent aware class. To differentiate between various awareness-driven behavioral states and their matching transmission coefficients, these compartments are necessary. Because the effects of awareness-based and treatment controls on HSV-II transmission dynamics may not be adequately modelled within a single infectious class, the extended compartmental structure is crucial. Equally important, contact patterns and treatment adherence are directly impacted by the inclusion of aware compartments, which represent physiologically and behaviorally unique states resulting from diagnosis, health education, or symptom awareness. Consistent with awareness-driven regulatory mechanisms, individuals in aware classes exhibit altered transmission and advancement rates when compared to unaware individuals. This necessitates capturing these varied reactions and their varying consequences on disease dynamics and control outcomes, which cannot be sufficiently represented in a reduced compartmental framework.

Furthermore, vaccination recipients may experience a lower incidence of breakthrough infection because it is presumed that the vaccine is defective. The corresponding force of infection ${\lambda }_v$ is given as follows:

$${\lambda }_v={\displaystyle \frac{\beta [H_u+{\zeta }_1{H}_A+{\eta }_1{H}_v+\theta (Q_u+{\zeta }_2{Q}_A+{\eta }_2{Q}_v)]}{N}},$$

with $0<{\eta }_1,{\eta }_2,{\zeta }_1,{\zeta }_2,\theta <1$ as the modification parameters that account for the decrease in infectiousness caused by the vaccine for people in the $H_v$ and $E_v$ classes as well as the decrease in infectiousness of people in the aware states relative to infectious people who have not received a vaccination in the $H_u$ and $E_u$ classes, respectively.

Consequently, the total population for the vaccinated and susceptible populations is given by the following:

$$N\left(t\right)=S\left(t\right)+V\left(t\right)+E_u\left(t\right)+E_v\left(t\right)+H_u\left(t\right)+H_A\left(t\right)+H_v\left(t\right)+Q_u\left(t\right)+Q_v\left(t\right)+Q_A\left(t\right)+T_H\left(t\right),$$

and thus their rates of change are, respectively, given by the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \mathrm{\Pi }(1-p\in )-{\lambda }_{v}S+\varpi V-(\xi +\mu )S,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& \mathrm{\Pi }p\in -(1-\mathrm{\Psi }){\lambda }_{v}V+\xi S-(\varpi +\mu )V.\hfill \end{array}$$

On the other hand, an exposed asymptomatic unvaccinated population in the asymptomatic exposed condition ($E_u$) is created by the force of infection for the unvaccinated ${\lambda }_v$. This population is subsequently reduced as symptoms appear (at a rate of $\sigma$) and by natural death ($\mu$):

$${\displaystyle \frac{d{E}_u}{dt}}={\lambda }_{v}S-({\sigma }_1+\mu )E_u.$$

The emergence of symptoms at a rate of ${\sigma }_2$ and the natural death rate at a rate of $\mu$ lower the population of vaccinated individuals who do not show symptoms ($E_v$), which is generated by breakthrough infection at a rate of $(1-\mathrm{\Psi }){\lambda }_v$ and is given thus:

$${\displaystyle \frac{d{E}_v}{dt}}=(1-\mathrm{\Psi }){\lambda }_{v}V-({\sigma }_2+\mu )E_v.$$

Both the asymptomatic and symptomatic aware state ($H_A$) is propagated by screening at a rate ${\tau }_{1}e$ with some efficacy and increases by reactivation from the quiescent aware state ($Q_A$) at a rate $r_A$. It decreases as a result of the natural return to latency (at a rate $q_A$) and progression to treated class (at a rate ${\varphi }_1$ and natural death rate $\mu$). Hence,

$${\displaystyle \frac{d{H}_A}{dt}}={\tau }_{1}e{H}_u+r_A{Q}_A-(q_A+{\varphi }_1+\mu )H_A.$$

The quiescent aware class ($Q_A$) is generated by the rate of return to latency at a rate $q_A$ and also by the rate of screening with some efficacy ${\tau }_{2}e$ and decreases by the natural death rate ($\mu$) as well as the treatment rate ${\varphi }_2$ and the reactivation rate $r_A$. Hence,

$${\displaystyle \frac{d{Q}_A}{dt}}=q_A{H}_A+{\tau }_{2}e{Q}_u-(r_A+{\varphi }_2+\mu )Q_A.$$

The treatment class ($T_H$) is generated by progression from $H_A, Q_A$ at a rate ${\varphi }_1$ and ${\varphi }_2$, respectively. It decreases by natural death rate, $\mu$. Thus,

$${\displaystyle \frac{d{T}_H}{dt}}={\varphi }_1{H}_A+{\varphi }_2{Q}_A-\mu T_H.$$

The reactivation of vaccinated persons in the quiescent state and the development of exposed vaccinated individuals at a rate of ${\sigma }_2$ result in infectious vaccinated individuals ($H_v$). The pace at which quiescence advances, $r_v$, and the natural mortality rate, $\mu$, both lower this population. So,

$${\displaystyle \frac{d{H}_v}{dt}}={\sigma }_2{E}_v+r_v{Q}_v-(q_v+\mu )H_v.$$

The population of unvaccinated infectious individuals $H_u$ is generated by a progression for the unvaccinated asymptomatic class $E_u$ at a rate ${\sigma }_1$, reactivation rate from unvaccinated quotient class $Q_u$ at a rate $r_u$, and decrease by the natural and disease-induced death rate, $\mu$ and ${\delta }_u$, respectively. Thus,

$${\displaystyle \frac{H_u}{dt}}={\sigma }_1{E}_u+r_u{Q}_u-(q_u+{\delta }_u+\mu +{\tau }_{1}e)H_u.$$

The population of quotient unvaccinated infectious individuals in the quotient state is increased by a rate of return to latency, $q_u$, and the rate of waning of the vaccination effect for the vaccinated quotient infectious class, $Q_v$, and decreased by the natural and disease-induced death rates $\mu$ and ${\delta }_u$, respectively. Thus, we have the following:

$${\displaystyle \frac{Q_u}{dt}}=q_u{H}_u+{\alpha }_1{Q}_v-(r_u+{\tau }_{2}e+\mu +{\delta }_{qu})Q_u.$$

Finally, the population of vaccinated infectious individuals in the quiescent state, $Q_v$, is increased by the progression to quiescence of infectious vaccinated individuals at the rate $q_v$. The rate of change of this population is reduced by the reactivation rate $r_v$, the loss of vaccine-induced immunity rate ${\alpha }_1$, and the natural death rate $\mu$. Individuals in the $Q_v$ class who lose their vaccine-induced immunity are moved to the $Q_u$ class at the rate ${\alpha }_1$.

Therefore,

$${\displaystyle \frac{Q_v}{dt}}=q_v{H}_v-(r_v+{\alpha }_1+\mu )Q_u.$$

The schematic diagram of the model is given in Figure 1 and the description of parameters is presented in

Table 1. Parameter values used for the simulation result of the HSV-II model.

Parameter	Description	Value	Source
${\mathrm{\Pi }}_H$	Recruitment rate of susceptible individuals	10,000	[5]
$\mu$	Natural death rate	0.143	[5]
$\beta$	Effective contact rate	0.3	Assumed
$\xi$	Vaccination rate of susceptible individuals	0.6	[5]
$\mathrm{\Psi }$	Efficacy of vaccine	0.65	[5]
$\varpi$	Waning rate of vaccine	$1/15$	Assumed
$r_H$	Reactivation rate in unvaccinated individuals	$365/4$	[5]
$r_u$	Reactivation rate in unvaccinated individuals	$365/4$	[5]
$r_v$	Reactivation rate in vaccinated individuals	$365/4$	[5]
$r_A$	Reactivation rate in aware individuals	$365/4$	Assumed
p	Proportion of newly recruited individuals vaccinated	0.0960	[5]
$q_u$	Rate of return to latency in unvaccinated unaware individuals	$365/2$	[5]
$q_v$	Rate of return to latency in vaccinated individuals	$365/4$	[5]
$q_A$	Rate of return to latency in aware individuals	$365/3$	[5]
${\sigma }_1$	Progression rate to symptoms (unvaccinated asymptomatic)	$365/15$	[5]
${\sigma }_2$	Progression rate to symptoms (vaccinated asymptomatic)	$365/18$	[5]
$\theta$	Modification parameter for lower infectiousness of aware classes	0.5	[5]
${\alpha }_1$	Progression rate related to quiescent infectious classes	$1/2$	[5]
$\epsilon$	Proportion of recruited susceptible individuals in whom the vaccine takes effect	0.6	[5]
${\eta }_1, {\eta }_2$	Modification for reduced infectiousness of vaccinated individuals	0.2, 0.1	[5]
${\delta }_u$	Disease-induced death rate for unvaccinated symptomatic individuals	0.0004	[5]
${\delta }_{qu}$	Disease-induced death rate for unvaccinated quiescent individuals	0.0003	[5]
${\tau }_1$	Screening rate of unvaccinated asymptomatic individuals	0.85	Assumed
${\tau }_2$	Screening rate of individuals in quiescent states	0.85	Assumed
${\varphi }_1, {\varphi }_2$	Treatment rates for aware symptomatic and quiescent individuals	variable	–

.

2.2. HSV-II Model Equations

The HSV-II model equations are given as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \mathrm{\Pi }(1-p\in )-{\lambda }_{v}S+\varpi V-(\xi +\mu )S,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& \mathrm{\Pi }p\in -(1-\mathrm{\Psi }){\lambda }_{v}V+\xi S-(\varpi +\mu )V,\hfill \\ \hfill {\displaystyle \frac{d{E}_u}{dt}}& =& {\lambda }_{v}S-({\sigma }_1+\mu )E_u,\hfill \\ \hfill {\displaystyle \frac{d{E}_v}{dt}}& =& (1-\mathrm{\Psi }){\lambda }_{v}V-({\sigma }_2+\mu )E_v,\hfill \\ \hfill {\displaystyle \frac{H_u}{dt}}& =& {\sigma }_1{E}_u+r_u{Q}_u-(q_u+{\delta }_u+\mu +{\tau }_{1}e)H_u,\hfill \\ \hfill {\displaystyle \frac{d{H}_v}{dt}}& =& {\sigma }_2{E}_v+r_v{Q}_v-(q_v+\mu )H_v,\hfill \\ \hfill {\displaystyle \frac{d{H}_A}{dt}}& =& {\tau }_{1}e{H}_u+r_A{Q}_A-(q_A+{\varphi }_1+\mu )H_A,\hfill \\ \hfill {\displaystyle \frac{Q_u}{dt}}& =& q_u{H}_u+{\alpha }_1{Q}_v-(r_u+{\tau }_{2}e+\mu +{\delta }_{qu})Q_u,\hfill \\ \hfill {\displaystyle \frac{Q_v}{dt}}& =& q_v{H}_v-(r_v+{\alpha }_1+\mu )Q_u,\hfill \\ \hfill {\displaystyle \frac{d{Q}_A}{dt}}& =& q_A{H}_A+{\tau }_{2}e{Q}_u-(r_A+{\varphi }_2+\mu )Q_A,\hfill \\ \hfill {\displaystyle \frac{d{T}_H}{dt}}& =& \varphi H_A+{\varphi }_2{Q}_A-\mu T_H,\hfill \end{array}$$

(1)

with inital conditions:

$$\begin{array}{c}S\left(0\right)=S_0, V\left(0\right)=V_0, E_u\left(0\right)=E_{u0}, E_v\left(0\right)=E_{v0}, H_u\left(0\right)=H_{u0}, H_v\left(0\right)=H_{v0}, \hfill \\ H_A\left(0\right)=H_{A0}, Q_u\left(0\right)=Q_{u0}, Q_v\left(0\right)=Q_{v0}, Q_A\left(0\right)=Q_{A0}, T_H\left(0\right)=T_{H0}.\hfill \end{array}$$

3. Results

3.1. Basic Properties of the HSV-II Model

This subsection provides the region where the model (1) has a bounded solution. To do this, we first assumed that $N_H\left(t\right)$ represented the entire population of herpes simplex virus II, where

$$N_H\left(t\right)=S\left(t\right)+V\left(t\right)+E_u\left(t\right)+E_v\left(t\right)+H_u\left(t\right)+H_v\left(t\right)+H_A\left(t\right)+Q_u\left(t\right)+Q_v\left(t\right)+Q_A\left(t\right)+T_H\left(t\right),$$

such that

$${\displaystyle \frac{d{N}_H}{dt}}=\mathrm{\Pi }-\mu N_H-{\delta }_u{H}_u-{\delta }_{qu}{Q}_u,$$

(2)

defines the total human population in the model system (1).

