A Dynamics and Control Study of the New H1N1 Influenza with Two Roots of Infection: The Impact of Optimal Vaccination and Treatment

Abstract

H1N1 influenza, also known as swine flu, is a subtype of the influenza A virus that can infect humans, pigs, and birds. Sensitivity analysis and optimal control studies play a crucial role in understanding the dynamics of H1N1 influenza. In this study, we have derived a mathematical model incorporating both symptomatic and asymptomatic infections, as well as vaccination, to assess the impact of key parameters on disease transmission. Also, we have assumed a density-dependent infection transmission in the modeling process of H1N1 dynamics. We determine the basic reproduction number using the next-generation matrix method and found that the disease-free equilibrium is stable when the basic reproduction number $R_0<1$ and the endemic equilibrium exists and is stable globally when $R_0>1$. By performing sensitivity analysis, the most influential factors affecting infection spread are identified, aiding in targeted intervention strategies. Optimal control techniques are then applied to determine the best approaches to minimize infections while considering resource constraints. The findings provide valuable insights for public health policies, offering effective strategies for mitigating H1N1 outbreaks and enhancing disease management efforts using optimal vaccination.

1. Introduction

The H1N1 virus exhibits symptoms similar to other influenza strains and belongs to type A, which along with type B primarily infects humans [1]. Types C and D cause mild illness, mainly in animals. Seasonal influenza affects one in six people annually, resulting in 3–5 million severe cases and up to 650,000 respiratory deaths globally [2,3]. H1N1 spreads via respiratory droplets, contact, and contaminated surfaces, with symptoms such as fever, cough, sore throat, and gastrointestinal issues, especially in children [1,4]. The WHO declared the H1N1 pandemic on 11 June 2009, attributing 284,400 deaths to the virus that year [5]. A list of past pandemic sizes is shown in Table 1. The pandemic was declared over in August 2010, though H1N1 remains a seasonal flu strain. Transmission is heightened in crowded settings; preventive measures include hand hygiene and respiratory etiquette [6].

Vaccination is the primary defense; however, producing vaccines takes at least six months, and shortages frequently occur [7]. Non-pharmaceutical interventions—such as school closures, airport screenings, and face mask use—were employed during the 2009 pandemic [8,9]. Face mask adoption was notable during the 2003 SARS outbreak, with 76% usage reported in Hong Kong [10]. The WHO advises vaccination prior to peak influenza periods, beginning in April or October [11]. It coordinates global efforts in vaccine development, distribution, and regulation, holding biannual consultations to recommend strains for seasonal vaccines [12]. Vaccination not only reduces disease burden but also mitigates societal and economic impacts [13,14,15]. Mathematical models assist in evaluating such measures.

Mathematical modeling is a helpful tool for investigating transmission patterns and assessing the outcomes of intervention initiatives. The mathematical models of influenza outbreaks have been studied and evaluated by several academics (since influenza is an airborne disease) [16,17]. Tracht et al. [18] studied the effectiveness of face masks in controlling the spread of the H1N1 influenza virus. Tchuenche et al. [19] studied the public views and reactions to infectious illnesses and the impact of media coverage. A study of H1N1 virus disease dynamics and sensitivity was performed by Krishnapriya et al. [20]. Kanyiri et al. [21] studied a mathematical analysis of influenza transmission and viral control drug resistance. The development of an age-dependent vaccination strategy for the 2009 A/H1N1 influenza epidemic in the Republic of Korea was studied by Kim et al. [22]. Dalal et al. [23] evaluated the process of creating the genosensor and its effectiveness in identifying the H1N1 virus. Ratre et al. [24] studied the role of children in the spread of the influenza pandemic and the