3.2. Positivity of Solution

In this subsection, it is demonstrated that given any positive initial values of each state variable, all solutions to (1) remain positive over time.

Theorem 1.

Let the initial data be $S\left(0\right) > 0, V\left(0\right) \ge 0, E_u\left(0\right) \ge 0, E_v\left(0\right) \ge 0, H_u\left(0\right) \ge 0, H_v\left(0\right) \ge 0, H_A\left(0\right) \ge 0, Q_u\left(0\right) \ge 0, Q_v\left(0\right) \ge 0, Q_A\left(0\right) \ge 0, T_H\left(0\right) \ge 0$; then the solutions $(S\left(t\right), V\left(t\right), E_u\left(t\right), E_v\left(t\right), H_u\left(t\right), H_v\left(t\right), H_A\left(t\right), Q_u\left(t\right), Q_v\left(t\right), Q_A\left(t\right),T_H\left(t\right))$ of the model (1) are non-negative for all time, $t > 0$.

Proof. 

Let $T_1=\sup \{t > 0:S > 0, V > 0, E_u > 0, E_v > 0, H_u > 0, H_v > 0, H_A > 0, Q_u > 0, Q_A > 0, Q_v > 0\}$, gives $T_1>0$.

Then it follows from the first equation of the model (1), after eliminating, that

$${\displaystyle \frac{dS}{dt}}=\mathrm{\Pi }pϵ+\varpi V-({\lambda }_v+\xi +\mu )S\ge -({\lambda }_v+\xi +\mu )S.$$

(3)

Applying the method of separation of variables on Equation (3), we have

$${\displaystyle \frac{dS}{S}}\ge -({\lambda }_v+\xi +\mu ).$$

(4)

Integrating Equation (4), we get

$$\ln S\left(t\right)\ge -{\int }_0^{T_1}{\lambda }_v\left(t\right)dt-(\xi +\mu ){\int }_0^{t}dt+C,C\mathrm{constant}.$$

(5)

Taking the exponential of both sides of Equation (5), we have

$$S\left(t\right)\ge \exp \left[-{\int }_0^{T_1}{\lambda }_v\left(t\right)dt-(\xi +\mu )t+C\right],$$

(6)

Simplifying Equation (6) further gives

$$S\left(t\right)\ge C_1\exp \left[-{\int }_0^{T_1}{\lambda }_v\left(t\right)dt-(\xi +\mu )t\right],C_1=\exp C.$$

(7)

Applying the initial time, $t=0$, on Equation (7) gives $S\left(0\right)=C_1$, and substituting in (7) we get

$$S\left(t\right)\ge S\left(0\right)\exp \left[-{\int }_0^{T_1}{\lambda }_v\left(t\right)dt-(\xi +\mu )t\right]>0.$$

(8)

Similarly, from the second equation of model (1), we obtain

$${\displaystyle \frac{dV}{dt}}\ge -[(1-\mathrm{\Psi }){\lambda }_{v}V+(\varpi +\mu )]V.$$

(9)

Separating the variables and integrating Equation (9) yields

$$V\left(t\right)\ge C_2\exp \left[-(1-\mathrm{\Psi }){\int }_0^{T_1}{\lambda }_v\left(t\right)dt-(\varpi +\mu )t\right],C_2\mathrm{constant}.$$

(10)

Applying the initial condition on Equation (10) gives $V\left(0\right)=C_2$, and substituting in (10) we get

$$V\left(t\right)\ge V\left(0\right)\exp \left[-(1-\mathrm{\Psi }){\int }_0^{T_1}{\lambda }_v\left(t\right)dt-(\varpi +\mu )t\right]\ge 0.$$

(11)

Applying the same approach to the remaining state variables, $E_u\left(\mathrm{t}\right), E_v\left(\mathrm{t}\right), H_u\left(\mathrm{t}\right), H_v\left(\mathrm{t}\right), H_A\left(\mathrm{t}\right), Q_u\left(\mathrm{t}\right), Q_A\left(\mathrm{t}\right), Q_v\left(\mathrm{t}\right)$ of system (1), we see that the solutions are all positive at all times, $t>0$.   □

3.3. Invariant Region of HSV-II Model

Theorem 2.

The closet $\mathrm{\Omega }=\left(S,V,E_u,E_v,H_u,H_v,H_A,Q_u,Q_v,Q_A,T_H\in {\mathcal{R}}_{+}^{11};N_H\le {\displaystyle \frac{\mathrm{\Pi }}{\mu }}\right)$ is positively invariant and attracting with respect to the model (1).

Proof. 

In the absence of disease-induced death (i.e., ${\delta }_u={\delta }_{qu}=0$) and by comparison theorem, (2) becomes

$${\displaystyle \frac{d{N}_H}{dt}}=\mathrm{\Pi }-\mu N_H.$$

(12)

Solving (12) by the method of integrating factor and applying initial conditions results in

$$N_H\left(t\right)\le {\displaystyle \frac{\mathrm{\Pi }}{\mu }}+\left(N_H\left(0\right)-{\displaystyle \frac{\mathrm{\Pi }}{\mu }}\right)e^{-\mu t}.$$

Simplifying further, we have the following:

$$N_H\left(t\right)\le N_H\left(0\right)e^{-\mu t}+{\displaystyle \frac{\mathrm{\Pi }}{\mu }}(1-e^{-\mu t}).$$

As $t⟶\infty$, $N_H\left(t\right)⟶{\displaystyle \frac{\mathrm{\Pi }}{\mu }}$.

Thus,

$$0\le N_H\left(t\right)\le {\displaystyle \frac{\mathrm{\Pi }}{\mu }}.$$

Therefore, the feasibility region for the system (1) is $\mathrm{\Omega }$.

Hence, the model (1) is epidemiologically and biologically meaningful.   □

4. Existence of Equilibrium

4.1. HSV-II-Free Equilibrium Point

The HSV-II-free equilibrium point is a steady-state solution where there is no HSV-II infection in the population. Thus, in the absence of HSV-II, it means that $H_u=H_v=H_A=Q_u=Q_v=Q_A=0$ (the infected populations). Thus, the HSV-II-free equilibrium of the model system (1) is given by the following:

$$E_0=\left({\displaystyle \frac{[\mathrm{\Pi }(1-p\in )k_1+p\in \varpi ]}{k_1\mu +\xi \mu }},{\displaystyle \frac{[\mathrm{\Pi }(1-p\in )\xi +p\in k_0]}{k_1\mu +\xi \mu }},0,0,0,0,0,0,0,0,0\right),$$

(13)

where

$$\begin{array}{c}{k}_0=\xi +\mu ,\\ k_1=\varpi +\mu .\end{array}$$

4.2. Basic Reproduction Number of HSV-II Model

The effective reproduction number of the model (1) is computed in this section. We use the next generation matrix method as outlined in [19] to determine the effective reproduction number. The associated incidence term $F_u, F_v$ and the transfer terms $V_u, V_v$ for the HSV-II model for the unvaccinated and vaccinated proportions are, respectively, obtained as follows:

$$F_u=\left(\begin{array}{c}{E}_u\\ H_u\\ H_A\\ Q_u\\ Q_A\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{{\beta }_u[H_u+{\zeta }_1{H}_A+\theta (Q_u+{\zeta }_2{Q}_A)]S}{N}}\\ 0\\ 0\\ 0\\ 0\end{array}\right),$$

$$V_u=\left(\begin{array}{c}({\sigma }_1+\mu )E_u\\ -{\sigma }_1{E}_u-r_u{Q}_u+({\tau }_{1}e+q_u+{\delta }_u+u)H_u\\ -{\tau }_{1}e{H}_u-r_A{Q}_A+(q_A+\mu +{\varphi }_1)H_A\\ -q_u{H}_u-{\alpha }_1{Q}_v+(r_u+{\tau }_{2}e+{\delta }_{qu}+\mu )Q_u\\ -q_A{H}_A-{\tau }_{2}e{Q}_u+(r_A+{\varphi }_2+\mu )Q_A\end{array}\right),$$

$$F_v=\left(\begin{array}{c}{E}_v\\ H_v\\ Q_v\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{\beta ({\eta }_1{H}_v+\theta {\eta }_2{Q}_v)}{N}}0\\ 0\end{array}\right),V_v=\left(\begin{array}{c}({\sigma }_2+\mu )E_v\\ -{\sigma }_2{E}_v-r_v{Q}_v+(q_v+\mu )H_v\\ -q_v{H}_v+(r_v+{\alpha }_1+\mu )Q_v\end{array}\right).$$

Thus, the effective reproduction number for the system (1) is the spectral radius $\rho \left(F{V}^{-1}\right)$, given as follows:

$$\begin{array}{ccc}\hfill R_{0u}& =& {\displaystyle \frac{\beta S_0{\sigma }_1[(k_4+\theta q_u)(k_3{k}_5-q_A{r}_A)+{\zeta }_1({\tau }_{1}e{k}_4{k}_5+{\tau }_{2}e{r}_A{q}_u)+\theta {\zeta }_2({\tau }_{1}e{k}_4{q}_A+{\tau }_{2}e{k}_3{q}_u)]}{N^0(k_3{k}_5-q_A{r}_A)(k_2{k}_4-q_u{r}_u)}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill R_{0v}& =& {\displaystyle \frac{\beta V_0{\sigma }_2(1-\mathrm{\Psi })({\eta }_1{k}_8+\theta {\eta }_2{q}_v)}{N{k}_6(k_7{k}_8-q_v{r}_v)}}.\hfill \end{array}$$

(14)

Hence,

$$R_0^{HSV}=R_{0u}+R_{0v}.$$

Therefore,

$$R_0^{HSV}={\displaystyle \frac{k_6(k_7{k}_8-q_v{r}_v)\beta S_0{\sigma }_1{A}_1+(k_3{k}_5-q_A{r}_A)(k_2{k}_4-q_u{r}_u)(1-\mathrm{\Psi })\beta V_0{\sigma }_2{B}_1}{N_0{k}_6(k_3{k}_5-q_A{r}_A)(k_2{k}_4-q_u{r}_u)(k_7{k}_8-q_v{r}_v)}}.$$

(15)

where

$$\begin{array}{c}{A}_1=((k_4+\theta q_u)(k_3{k}_5-q_A{r}_A)+{\zeta }_1({\tau }_{1}e{k}_4{k}_5+{\tau }_{2}e{r}_A{q}_u)+\theta {\zeta }_2({\tau }_{1}e{k}_4{q}_A+{\tau }_{2}e{k}_3{q}_u)),\hfill \\ B_1={\eta }_1{k}_8+{\theta }_2{q}_v.\hfill \end{array}$$

It should be noted that

$$\begin{array}{c}{k}_7{k}_8-q_v{r}_v>0,\hfill \\ k_3{k}_5-q_A{r}_A=q_A({\varphi }_2+\mu )+({\varphi }_1+\mu )k_5>0,\hfill \\ k_2{k}_4-q_u{r}_u=q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4>0.\hfill \end{array}$$

The estimated number of secondary cases produced by a single HSV-infected person throughout the infectiousness period in a community of susceptible individuals is known as the effective reproduction number, $R_0^{HSV}$ [19].

4.3. Endemic Equilibrium Points of HSV-II Model

The steady-state conditions under which the population is infected with HSV-II is known as the HSV endemic equilibrium point. This means that the infectious states are all greater than zero ($H_u\ne 0, H_v\ne 0, H_A\ne 0, Q_u\ne 0, Q_v\ne 0, Q_A\ne 0$). In order to prove that our HSV-II model has endemic equilibria, we simultaneously solve the model (1) equations at steady state, where the force of infection is

$${\lambda }_v^{\ast }={\displaystyle \frac{\beta [H_u^{\ast }+{\zeta }_1{H}_A^{\ast }+{\eta }_1{H}_v^{\ast }+\theta (Q_u^{\ast }+{\zeta }_2{Q}_v^{\ast }+{\eta }_2{Q}_A^{\ast })]}{N^{\ast }}}$$

(16)

and the endemic equilibrium point is obtained to be

$$E_1=(S^{\ast },V^{\ast },E_u^{\ast },E_v^{\ast },H_u^{\ast },H_v^{\ast },H_A^{\ast },Q_u^{\ast },Q_v^{\ast },Q_A^{\ast },T_H^{\ast }).$$

(17)

with

$$\begin{array}{cc}\hfill S^{\ast }& ={\displaystyle \frac{\mathrm{\Pi }(R_0-1)(1-p\epsilon )k_{{\eta }_1}+\mathrm{\Pi }\varpi p\epsilon }{({\gamma }_H,Y_H-\xi \varpi )R_0}},\hfill \\ \hfill V^{\ast }& ={\displaystyle \frac{(R_0-1)\mathrm{\Pi }(p\epsilon +\xi (1-p)\epsilon )}{({\gamma }_H,Y_H-\xi \varpi )R_0}},\hfill \\ \hfill E_u^{\ast }& ={\displaystyle \frac{(R_0-1)\left(\mathrm{\Pi }(1-p)\epsilon +\varpi p\epsilon \right)}{k_1({\gamma }_H,Y_H-\xi \varpi )R_0}},\hfill \\ \hfill E_v^{\ast }& ={\displaystyle \frac{(1-\psi )(R_0-1)\mathrm{\Pi }(p\epsilon +\xi (1-p)\epsilon )}{k_6({\gamma }_H,Y_H-\xi \varpi )R_0}},\hfill \\ \hfill H_u^{\ast }& ={\displaystyle \frac{k_{{\sigma }_1}{k}_1(R_0-1)k_{\delta }({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_{{\gamma }_H}-r_{q_1})\left(\mathrm{\Pi }{\gamma }_H(1-p\epsilon )+\mathrm{\Pi }\varpi p\epsilon \right)+M}{k_1({\gamma }_H,Y_H-\xi \varpi )(k_2{k}_4-q_{u_{1v}})k_6({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_{{\gamma }_H}-r_{q_1})}},\hfill \\ \hfill H_v^{\ast }& ={\displaystyle \frac{k_{\delta }{\sigma }_2(1-\psi )(R_0-1)(k_{{\eta }_1}\mathrm{\Pi }p\epsilon +\mathrm{\Pi }(1-p\epsilon ))}{k_6({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_{{\gamma }_H}-r_{q_v})}},\hfill \\ \hfill H_a^{\ast }& ={\displaystyle \frac{M_3+\mathrm{\Pi }\varpi p\epsilon +k_1({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_4-q_{r_{1v}})M+M_2}{k_1({\gamma }_H,Y_H-\xi \varpi )\left[(k_3{k}_5-r_{q_A})(k_2{k}_4-q_{u_{1v}})(k_5{k}_{{\gamma }_H}-r_{q_1})k_6(k_5{k}_4-q_{u_{1v}})\right]}},\hfill \\ \hfill Q_{u_1}^{\ast }& ={\displaystyle \frac{q_{u_1}{\sigma }_1(R_0-1)\frac{\mathrm{\Pi }}{Y_H}(1-p\epsilon )+\mathrm{\Pi }\varpi p\epsilon }{k_1({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_4-q_{u_{1v}})}},\hfill \\ \hfill Q_{v_1}^{\ast }& ={\displaystyle \frac{M_4+\mathrm{\Pi }\varpi p\epsilon +k_1({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_4-q_{u_{1v}})M+M_1}{k_1({\gamma }_H,Y_H-\xi \varpi )\left[(k_3{k}_5-r_{q_A})(k_2{k}_4-q_{u_{1v}})(k_5{k}_{{\gamma }_H}-r_{q_1})k_6(k_5{k}_4-q_{u_{1v}})\right]}},\hfill \\ \hfill Q_v^{\ast }& ={\displaystyle \frac{q_v{\sigma }_2(1-\psi )(R_0-1)(k_{{\eta }_1}\mathrm{\Pi }p\epsilon +\mathrm{\Pi }(1-p\epsilon ))}{k_6({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_{{\gamma }_H}-r_{q_v})}},\hfill \\ \hfill T_H^{\ast }& ={\displaystyle \frac{{\varphi }_1(M_3+\mathrm{\Pi }\varpi p\epsilon )+k_1({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_4-q_v)M+M_2+{\varphi }_2(M_4+\mathrm{\Pi }\varpi p\epsilon )+k_1({\gamma }_H,Y_H-\xi \varpi )(k_5{k}_4-q_v)M+M_1}{\mu k_1({\gamma }_H,Y_H-\xi \varpi )\left[(k_3{k}_5-r_{q_A})(k_2{k}_4-q_v)(k_5{k}_{{\gamma }_H}-r_{q_1})k_6(k_5{k}_4-q_v)\right]}}.\hfill \end{array}$$

$$\begin{array}{c}{k}_2={\tau }_{1}e+q_u+q_u+\mu ,\hfill \\ k_3=q_A+{\varphi }_1+\mu ,\hfill \\ k_4=r_u+{\tau }_{2}e+q_{qu}+\mu ,\hfill \\ k_5=r_A+{\varphi }_2+\mu ,\hfill \\ k_6={\sigma }_2+\mu ,\hfill \\ k_7=q_v+\mu ,\hfill \\ k_8=r_v+\mu ,\hfill \\ k_9={\sigma }_1+\mu A={\varphi }_1(M_3+\mathrm{\Pi }\varpi p\in )+k_9({\gamma }_{H_2}{\gamma }_{H_1}-\xi \varpi )(k_2{k}_4-q_u{r}_u)M+M_2,\hfill \\ B={\varphi }_2(M_4+\mathrm{\Pi }\varpi p\in )+k_9({\gamma }_{H_2}{\gamma }_{H_2}-\xi \varpi )(k_2{k}_4-q_u{r}_u)M+M_1.\hfill \end{array}$$

4.4. Local Stability of HSV-II-Free Equilibrium

Theorem 3.

The HSV-II-free equilibrium $E_0$ point of model (1) is locally asymptotically stable (LAS) if $R_0^{HSV}<1$ and unstable if $R_0^{HSV}>1$.

Proof. 

To show the local stability of $E_0$, we first obtain the Jacobian matrix of the system at $E_0$. Thus,

$$\left(\begin{array}{ccccccccccc}-k_0& \varpi & 0& -{\displaystyle \frac{\beta S_0}{N^0}}& -{\displaystyle \frac{\beta {\zeta }_1{S}_0}{N^0}}& -{\displaystyle \frac{\beta \theta S_0}{N^0}}& -{\displaystyle \frac{\beta \theta {\zeta }_2{S}_0}{N^0}}& 0& {\displaystyle \frac{\beta {\eta }_1{S}_0}{N^0}}& {\displaystyle \frac{\beta \theta {\eta }_2{S}_0}{N^0}}& 0\\ \xi & -k_1& 0& -{\displaystyle \frac{a_1}{N^0}}& -{\displaystyle \frac{a_2}{N^0}}& -{\displaystyle \frac{a_3}{N^0}}& -{\displaystyle \frac{a_4}{N^0}}& 0& -{\displaystyle \frac{a_5}{N^0}}& -{\displaystyle \frac{a_6}{N^0}}& 0\\ 0& 0& -k_9& -{\displaystyle \frac{\beta S_0}{N^0}}& -{\displaystyle \frac{\beta {\zeta }_1{S}_0}{N^0}}& -{\displaystyle \frac{\beta \theta S_0}{N^0}}& -{\displaystyle \frac{\beta \theta {\zeta }_2{S}_0}{N^0}}& 0& {\displaystyle \frac{\beta {\eta }_1{S}_0}{N^0}}& {\displaystyle \frac{\beta \theta {\eta }_2{S}_0}{N^0}}& 0\\ 0& 0& {\sigma }_1& -k_2& 0r_u& 0& 0& 0& 0& 0\\ 0& 0& 0& {\tau }_{1}e& -k_3& 0& r_A& 0& 0& 0& 0\\ 0& 0& 0& q_u& 0& -k_4& 0& 0& 0& {\alpha }_1& 0\\ 0& 0& 0& 0& q_A& {\tau }_{2}e& -k_5& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{a_1}{N^0}}& {\displaystyle \frac{a_2}{N^0}}& {\displaystyle \frac{a_3}{N^0}}& {\displaystyle \frac{a_4}{N^0}}& -k_6& {\displaystyle \frac{a_5}{N^0}}& {\displaystyle \frac{a_6}{N^0}}& 0\\ 0& 0& 0& 0& 0& 0& 0& {\sigma }_2& -k_7& r_v& 0\\ 0& 0& 0& 0& 0& 0& 0& q_v& 0& -k_8& 0\\ 0& 0& 0& 0& {\varphi }_1& 0& {\varphi }_2& 0& 0& 0& -\mu \end{array}\right).$$

(18)

where

$$\begin{array}{c}{a}_1=(1-\mathrm{\Psi })\beta V_0,\hfill \\ a_2=(1-\mathrm{\Psi })\beta {\zeta }_1{V}_0,\hfill \\ a_3=(1-\mathrm{\Psi })\beta \theta V_0,\hfill \\ a_4=(1-\mathrm{\Psi })\beta \theta {\zeta }_2{V}_0,\hfill \\ a_5=(1-\mathrm{\Psi })\beta {\eta }_1{V}_0,\hfill \\ a_6=(1-\mathrm{\Psi })\beta \theta {\eta }_2{V}_0.\hfill \end{array}$$

Using the approach used in [5], the characteristic equation of the Jacobian (18) at the DFE, $E_0$ is given by

$$J\left(E_0\right)=\left[G_{6\times 10}^{\ast }{G}_{4\times 10}^{\ast \ast }\right].$$

Thus, the following characteristic equation is obtained:

$$({\lambda }^6+B_1{\lambda }^5+B_2{\lambda }^4+B_3{\lambda }^2+B_4{\lambda }^2+B_5{\lambda }^1+B_6)({\lambda }^4+P_1{\lambda }^3+P_2{\lambda }^2+P_3\lambda +P_4)=0.$$

(19)

where

$$\begin{array}{cc}\hfill B_1& =k_9+k_2+k_3+k_4+k_0+k_1,\hfill \\ \hfill B_2& =(k_9+k_2+k_3+k_4++k_1)k_0+(k_9+k_2+k_3+k_4)k_1\hfill \\ & +(k_2+k_3+k_4)k_9+k_2{k}_3+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{2}e+{\delta }_u+\mu )k_4\hfill \\ & -\varpi \epsilon -d_1{\sigma }_1+k_3{k}_4,\hfill \\ \hfill B_3& =[(k_9+k_2+k_4+k_1)k_0+(k_9+k_2+k_4)k_1+(k_2+k_4)k_9-\varpi \epsilon \hfill \\ & +q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4-d_1{\sigma }_1]k_3+[(k_9+k_2+k_4)k_1\hfill \\ & +(k_2+k_4)k_9+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4-d_1{\sigma }_1]k_0\hfill \\ & +[(k_2+k_4)k_9+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4-d_1{\sigma }_1]k_1\hfill \\ & +[q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+q_u({\tau }_{1}e+{\delta }_u+\mu )k_4-\varpi \epsilon ]k_9-(\varpi \epsilon +d_1{\sigma }_1)k_4\hfill \\ & -\varpi \epsilon k_2-{\sigma }_1(c_1{d}_2+d_3{q}_u),\hfill \\ \hfill B_4& =[(k_9+k_2+k_4)k_1+(k_9{k}_4+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4\hfill \\ & -d_1{\sigma }_1+k_9{k}_2)+((k_9{k}_4+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4-d_1{\sigma }_1\hfill \\ & +k_9{k}_2)k_1-\varpi \epsilon -d_1{\sigma }_1+k_9{k}_2)k_4+(\varpi \epsilon -q_u{r}_u)k_9-\varpi \epsilon k_2-d_3{q}_u{\sigma }_1)k_3\hfill \\ & +((k_9{k}_4+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+q_u({\tau }_{1}e+{\delta }_u+\mu )k_4-d_1{\sigma }_1+k_9{k}_2)k_1\hfill \\ & -d_1{\sigma }_1+k_9(q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+q_u({\tau }_{1}e+{\delta }_u+\mu )k_4)-{\sigma }_1(c_1{d}_2+d_3{q}_u))k_0\hfill \\ & +(k_9(q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+q_u({\tau }_{1}e+{\delta }_u+\mu )k_4)-{\sigma }_1(c_1{d}_2+d_3{q}_u))k_1\hfill \\ & -(\varpi \epsilon -\varpi \epsilon k_2-c_1{d}_2{\sigma }_1)k_4-\varpi \epsilon (d_1{\sigma }_1+k_9{k}_2+q_u{r}_u)],\hfill \\ \hfill B_5& =\left(\right(((k_9{k}_4+q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4-d_1{\sigma }_1+k_9{k}_2)k_1\hfill \\ & -(d_1{k}_4+d_3{q}_u){\sigma }_1+k_9(k_2{k}_4+q_u{r}_u))k_0-((d_1{k}_4+d_3{q}_u){\sigma }_1\hfill \\ & +k_9(q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu ))k_1-((k_9+k_2)k_4+q_u{r}_u\hfill \\ & +d_1{\sigma }_1+k_9{k}_2\left)\varpi \epsilon \right)k_3\left(\right((c_1{d}_2+d_1{k}_4+d_3{q}_u){\sigma }_1+k_9{k}_4(k_2{k}_4+q_u{r}_u)\hfill \\ & -c_1{d}_2{k}_4{\sigma }_1)k_0-c_1{d}_2{k}_4{k}_1{\sigma }_1+((c_1{d}_2+d_1{k}_4+d_3{q}_u){\sigma }_1\hfill \\ & -k_9(q_u({\tau }_{2}e+{\delta }_{qu}+\mu )+({\tau }_{1}e+{\delta }_u+\mu )k_4))\varpi \epsilon ,\hfill \\ \hfill B_6& =(\varpi \epsilon +k_0{k}_1)(((d_1{k}_4+d_3{q}_u){\sigma }_1+k_9(k_2{k}_4+q_u{r}_u))k_3+c_1{d}_2{k}_4{\sigma }_1)\hfill \\ \hfill P_1& =k_5+k_6+k_7+k_8,\hfill \\ \hfill P_2& =(k_6+k_7+k_8)+(k_7+k_8)k_6-c_5{\sigma }_2-q_v{c}_6+k_7{k}_8,\hfill \\ \hfill P_3& =((k_6+k_8)k_7-e_5{\sigma }_2-q_v{e}_6+k_6{k}_8)k_5+(k_6{k}_8-e_6{q}_u)k_7-e_5(k_8{\sigma }_2+q_v{r}_v),\hfill \\ \hfill P_4& =(k_8(e_5{\sigma }_2+k_6{k}_7)-(e_5{r}_v+e_6{k}_7))k_5.\hfill \end{array}$$

We use the Routh–Hurwitz criterion, which states that all roots of the characteristic Equation (19) have negative real part if and only if the coefficients $B_i$ and $P_i$ are positive and the determinant of the matrices $H_i>0$ for $i=0,1,2,3,...,6$ are all positive and $H_j>0$ for $j=1,2,3,4$.

Thus, from Equation (19), we have

$$H_6=\left(\begin{array}{cccccc}{B}_1& 1& 0& 0& 0& 0\\ B_3& B_2& B_1& 1& 0& 0\\ B_5& B_4& B_3& B_2& B_1& 1\\ 0& B_6& B_5& B_4& B_3& B_2\\ 0& 0& 0& B_6& B_5& B_4\\ 0& 0& 0& 0& 0& B_6\end{array}\right).$$

(20)

H is called the Hurwitz matrix, and the principle minors are

$$\begin{array}{ccc}\hfill H_1& =& B_1,\hfill \\ & =& k_4+k_3+k_2+k_1+k_0>0.\hfill \end{array}$$

$$H_2=\left(\begin{array}{cc}{B}_1& 1\\ 0& B_2\end{array}\right)=B_1{B}_2>0.$$

(21)

$$H_3=\left(\begin{array}{ccc}{B}_1& 1& 0\\ B_3& B_2& B_1\\ 0& 0& B_2\end{array}\right)=B_1{B}_2{B}_3-B_3^2>0,\mathrm{provided}{B}_1{B}_2{B}_3>B_3^2.$$

(22)

$$H_4=\left(\begin{array}{cccc}{B}_1& 1& 0& 0\\ B_3& B_2& B_1& 1\\ 0& B_4& B_3& B_2\\ 0& 0& 0& B_4\end{array}\right)=B_1{B}_2{B}_3{B}_4-B_1^2{B}_4^2-B_3^2{B}_4>0,\mathrm{provided}{B}_1{B}_2{B}_3{B}_4>B_1^2{B}_4^2+B_3^2{B}_4.$$

(23)

$$\begin{array}{ccc}\hfill H_5& =& \left(\begin{array}{ccccc}{B}_1& 1& 0& 0& 0\\ B_3& B_2& B_1& 1& 0\\ B_5& B_4& B_3& B_2& B_1\\ 0& 0& B_5& B_4& B_3\\ 0& 0& 0& 0& B_5\end{array}\right)\hfill \\ & =& B_1{B}_2{B}_3{B}_4{B}_5+2B_1{B}_4{B}_5^2+B_2{B}_3{B}_5^2-B_3^2{B}_4{B}_5-B_1^2{B}_4{B}_5-B_1{B}_2^2{B}_5^2-B_5^3>0,\hfill \end{array}$$

provided $B_1{B}_2{B}_3{B}_4{B}_5+2B_1{B}_4{B}_5^2+B_2{B}_3{B}_5^2>B_3^2{B}_4{B}_5+B_1^2{B}_4{B}_5+B_1{B}_2^2{B}_5^2+B_5^3$.

$$H_6=\left(\begin{array}{cccccc}{B}_1& 1& 0& 0& 0& 0\\ B_3& B_2& B_1& 1& 0& 0\\ B_5& B_4& B_3& B_2& B_1& 1\\ 0& B_6& B_5& B_4& B_3& B_2\\ 0& 0& 0& B_6& B_5& B_4\\ 0& 0& 0& 0& 0& B_6\end{array}\right)>0,$$

(24)

provided

$$\begin{array}{c}2B_1^2{B}_2{B}_5{B}_6+B_1^2{B}_3{B}_4{B}_6^2+B_1{B}_2{B}_3{B}_4{B}_5{B}_6+2B_1{B}_4{B}_5^2{B}_6+B_2{B}_3{B}_5^2{B}_6\hfill \\ >B_5{B}_6^3+B_1^2-B_4^2{B}_5{B}_6+B_1{B}_2^2{B}_5^2{B}_6+B_1{B}_2{B}_3^2{B}_6^2+3B_1{B}_2{B}_3{B}_5{B}_6^2{B}_3^2{B}_6+B_5{B}_6.\hfill \end{array}$$

Thus, $H_i>0,i=1,2,3,4,5,6$. Similarly, from the second part of Equation (19), the second characteristic equation is given by

$$P_0{\lambda }^4+P_1{\Lambda }^3+P_2{\Lambda }^2+P_3\lambda +P_4=0,$$

(25)

where

$$\begin{array}{c}{P}_0=1,\hfill \\ P_1=k_5+k_6+k_7+k_8,\hfill \\ P_2=k_6+k_7+k_8+k_6(k_7+k_8)-c_5{\sigma }_2-q_v{c}_6+k_7{k}_8,\hfill \\ P_3=(k_7(k_6+k_8)-e_5{\sigma }_2-q_v{e}_6+k_6{k}_8)k_5+(k_6{k}_8-e_6{q}_u)k_7-e_5(k_8{\sigma }_2+q_v{r}_v),\hfill \\ P_4=(k_8(e_5{\sigma }_2+k_6{k}_7)-(e_5{r}_v+e_6{k}_7))k_5\hfill \end{array}$$

$$H_4=\left(\begin{array}{cccc}{P}_1& 1& 0& 0\\ P_3& P_2& P_1& 1\\ P_5& P_4& P_3& P_2\\ 0& 0& P_5& P_4\end{array}\right),$$

(26)

H is called the Hurwitz matrix, and the principal minors are:

$$H_1=P_1>0,$$

(27)

$$H_2=\left(\begin{array}{cc}{P}_1& 1\\ P_0& P_2\end{array}\right)=P_1{P}_2>0, \mathrm{provided}{P}_1{P}_2>0$$

(28)

$$H_3=\left(\begin{array}{ccc}{P}_1& 1& 0\\ P_3& P_2& P_1\\ 0& 0& P_3\end{array}\right)=P_1{P}_2{P}_3-P_1{P}_3>0, \mathrm{provided}{P}_1{P}_2{P}_3>P_1{P}_3,$$

(29)

$$H_4=\left(\begin{array}{cccc}{P}_1& 1& 0& 0\\ P_3& P_2& P_1& 1\\ P_5& P_4& P_3& P_2\\ 0& 0& 0& P_4\end{array}\right)=P_1{P}_2{P}_3{P}_4-P_1^2{P}_4^2-P_3^2{P}_4>0, \mathrm{provided}{P}_1{P}_2{P}_3{P}_4>P_1^2{P}_4^2+P_3^2{P}_4.$$

(30)

Therefore, $H_j>0,j=1,2,3,4$. Since $H_i>,i=1,2,3,4,5,6$ and $H_j>0,j=1,2,3,4$. This means that all the eigenvalues of the characteristic Equation (19) have negative real parts, implying that ${\lambda }_i$ is less that zero, for $i=1,2,3,...,11$ when $R_0^{HSV}<1$.

Therefore, we conclude that the DFE, $E_0$, is locally asymptotically stable.   □

Figure 2 illustrates the essential threshold dynamics of HSV-II virus persistence by displaying the link between the basic reproduction number ($R_0$) and the effective contact rate ($\beta$). Each infected person typically produces less than one secondary case when $R_0=0.99$, which is slightly below unity. This suggests that the disease will eventually die out under a control strategy. On the other hand, when $R_0=1.01$, which is marginally above unity, the infection persists in the population since each infected person generates multiple new cases. The sensitivity of HSV-II dynamics to variations in $\beta$ is highlighted by this small margin around the threshold value of $R_0=1$, underscoring the significance of successful interventions. Therefore, maintaining $R_0<1$ through these strategies is essential to suppress HSV-II transmission and prevent long-term endemicity.

4.5. Global Stability of the HSV-II-Free Equilibrium Point

Using the theorem by [20], the global stability of $E_0$ for model (1) is examined in this section.

Lemma 1.

Consider the model system (1) written in the form

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dX}{dt}}& =& F(X,Z),\hfill \\ \hfill {\displaystyle \frac{dZ}{dt}}& =& G(X,Z),(X,0)=0.\hfill \end{array}$$

(31)

where $X=(S,V,T_H)$ and $Z=(E_u,E_v,H_u,H_v,H_A,Q_u,Q_v,Q_A)$, with the components of $X\in {\mathcal{R}}^3$ denoting the uninfected population and the components of $Z\in {\mathcal{R}}^8$ denoting the infected population. The DFE is now denoted as follows:

$$E_0=\left(\begin{array}{c}{\displaystyle \frac{\mathrm{\Pi }(1-p\in )k_1+p\in \varpi }{k_1\mu +\xi \mu }}\\ {\displaystyle \frac{\mathrm{\Pi }(1-p\in )\xi +p\in k_0}{k_1\mu +\xi \mu }}\\ 0\end{array}\right).$$

(32)

To ensure global asymptotic stability, the conditions $H_1$ and $H_2$ below must be satisfied.

$$\begin{gathered} H_1:{\displaystyle \frac{dX}{dt}}=F(X,0),\mathit{\text{the DFE is globally asymptotically stable.}} \\ {H}_2:\widehat{G}(X,Z)=PZ-G(X,Z),\widehat{G}(X,Z)\ge 0,for(X,Z)\in \mathrm{\Omega } \end{gathered}$$

where $P=D_{z}G(X^0,0)$ is an M-matrix (the off-diagonal elements of P are non-negative) and Ω is the region where the model makes biological sense. Then, the DFE $X_0$ is globally asymptotically stable provided that $R_0<1$ [20].

From the model system (1), we check the conditions in Lemma 1. Thus, we have

$${\displaystyle \frac{dX}{dt}}=F(X,Z)=\left(\begin{array}{c}\mathrm{\Pi }(1-p\in )+\varpi V-{\lambda }_{v}S-k_{0}S\\ \mathrm{\Pi }p\in +\xi S-(1-\mathrm{\Psi }){\lambda }_{v}V-k_{1}V\\ {\varphi }_1{H}_A+{\varphi }_2{Q}_A-\mu T_H\end{array}\right),$$

(33)

$${\displaystyle \frac{dZ}{dt}}=G(X,Z)=\left(\begin{array}{c}{\lambda }_{v}S-k_9{E}_u\\ {\sigma }_1{E}_u+r_u{Q}_u-k_2{H}_u\\ {\tau }_{1}e{H}_u+r_A{Q}_A-k_3{H}_A\\ q_u{H}_u+{\alpha }_1{Q}_v-k_4{Q}_u\\ q_A{H}_A+q_T{T}_H-k_5{Q}_A\\ (1-\mathrm{\Psi }){\lambda }_{v}V-k_6{E}_v\\ {\sigma }_2{E}_v+r_v{Q}_v-k_7{H}_v\\ q_v{H}_v-k_8{Q}_v\end{array}\right).$$

(34)

At disease-free equilibrium, (33) becomes

$${\displaystyle \frac{dX}{dt}}=F(X,0)=\left(\begin{array}{c}\mathrm{\Pi }(1-p\in )+\varpi V-k_{0}S\\ \mathrm{\Pi }p\in +\xi S-k_{1}V\\ -\mu R\end{array}\right).$$

(35)

Thus, from (35), we see that $X^0=\left({\displaystyle \frac{\mathrm{\Pi }[(1-p\in )k_1+p\in \varpi ]}{k_1\mu +\xi \mu }},{\displaystyle \frac{\mathrm{\Pi }[(1-p\in )\xi +p\in k_0]}{k_1\mu +\xi \mu }},0\right)$ is locally asymptotically stable.

This can be verified from the solutions, namely,

$$\begin{array}{ccc}\hfill S\left(t\right)& =& {\displaystyle \frac{\mathrm{\Pi }[(1-p\in )k_1+p\in \varpi ]}{k_1\mu +\xi \mu }}+\left[S\left(0\right)-{\displaystyle \frac{\mathrm{\Pi }[(1-p\in )k_1+p\in \varpi ]}{k_1\mu +\xi \mu }}\right]e^{-(\xi +\mu )t},\hfill \\ \hfill V\left(t\right)& =& {\displaystyle \frac{\mathrm{\Pi }[(1-p\in )\xi +p\in k_0]}{k_1\mu +\xi \mu }}+\left[V\left(0\right)-{\displaystyle \frac{\mathrm{\Pi }[(1-p\in )\xi +p\in k_0]}{k_1\mu +\xi \mu }}\right]e^{-(\varpi +\mu )t}.\hfill \end{array}$$

(36)

$T_h\left(t\right)=T_h\left(0\right)e^{ut}$ and as $t⟶\infty ,$ the solutions

$$S(\infty ),V(\infty ),T_h(\infty )⟶\left({\displaystyle \frac{\mathrm{\Pi }[(1-p\in )k_1+p\in \varpi ]}{k_1\mu +\xi \mu }},{\displaystyle \frac{\mathrm{\Pi }[(1-p\in )\xi +p\in k_0]}{k_1\mu +\xi \mu }},0\right),$$

which implies a global convergence of system (1) in ${\mathrm{\Omega }}_H$, and this satisfies condition $H_1$.

The second condition demands that

$$\widehat{G}(G,Z)=PZ-G(X,Z),$$

(37)

and that

$$\widehat{G}(G,Z)\ge 0,\forall (X,Z)\in {\mathrm{\Omega }}_H.$$

(38)

We already know that from Equation (34), $\widehat{G}(G,Z)=PZ-G(X,Z)$, where P is an n × n matrix, Z is a column vector, and G(X, Z) is a column vector formed from the infectious compartments. Computing the Jacobian of Equation (34) and evaluating at the disease-free equilibrium gives the following:

$$P_{DFE}=\left(\begin{array}{cccccccc}-k_9& {\displaystyle \frac{\beta S_0}{N}}& {\displaystyle \frac{\beta {\zeta }_1{S}_0}{N}}& {\displaystyle \frac{\beta \theta S_0}{N}}& {\displaystyle \frac{\beta {\zeta }_1{S}_0}{N}}& {\displaystyle \frac{\beta \theta {\zeta }_2{S}_0}{N}}& {\displaystyle \frac{\beta {\eta }_1{S}_0}{N}}& {\displaystyle \frac{\beta \theta {\eta }_2{S}_0}{N}}\\ {\sigma }_1& -k_2& 0& r_u& 0& 0& 0& 0\\ 0& {\tau }_{1}e& -k_3& 0& r_A& 0& 0& 0\\ 0& q_u& 0& -k_4& 0& 0& 0& {\alpha }_1\\ 0& 0& q_A& {\tau }_{2}e& -k_5& 0& 0& 0\\ 0& {\displaystyle \frac{(1-\mathrm{\Psi })\beta V_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\zeta }_1{V}_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta \theta V_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\zeta }_2{V}_0}{N}}& -k_6& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\eta }_1{V}_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\eta }_2{V}_0}{N}}\\ 0& 0& 0& 0& 0& {\sigma }_2& -k_7& r_v\\ 0& 0& 0& 0& 0& 0& q_v& -k_8\end{array}\right).$$

(39)

It can be observed that the matrix P is an M-matrix because all its off-diagonal elements are non-negative. Thus, multiplying the matrix P and Z gives the following:

$$PZ=\left(\begin{array}{cccccccc}-k_9& {\displaystyle \frac{\beta S_0}{N}}& {\displaystyle \frac{\beta {\zeta }_1{S}_0}{N}}& {\displaystyle \frac{\beta \theta S_0}{N}}& {\displaystyle \frac{\beta {\zeta }_1{S}_0}{N}}& {\displaystyle \frac{\beta \theta {\zeta }_2{S}_0}{N}}& {\displaystyle \frac{\beta {\eta }_1{S}_0}{N}}& {\displaystyle \frac{\beta \theta {\eta }_2{S}_0}{N}}\\ {\sigma }_1& -k_2& 0& r_u& 0& 0& 0& 0\\ 0& {\tau }_{1}e& -k_3& 0& r_A& 0& 0& 0\\ 0& q_u& 0& -k_4& 0& 0& 0& {\alpha }_1\\ 0& 0& q_A& {\tau }_{2}e& -k_5& 0& 0& 0\\ 0& {\displaystyle \frac{(1-\mathrm{\Psi })\beta V_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\zeta }_1{V}_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta \theta V_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\zeta }_2{V}_0}{N}}& -k_6& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\eta }_1{V}_0}{N}}& {\displaystyle \frac{(1-\mathrm{\Psi })\beta {\eta }_2{V}_0}{N}}\\ 0& 0& 0& 0& 0& {\sigma }_2& -k_7& r_v\\ 0& 0& 0& 0& 0& 0& q_v& -k_8\end{array}\right)\left(\begin{array}{c}{E}_u\\ H_u\\ H_A\\ Q_u\\ Q_A\\ E_v\\ H_v\\ Q_v\end{array}\right).$$

(40)

Evaluating Equation (40), we have

$$PZ=\left(\begin{array}{c}[H_u+{\eta }_1{H}_v+{\zeta }_1{H}_A+\theta (Q_u+{\eta }_2{Q}_v+{\zeta }_2{Q}_A)]{\displaystyle \frac{S_0}{N^0}}-k_9{k}_u\\ {\sigma }_1{E}_u+r_u{Q}_u-k_2{H}_u\\ {\tau }_{1}e{H}_u+r_A{Q}_A-k_3{H}_A{q}_u{H}_u+{\alpha }_1{Q}_v-k_4{Q}_u\\ q_A{H}_A+{\tau }_{2}e{Q}_u-k_5{Q}_A\\ (1-\mathrm{\Psi }){\lambda }_v{V}_0-k_6{E}_v\\ {\sigma }_2{E}_v+r_v{Q}_v-k_7{H}_v\\ q_v{H}_v-k_8{Q}_v\end{array}\right)$$

(41)

Now, using $G(X,Z)=PZ-G(X,Z)$ the following is obtained:

$$G(X,Z)=\left(\begin{array}{c}[H_u+{\eta }_1{H}_v+{\zeta }_1{H}_A+\theta (Q_u+{\eta }_2{Q}_v+{\zeta }_2{Q}_A)]{\displaystyle \frac{S_0}{N^0}}-k_9{k}_u\\ 0\\ 0\\ 0\\ 0\\ (1-\mathrm{\Psi })[H_u+{\eta }_1{H}_v+{\zeta }_1{H}_A+\theta (Q_u+{\eta }_2{Q}_v+{\zeta }_2{Q}_A)]\left[{\displaystyle \frac{V_0}{N^0}}-{\displaystyle \frac{V}{N}}\right]\\ 0\\ 0\end{array}\right).$$

(42)

Hence, it is clear that the matrix $G(X,Z)\ngeqq 0$. $H_2$ is not satisfied. This suggests that backward bifurcation may occur at $E_0$ when $R_0^{HSV}<1$. Thus, we conclude the HSV-free equilibrium point $E_0=(X^0,0)$ of system (1) is not globally asymptotically stable when $R_0^{HSV}<1$.

It should be noted that the failure to satisfy $H_2$ does not automatically imply that the DFE is not GAS in general. It simply means that the method cannot be used to conclude GAS. Additional techniques or alternative Lyapunov functions can be explored to assess global stability. In other words, the absence of $H_2$ validation only shows that the sufficient conditions for GAS as described in [19,21] are not met.

5. Sensitivity Analysis

This part examines the sensitivity analysis of the reproduction number’s parameters. This way, it will be simpler to confirm and identify the variables affecting the basic reproductive number. By partially differentiating $R_0^{HSV}$ with respect to the model parameters in Table 1, the impact was computed. The sensitivity index was calculated using the method described by [13]. This method creates a formula to determine each basic parameter’s sensitivity index, which is specified as

$${\mathrm{\Delta }}^{R_0^{HSV}}={\displaystyle \frac{\partial R_0^{HSV}}{\partial x}}\times {\displaystyle \frac{x}{R_0^{HSV}}}.$$

(43)

where x represents all the basic parameters to be tested ($\beta ,\xi ,{\zeta }_1,{\zeta }_2,{\varphi }_1,{\varphi }_2,{\tau }_1,{\tau }_2,\theta ,\varpi ,e,p,\in$). By utilizing the effective reproduction number provided below, the sensitivity of each of the 12 parameters listed in Table 1 to $R_0^{HSV}$ is determined, where $R_0^{HSV}$ is given as

$$R_0^{HSV}={\displaystyle \frac{k_6(k_7{k}_8-q_v{r}_v)\beta S_0{\sigma }_1{A}_1+(k_3{k}_5-q_A{r}_A)(k_2{k}_4-q_u{r}_u)(1-\mathrm{\Psi })\beta V_0{\sigma }_2{B}_1}{N{k}_6(k_3{k}_5-q_A{r}_A)(k_2{k}_4-q_u{r}_u)(k_7{k}_8-q_v{r}_v)}}.$$

The sensitivity index was used to infer parameter values, which are found in Table 1.

Table 2. Sensitivity indices for the parameters of $R_0^{HSV}$.

Parameter	Baseline Value	References	Sensitivity Index
$\mathrm{\Pi }$	10,000	[5]	+1.0000
$\beta$	0.3000	[5]	+0.7510
$\varpi$	1/15	[8]	+0.4965
$\xi$	0.6000	[5]	−0.1661
∈	0.6000	[5]	−0.4285
${\zeta }_1$	0.5000	Assumed	+0.0052
${\zeta }_2$	0.0600	Assumed	+0.0676
${\varphi }_1$	0.5	Assumed	−0.1857
${\varphi }_2$	0.4	Assumed	−0.2472
${\tau }_1$	0.5	Assumed	−0.0557
${\tau }_2$	0.5	Assumed	−0.0885
e	0.6	Assumed	+0.1466

shows how the parameters have both positive and negative effects on $R_0^{HSV}$. An increase in $R_0^{HSV}$, as shown by positive SI values like $\beta , {\zeta }_1, {\zeta }_2, \theta , e$, and $\varpi$, is what causes the virus to infect the population. Even though $\xi , {\varphi }_1, {\varphi }_2, {\tau }_1, {\tau }_2, \theta , e, p$, and ∈ have negative SI, the infection eventually vanishes from the population as an increase in these parameter values decreases $R_0^{HSV}$.

6. Optimality Control for HSV-II Model

Health officials face a great deal of difficulty in controlling infectious disease epidemics. Adopting the right control measures to eliminate or reduce disease in a community is a difficult task. An appropriate optimal control that can be successfully executed to stop the spread of disease can be set using the guidance provided by the optimal control analysis. In this section, three control variables are taken into consideration in order to build an optimal control problem for the HSV-II model (1). The sensitivity data from the previous section are taken into account while adopting the control measures. Reducing the likelihood of transmission per interaction between infected individuals and the environment is the first practical measure that will be useful to minimize the occurrence of HSV-II, according to the sensitivity indices. The effective contact rates, or ${\eta }_1, {\eta }_2, {\zeta }_1, {\zeta }_2$, will decrease as a result of this action.The second useful control variable is raising $\xi$, or the vaccine’s efficacy. The last control variable in this study is to ensure or maximize the rate at which people voluntarily consent to laboratory testing in order to find out if they have HSV-II infection and receive treatment (i.e., to improve ${\tau }_1, {\tau }_2, {\varphi }_1, {\varphi }_2$). The HSV-II model (1) was supplemented with three time-dependent variables in order to create the mathematical control model that would evaluate the impact of all the previously proposed control measures on the disease dynamics. The three control variables denoted by the symbols $u_1\left(t\right), u_2\left(t\right)$, and $u_3\left(t\right)$ are the effective use of condoms, the HSV-II vaccine, laboratory testing, and therapy, respectively. The control system is so described as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}& =\mathrm{\Pi }(1-p\in )+\varpi V-(1-u_1){\lambda }_{v}S-(u_2\xi +\mu )S,\hfill \\ \hfill {\displaystyle \frac{dV\left(t\right)}{dt}}& =\mathrm{\Pi }p\in +u_2\xi S-(1-u_1)(1-\mathrm{\Psi }){\lambda }_{v}V-(\varpi +\mu )V,\hfill \\ \hfill {\displaystyle \frac{d{E}_u\left(t\right)}{dt}}& =(1-u_1){\lambda }_{v}S-({\sigma }_1+\mu )E_u,\hfill \\ \hfill {\displaystyle \frac{d{E}_v\left(t\right)}{dt}}& =(1-u_1)(1-\mathrm{\Psi }){\lambda }_{v}V-({\sigma }_2+\mu )E_v,\hfill \\ \hfill {\displaystyle \frac{d{H}_u\left(t\right)}{dt}}& ={\sigma }_1{E}_u+r_u{Q}_u-(u_3{\tau }_{1}e{q}_u+{\delta }_u+\mu )H_u,\hfill \\ \hfill {\displaystyle \frac{d{H}_v\left(t\right)}{dt}}& ={\sigma }_2{E}_v+r_v{Q}_v-(q_v+\mu )H_v,\hfill \\ \hfill {\displaystyle \frac{d{H}_A\left(t\right)}{dt}}& =u_3{\tau }_{1}e{H}_u+r_A{Q}_A-(q_A+u_3{\varphi }_1+\mu )H_A,\hfill \\ \hfill {\displaystyle \frac{d{Q}_u\left(t\right)}{dt}}& =q_u{H}_u+{\alpha }_1{Q}_v-(r_u+u_3{\tau }_{2}e+\mu +{\delta }_{qu})Q_u,\hfill \\ \hfill {\displaystyle \frac{d{Q}_v\left(t\right)}{dt}}& =q_v{H}_v-(r_v+{\alpha }_1+\mu )Q_u,\hfill \\ \hfill {\displaystyle \frac{d{Q}_A\left(t\right)}{dt}}& =q_A{H}_A+u_3{\tau }_{2}e{Q}_u-(r_A+u_3{\varphi }_2+\mu )Q_A,\hfill \\ \hfill {\displaystyle \frac{d{T}_H\left(t\right)}{dt}}& =u_3\varphi H_A+u_3{\varphi }_2{Q}_A-\mu T_H,\hfill \end{array}$$

(44)

with inital conditions:

$$\begin{array}{c}S\left(0\right)=S_0,V\left(0\right)=V_0, E_u\left(0\right)=E_{u0}, E_v\left(0\right)=E_{v0}, H_u\left(0\right)=H_{u0}, H_v\left(0\right)=H_{v0}, H_A\left(0\right)=H_{A0}, Q_u\left(0\right)=Q_{u0}, \hfill \\ Q_v\left(0\right)=Q_{v0}, Q_A\left(0\right)=Q_{A0}, T_H\left(0\right)=T_{H0}.\hfill \end{array}$$

where

$${\lambda }_v={\displaystyle \frac{\beta (H_u+{\zeta }_1{H}_A+{\eta }_1{H}_v+\theta (Q_u+{\zeta }_2{Q}_A+{\eta }_2{Q}_v))}{N}}.$$

The control functions $u_1,u_2,u_3$ are Lebesque measurable and bounded at $\{0\le u_1\le 1,$$0\le u_2\le \mathrm{\Psi }, 0\le u_3\le e, 0\le t\le T\}$. Finding the best levels of effort required to regulate HSV-II transmission and related state variables in order to maximize the objective function is the primary goal. The objective functional is given by

$$J_H=\underset{u_i}{\min }{\int }_0^T\left[B_1{I}_E+B_2{I}_v+B_3{I}_T+{\displaystyle \frac{1}{2}}{W}_1{u}_1+{\displaystyle \frac{1}{2}}{W}_2{u}_2+{\displaystyle \frac{1}{2}}{W}_3{u}_3\right]dt.$$

(45)

subject to $I_E=(E_u+E_v), I_v=V, I_T=(H_A+Q_A+H_u+Q_u)$, where the relative cost of putting the various intervention options into practice over a period of $[0,T]$ is measured by the positive weights $B_i$. The cost of intervention options is measured by ${\displaystyle \frac{1}{2}}{W}_i{u}_i^2$. Following the recommendations of [22], a quadratic cost functional was selected. The optimal control $u_1^{\ast }, u_2^{\ast },u_3^{\ast }$ is thus sought after so that

$$J_H(u_1^{\ast },u_2^{\ast },u_3^{\ast })=\min \left\{J_H(u_1,u_2,u_3)\right|(u_1,u_2,u_3)\in U_H\},$$

(46)

where $U_H=f(u_1, u_2, u_3)$ such that $u_i$ is measurable with $0\le u_1\le 1, 0\le u_2\le \mathrm{\Psi },$$0\le u_3\le e:\forall t\in [0,T]$ is the set of admissible control.

Existence of the Control Problem

As $f=(f_1,f_2,f_3,...,F_{11})$ and the right side of the optimality control problem (44) consists of continuously differentiable functions and is concave in $(S,V,E_u,E_v,H_u,H_v,H_A,Q_u,Q_v,$$Q_A,T_H)$ and $u=(u_1,u_2,u_3)$, it can be concluded that the optimal control exists. We now apply the well-known Pontryagin’s maximum principle, which gives the required condition for the controls’ optimality system. Using the principle, the system (44) is converted into a minimization problem, with respect to the controls $u_i$. Thus, the Hamiltonian $H_H$ is given by

$$H_H=B_1{I}_E+B_2{I}_v+B_3{I}_T+{\displaystyle \frac{W_1{u}_1^2}{2}}+{\displaystyle \frac{W_2{u}_2^2}{2}}+{\displaystyle \frac{W_3{u}_3^2}{2}}+〈{\lambda }_j\times x_j〉,$$

(47)

where $j=1,2,...,11$ and ${\lambda }_j$ are the adjoint variables while $x_j$ describe the state system (human compartments) [22].

Theorem 4.

If $u_i^{\ast }\left(t\right)$ and $x^{\ast }\left(t\right)$ are the optimal problem (objective function), then there exists a piecewise differentiable adjoint variable $\lambda \left(t\right)$ such that

$$H_H(t,x^{\ast }\left(t\right),u^{\ast }\left(t\right),\lambda \left(t\right))\le H_H(t,x^{\ast }\left(t\right),u^{\ast }\left(t\right),\lambda \left(t\right))$$

for all controls $u_i$ at each time t, where the Hamiltonian $H_H$ is

$$H_H=f(t,x\left(t\right),u\left(t\right)+\lambda \left(t\right)g(t,x\left(t\right),u\left(t\right))).\mathit{and}$$

$$\begin{array}{ccc}\hfill \lambda \prime \left(t\right)& =& {\displaystyle \frac{\partial H_H(t,x^{\ast }\left(t\right),\lambda \left(t\right))}{\partial x}},\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill \lambda \left(t\right)& =& 0.\hfill \end{array}$$

(49)

Thus, the Hamilton is

$$\begin{array}{ccc}\hfill H_H& =& B_1{I}_E+B_2{I}_v+B_3{I}_T+{\displaystyle \frac{W_1{u}_1^2}{2}}+{\displaystyle \frac{W_2{u}_2^2}{2}}+{\displaystyle \frac{W_3{u}_3^2}{2}}+{\lambda }_S(\mathrm{\Pi }(1-p\in )\hfill \\ & +& \varpi V-(1-u_1){\lambda }_v^{\ast }S-u_2\xi S-\mu S)+{\lambda }_v(\mathrm{\Pi }p\in +u_2\xi S-(1-u_1)(1-\mathrm{\Psi }){\lambda }_v^{\ast }V\hfill \\ & -& (\varpi +\mu )V)+{\lambda }_{E_u}((1-u_1){\lambda }_v^{\ast }S-({\sigma }_1+\mu )E_u)+{\lambda }_{E_v}((1-u_1)(1-\mathrm{\Psi }){\lambda }_v^{\ast }V\hfill \\ & -& ({\sigma }_2+\mu )E_v)+{\lambda }_{H_u}({\sigma }_1{E}_u+r_u{Q}_u-u_3{\tau }_{1}e{H}_u-(q_u+{\delta }_u+u)H_u)\hfill \\ & +& {\lambda }_{H_v}({\sigma }_2{E}_v+r_v{Q}_v-(q_v+\mu )H_v)+{\lambda }_{H_A}(u_3{\tau }_{1}e{H}_u+r_A{Q}_A-(q_A+\mu )H_A\hfill \\ & -& u_3{\varphi }_1{H}_A)+{\lambda }_{Q_u}(q_u{H}_u+{\alpha }_1{Q}_v-u_3{\tau }_{2}e{Q}_u-(r_u+{\delta }_{qu}+\mu )Q_u)\hfill \\ & +& {\lambda }_{Q_v}(q_v{H}_v-(r_v+\mu +{\alpha }_1)Q_v)+{\lambda }_{Q_v}(q_A{H}_A+u_3{\tau }_{2}e{Q}_u-u_3{\varphi }_2{Q}_A\hfill \\ & -& (r_A+\mu )Q_A)+{\lambda }_{T_H}(u_3{\varphi }_1{H}_A+u_3{\varphi }_2{Q}_A-\mu T_H),\hfill \end{array}$$

where ${\lambda }_S,{\lambda }_V,{\lambda }_{E_u},{\lambda }_{E_v},{\lambda }_{H_u},{\lambda }_{H_v},{\lambda }_{H_A},{\lambda }_{Q_u},{\lambda }_{Q_v},{\lambda }_{Q_A},{\lambda }_{T_H}$ are the adjoint variables or co-state variable.

Theorem 5.

There exists an optimal control set $u_1,u_2,u_3$ and corresponding solution, $S,V,E_u,E_v,H_u,H_v,H_A,Q_u,Q_v,Q_A$ and $T_H$ that minimize $J(u_1,u_2,u_3)$ over $U_H$. Furthermore, there exist adjoint functions ${\lambda }_S,{\lambda }_V,{\lambda }_{E_u},{\lambda }_{E_v},{\lambda }_{H_u},{\lambda }_{H_v},{\lambda }_{H_A},{\lambda }_{Q_u},{\lambda }_{Q_v},{\lambda }_{Q_A},{\lambda }_{T_H}$ satisfying:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_S}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_S}}& =(1-u_1)({\lambda }_S-{\lambda }_{E_u}){\lambda }_v^{\ast }+u_2\xi ({\lambda }_S-{\lambda }_v)+\mu {\lambda }_S,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_v}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_v}}& =(1-u_1)(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v}){\lambda }_v^{\ast }-\varpi {\lambda }_S+(\varpi +\mu ){\lambda }_v-B_2,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{E_u}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{E_u}}}& =({\sigma }_1+\mu ){\lambda }_{E_u}-{\sigma }_1{\lambda }_{H_u}-B_1,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{E_v}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{E_v}}}& =({\sigma }_2+\mu ){\lambda }_{E_v}-{\sigma }_2{\lambda }_{H_v}-B_1,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{H_u}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{H_u}}}& =u_3{\tau }_{1}e{\lambda }_{H_u}+(q_u+{\delta }_u+\mu ){\lambda }_{H_u}-{\sigma }_2{\lambda }_{H_v}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta }{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi })({\lambda }_S-{\lambda }_{E_u})]-B_3,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{H_v}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{H_v}}}& =(q_v+\mu ){\lambda }_{H_v}-q_v{\lambda }_{Q_v}+(1-u_1){\displaystyle \frac{\beta {\eta }_1}{N}}[({\lambda }_S-{\lambda }_{E_u})\hfill \\ & +(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})],\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{H_A}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{H_A}}}& =(q_A+\mu ){\lambda }_{H_A}+u_3{\varphi }_1({\lambda }_{H_A}-{\lambda }_{T_H})-q_A{\lambda }_{Q_A}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta {\zeta }_1}{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})]-B_3,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{Q_u}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_u}}}& =u_3{\tau }_{2}e({\lambda }_{Q_u}-{\lambda }_{Q_A})+(r_u+{\delta }_{qu}+\mu )-r_u{\lambda }_{H_u}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta \theta }{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})]-B_3,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{Q_A}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_A}}}& =u_3{\varphi }_2({\lambda }_{Q_A}-{\lambda }_{T_H})+(r_A+\mu ){\lambda }_{Q_A}-r_A{\lambda }_{H_A}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta \theta {\zeta }_2}{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})]-B_3,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{Q_v}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_v}}}& =(r_v+\mu +{\alpha }_1){\lambda }_{Q_v}-{\alpha }_1{\lambda }_{Q_u}+(1-u_1){\displaystyle \frac{\beta \theta {\eta }_2}{N}}[({\lambda }_S-{\lambda }_{E_u})\hfill \\ & +(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})]-B_2,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{T_H}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{T_h}}}& =\mu {\lambda }_{T_H},\hfill \end{array}$$

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with the transversality condition: ${\lambda }_S\left(t_f\right)={\lambda }_V\left(t_f\right)={\lambda }_{E_u}\left(t_f\right)={\lambda }_{E_v}\left(t_f\right)={\lambda }_{H_u}\left(t_f\right)={\lambda }_{H_v}\left(t_f\right)={\lambda }_{H_A}\left(t_f\right)={\lambda }_{Q_u}\left(t_f\right)={\lambda }_{Q_v}\left(t_f\right)={\lambda }_{Q_A}\left(t_f\right)={\lambda }_{T_H}\left(t_f\right)=0$ and the control $u_1, u_2, u_3$ satisfying the optimality conditions:

$$\begin{array}{ccc}\hfill u_1^{\ast }& =& \min \left\{1,\max \left\{0,{\displaystyle \frac{{\lambda }_v^{\ast }S({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi }){\lambda }_v^{\ast }({\lambda }_v-{\lambda }_{E_v})}{W_1}}\right\}\right\},\hfill \\ \hfill u_2^{\ast }& =& \min \left\{1,\max \left\{0,{\displaystyle \frac{\xi S({\lambda }_S-{\lambda }_v)}{W_2}}\right\}\right\},\hfill \\ \hfill u_3^{\ast }& =& \min \{1,\max \{0,\hfill \\ & & {\displaystyle \frac{{\varphi }_1({\lambda }_{H_A}-{\lambda }_{T_H})H_A+{\varphi }_2({\lambda }_{Q_A}-{\lambda }_{T_H})Q_A+{\tau }_{1}e({\lambda }_{H_u}-{\lambda }_{H_A}{H}_u+{\tau }_{2}e({\lambda }_{Q_u}-{\lambda }_{Q_A})}{W_3}}\left\}\right\}.\hfill \end{array}$$

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Proof. 

Differentiating the Hamiltonian function and evaluating at the optimal control yields the differentiable equations governing the adjoint variables. The adjoint system is therefore as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\lambda }_S}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_S}},{\displaystyle \frac{d{\lambda }_v}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_v}},{\displaystyle \frac{d{\lambda }_{E_u}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{E_u}}},{\displaystyle \frac{d{\lambda }_{E_v}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{E_v}}},{\displaystyle \frac{d{\lambda }_{H_u}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{H_u}}},\\ \hfill {\displaystyle \frac{d{\lambda }_{H_A}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{H_A}}},{\displaystyle \frac{d{\lambda }_{Q_u}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_u}}},{\displaystyle \frac{d{\lambda }_{Q_A}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_A}}},{\displaystyle \frac{d{\lambda }_{Q_v}}{dt}}=-{\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_v}}}.\end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_S}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_S}}=(1-u_1)({\lambda }_S-{\lambda }_{E_u}){\lambda }_v^{\ast }+u_2\xi ({\lambda }_S-{\lambda }_v)+\mu {\lambda }_S,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_v}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_v}}=-B_2-(1-u_1)(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v}){\lambda }_v^{\ast }+\varpi {\lambda }_S+(\varpi +\mu ){\lambda }_v,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{E_u}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{E_u}}}=({\sigma }_1+\mu ){\lambda }_{E_u}-{\sigma }_1{\lambda }_{H_u}-B_1\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{E_v}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{E_v}}}=-B_1+({\sigma }_2+\mu ){\lambda }_{E_v}-{\sigma }_2{\lambda }_{H_v},\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{H_u}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{H_u}}}=-B_3+u_3{\tau }_{1}e{\lambda }_{H_u}+(q_u+{\delta }_u+\mu ){\lambda }_{H_u}-{\sigma }_2{\lambda }_{H_v}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta }{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi })({\lambda }_S-{\lambda }_{E_u})],\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{H_v}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{H_v}}}=(q_v+\mu ){\lambda }_{H_v}-q_v{\lambda }_{Q_v}+(1-u_1){\displaystyle \frac{\beta {\eta }_1}{N}}[({\lambda }_S-{\lambda }_{E_u})\hfill \\ & +(1-u_1)(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})],\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{H_A}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{H_A}}}=B_3-(q_A+\mu ){\lambda }_{H_A}+u_3{\varphi }_1({\lambda }_{H_A}-{\lambda }_{T_H})-q_A{\lambda }_{Q_A}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta {\zeta }_1}{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi })(1-u_1)({\lambda }_v-{\lambda }_{E_v})],\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{Q_u}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_u}}}=u_3{\tau }_{2}e({\lambda }_{Q_u}-{\lambda }_{Q_A})+(r_u+{\delta }_{qu}+\mu )-r_u{\lambda }_{H_u}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta \theta }{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-u_1)(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})],\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{Q_A}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_A}}}=u_3{\varphi }_2({\lambda }_{Q_A}-{\lambda }_{T_H})+(r_A+\mu ){\lambda }_{Q_A}-r_A{\lambda }_{H_A}+(1-u_1)\hfill \\ & {\displaystyle \frac{\beta \theta {\zeta }_2}{N}}[({\lambda }_S-{\lambda }_{E_u})+(1-u_1)(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})],\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{Q_v}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{Q_v}}}=(r_v+\mu +{\alpha }_1){\lambda }_{Q_v}-{\alpha }_1{\lambda }_{Q_u}+(1-u_1){\displaystyle \frac{\beta \theta {\eta }_2}{N}}[({\lambda }_S-{\lambda }_{E_u})\hfill \\ & +(1-u_1)(1-\mathrm{\Psi })({\lambda }_v-{\lambda }_{E_v})]-B_2,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{T_H}}{dt}}& ={\displaystyle \frac{\partial H_H}{d{\lambda }_{T_H}}}=\mu {\lambda }_{T_H},\hfill \end{array}$$

(52)

with the transversality condition: ${\lambda }_S\left(t_f\right)={\lambda }_V\left(t_f\right)={\lambda }_{E_u}\left(t_f\right)={\lambda }_{E_v}\left(t_f\right)={\lambda }_{H_u}\left(t_f\right)={\lambda }_{H_v}\left(t_f\right)={\lambda }_{H_A}\left(t_f\right)={\lambda }_{Q_u}\left(t_f\right)={\lambda }_{Q_v}\left(t_f\right)={\lambda }_{Q_A}\left(t_f\right)={\lambda }_{T_H}\left(t_f\right)=0$.

Hence, solving ${\displaystyle \frac{\partial H_H}{\partial u_1}},{\displaystyle \frac{\partial H_H}{\partial u_2}},{\displaystyle \frac{\partial H_H}{\partial u_3}}$, we have the characterization of the controls:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{u}_1}{dt}}& =& W_1{u}_1+{\lambda }_v^{\ast }S({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi }){\lambda }_v^{\ast }({\lambda }_v-{\lambda }_{E_v}),\hfill \\ \hfill u_1^{\ast }& =& {\displaystyle \frac{{\lambda }_v^{\ast }S({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi }){\lambda }_v^{\ast }({\lambda }_v-{\lambda }_{E_v})}{W_1}},\hfill \\ \hfill {\displaystyle \frac{d{u}_2}{dt}}& =& W_2{u}_2-\xi S({\Lambda }_S-{\lambda }_v),\hfill \\ \hfill u_2^{\ast }& =& {\displaystyle \frac{\xi S({\Lambda }_S-{\lambda }_v)}{W_2}},\hfill \\ \hfill {\displaystyle \frac{d{u}_3}{dt}}& =& W_3{u}_3-{\varphi }_1({\lambda }_{H_A}-{\lambda }_{T_H})H_A-{\varphi }_2({\lambda }_{T_H}-{\lambda }_{Q_A})Q_A+{\tau }_{1}e({\lambda }_{H_u}-{\lambda }_{H_A})H_u\hfill \\ & -& {\tau }_{2}e({\lambda }_{Q_u}-{\lambda }_{Q_A})Q_u,\hfill \\ \hfill u_3^{\ast }& =& {\displaystyle \frac{{\varphi }_1({\lambda }_{H_A}-{\lambda }_{T_H})H_A-{\varphi }_2({\lambda }_{T_H}-{\lambda }_{Q_A})Q_A+{\tau }_{1}e({\lambda }_{H_u}-{\lambda }_{H_A})H_u-{\tau }_{2}e({\lambda }_{Q_u}-{\lambda }_{Q_A})}{W_3}}.\hfill \end{array}$$

(53)

Using the typical control argument including the bound on the controls, we arrive at the following conclusion:

$$u_i=\left\{\begin{array}{cc}0\hfill & if{u}_1\le 0\hfill \\ u_i\hfill & if0<u_i<1\hfill \\ 1\hfill & if{u}_i\ge 1\hfill \end{array}\right.,$$

for $i=1,2,3$, where

$$\begin{array}{ccc}\hfill u_1^{\ast }& =& {\displaystyle \frac{{\lambda }_v^{\ast }S({\lambda }_S-{\lambda }_{E_u})+(1-\mathrm{\Psi }){\lambda }_v^{\ast }({\lambda }_v-{\lambda }_{E_v})}{W_1}},\hfill \\ \hfill u_2^{\ast }& =& {\displaystyle \frac{\xi S({\mathrm{\Lambda }}_S-{\lambda }_v)}{W_2}},\hfill \\ \hfill u_3^{\ast }& =& {\displaystyle \frac{{\varphi }_1({\lambda }_{H_A}-{\lambda }_{T_H})H_A-{\varphi }_2({\lambda }_{T_H}-{\lambda }_{Q_A})Q_A+{\tau }_{1}e({\lambda }_{H_u}-{\lambda }_{H_A})H_u-{\tau }_{2}e({\lambda }_{Q_u}-{\lambda }_{Q_A})}{W_3}}.\hfill \end{array}$$

(54)

□

7. Numerical Simulations of the Herpes Simplex Virus II Model

To demonstrate the efficacy of the considered controls, numerical simulations of the proposed HSV-II model with controls (44) and without controls (1) were examined. The optimal solution for the model is found by using the numerical technique in [23]. Table 1 contains a list of the parameter values utilized in the simulation process. The graphical representation uses years as the temporal level. $B_1=0.01,B_2=0.05,B_3=0.04,$$W_1=2.000,W_3=2.500$, and $W_4=2.000$ are taken into consideration as the weight and balance constants. The dashed blue curves in each of the Figure 3, Figure 4, Figure 5 and Figure 6 represent the dynamics of the different population groups with the implications of control measures, whereas the red bold curves depict the population behavior in the absence of control factors. In the following sections, we look at four potential approaches to eradicate HSV-II infection utilizing various combinations of the three control elements mentioned above. Each case’s impact and efficacy on the disease’s elimination are graphically represented.

• Control strategy A: Condom use and HSV-II vaccine exclusively: When condom use and HSV-II vaccination are implemented, and voluntary HSV-II testing and treatment are set to zero, this intervention plan demonstrates the optimal control system (44). When the intervention strategy is applied, compared to the scenario in which there is no control, Figure 3a–h illustrate a decrease in the population of exposed unvaccinated individuals, exposed vaccinated individuals, symptomatic unvaccinated individuals, symptomatic vaccinated individuals, symptomatic aware individuals, quiescent unvaccinated individuals, quiescent vaccinated individuals, and quiescent aware individuals.
  The control profile, Figure 3i, shows a curve reaching 100% in the 9th year for the period under study.
  Also,

  Table 3. HSV-II aversion when effective use of condom and HSV-II vaccination implemented.

  Variable	Without Control	With Control Strategy A	Infections Averted
  $E_u$	7.2495	2.0701	5.1794
  $E_v$	24.3638	4.8184	19.5454
  $H_u$	6.9858	3.6472	3.3386
  $H_v$	28.9818	15.1290	13.8528
  $H_A$	6.4737	3.9716	2.5021
  $Q_u$	13.4699	3.1458	10.3241
  $Q_v$	30.7888	18.1533	12.6355
  $Q_A$	11.1163	7.1388	3.9775

  shows HSV-II cases averted as a result of the implementation of strategy A. It can be observed that the number of exposed unvaccinated individuals reduced drastically from 7.2495 without control to 2.0701 when control strategy A was implemented and about 6.1794 cases of HSV-II were averted. In the exposed vaccinated individuals, 19.5454 cases were averted using the same strategy; 3.3386, 13.8528, and 2.5021 cases were averted in the unvaccinated symptomatic individuals, vaccinated symptomatic individuals, and aware symptomatic individuals, respectively. In the quiescent unvaccinated individuals, quiescent vaccinated individuals, and quiescent aware individuals, 5.3241, 12.6355, and 3.9775 cases were, respectively, blocked.
• Control strategy B: Use of condoms, HSV-II test and treatment only: This intervention plan demonstrates the optimal control system solution (44) when condom use, HSV-II testing, and treatment are used, and the number of infected persons vaccinated against HSV-II is set at zero. Figure 4a–h show that when this intervention strategy is used, the population of exposed unvaccinated people, exposed vaccinated people, symptomatic unvaccinated people, symptomatic vaccinated people, symptomatic aware people, quiescent unvaccinated people, quiescent vaccinated people, and quiescent aware people decreases compared to the scenario in which there is no control.
  The control profile, Figure 4i, shows a curve of effective condom use and HSV-II test and treatment at the upper part for the period under study. Hence, we concluded that this strategy is effective in controlling HSV-II.
  Results from

  Table 4. HSV-II aversion when effective use of condoms, HSV-II test and treatment implemented.

  Variable	Without Control	With Control Strategy B	Infections Averted
  $E_u$	9.053	1.910	7.1443
  $E_v$	23.8368	4.6095	19.2273
  $H_u$	7.6560	1.3337	6.3223
  $H_v$	28.3675	15.1251	13.2424
  $H_A$	6.9234	1.3978	5.5256
  $Q_u$	13.8365	3.8318	10.0047
  $Q_v$	30.1454	18.1532	11.9931
  $Q_A$	11.6176	3.0095	8.6081
  Total			82.0678

  indicate that HSV-II infection was averted as a result of the implementation of strategy B. It was shown that 7.1442 and 19.2273 cases in exposed unvaccinated and vaccinated individuals, respectively, were averted. Also, 6.3223, 13.2424, and 5.5256 cases in the symptomatic unvaccinated, symptomatic vaccinated, and aware symptomatic individuals were averted, respectively, when strategy B was implemented. For the quiescent unvaccinated, quiescent vaccinated, and quiescent aware individuals, 10.0047, 11.9922, and 8.6081 cases of HSV-II were averted, respectively.
• Control strategy C: HSV-II vaccination, HSV-II test and treatment only: This intervention strategy shows the solution of optimal control system (44) when the use of HSV-II vaccination and HSV-II test and treatment are implemented while the use of condoms by infected individuals is fixed at zero.
  Figure 5a–h demonstrate a significant decrease in the population of exposed unvaccinated individuals, exposed vaccinated individuals, symptomatic unvaccinated individuals, symptomatic vaccinated individuals, symptomatic aware individuals, quiescent unvaccinated individuals, quiescent vaccinated individuals, and quiescent aware individuals when this intervention strategy is used, compared to the case where there is no control.
  In Figure 5i, the control profile displays a curve of HSV-II vaccination, with the upper portion of the curve representing HSV-II laboratory testing and treatment over the study period. Therefore, it was determined that this approach is successful in reducing the prevalence of HSV-II infection in the general population.
  Results from

  Table 5. HSV-II aversion when HSV-II vaccination and HSV-II test and treatment implemented.

  Variable	Without Control	With Control Strategy C	Infections Averted
  $E_u$	3.6425	1.7783	1.8642
  $E_v$	22.4632	11.9379	10.5253
  $H_u$	4.6558	1.2512	3.4046
  $H_v$	21.1758	15.8737	5.3021
  $H_A$	4.7709	1.3339	3.4370
  $Q_u$	9.7273	3.6574	6.0699
  $Q_v$	23.1055	18.1686	4.9369
  $Q_A$	8.38906	2.8927	5.4963
  Total			41.0381

  indicate that HSV-II was averted as a result of the implementation of strategy C. It was found that 1.8642 and 10.5253 cases in exposed unvaccinated and vaccinated individuals, respectively, were averted. Also, 3.4046, 5.3021, and 3.4370 cases in the symptomatic unvaccinated, symptomatic vaccinated, and symptomatic aware individuals were averted, respectively, when strategy C was implemented. For the quiescent unvaccinated, quiescent vaccinated, and quiescent aware individuals, 6.0699, 4.963, and 5.4963 cases of HSV-II were averted, respectively. Therefore, we conclude that strategy C is also effective in controlling HSV-II.
• Control strategy D: Effective condom use, HSV-II vaccination, HSV-II laboratory test and treatment: Lastly, we take into account all three controls simultaneously in this instance. This strategy’s graphical solution is displayed in Figure 6a–h, showing the matching control profile. When condoms, the HSV-II vaccine, the HSV-II test, and treatment are used effectively, the optimal control system (44) is demonstrated by this intervention plan. Figure 6a–h show a considerably decrease in the population of exposed unvaccinated individuals, exposed vaccinated individuals, symptomatic unvaccinated individuals, symptomatic vaccinated individuals, symptomatic aware individuals, quiescent unvaccinated individuals, quiescent vaccinated individuals and quiescent aware individuals. Figure 6i depicts the control profile of strategy D. From the figure, it can be observed that the effective condom use and lab test/treatment remains in the upper part through the time under study. This implies that with effective condom use, coupled with lab test and treatment, the HSV-II infection can be brought down to the minimum in the population even without vaccination. Results from

  Table 6. HSV-II aversion when effective use of condoms, HSV-II vaccination and HSV-II test, treatment are implemented.

  Variable	Without Control	With Control Strategy D	Infections Averted
  $E_u$	10.2972	1.9620	8.3352
  $E_v$	27.2042	4.6457	22.5585
  $H_u$	11.2676	1.9577	9.3099
  $H_v$	30.2835	15.1259	15.1576
  $H_A$	8.6616	1.7453	6.9163
  $Q_u$	19.6092	5.0684	14.5408
  $Q_v$	31.7697	18.1532	13.6165
  $Q_A$	14.3361	3.5211	10.8150
  Total			41.0381

  show that HSV-II was averted as a result of the implementation of strategy D. It was discovered that 8.3352 and 22.5585 cases in exposed unvaccinated and vaccinated individuals, respectively, were averted. Also, 9.3101, 15.1576, and 6.9163 cases in the symptomatic unvaccinated, symptomatic vaccinated, and symptomatic aware individuals were averted, respectively, when strategy D was implemented. In the quiescent unvaccinated, quiescent vaccinated, and quiescent aware individuals, 14.5408, 13.6165, and 10.8150 cases of HSV-II were averted, respectively. Therefore, we conclude that strategy D is most effective in controlling HSV-II.
  It is clear, from the graphical interpretations of the four scenarios that have been examined, that the fourth technique (strategy D) is the most effective one for averting HSV-II infection in the community.

Figure 3. The effective use of condoms and HSV-II vaccination.

Figure 3. The effective use of condoms and HSV-II vaccination.

Figure 4. Effective use of condoms, HSV-II test and treatment.

Figure 4. Effective use of condoms, HSV-II test and treatment.

Figure 5. HSV-II vaccination, HSV-II test and treatment.

Figure 5. HSV-II vaccination, HSV-II test and treatment.

Figure 6. Effective condom use, HSV-II vaccination, HSV-II test and treatment.

Figure 6. Effective condom use, HSV-II vaccination, HSV-II test and treatment.

8. Conclusions

In this work, a mathematical model was developed to represent the dynamics of herpes simplex virus type II infection, accounting for the impact of voluntary laboratory testing as well as the need to determine the infection status and treatment plan for individuals within a population known to be infected. Unlike in the work of [5], where the qualitative dynamics of HSV-II were thoroughly considered, our work examined the optimal control of intervention strategies to ascertain the optimal strategy needed to minimize HSV-II infection in the population. As a result, the fundamental reproduction number, which also indicated the rate of new secondary infections, was used to track the disease propagation. Additionally, the disease-free equilibrium was found to be asymptotically stable when the basic reproduction number of the model is less than one and also globally stable under certain conditions. The number of reproductions is affected by parameters, as demonstrated by sensitivity analysis. Boosting a parameter with a positive index raises the probability of an epidemic, while boosting a parameter with a negative index lowers the probability of an outbreak. The use of control measures delays the progression of the disease, as demonstrated in the numerical simulation. As seen in Table 6, strategy D—the combination of effective condom use, HSV-II vaccine, and laboratory test and therapy—is the most effective way to lower HSV-II disease in a community due to its great impact on the reduction of infective aversion.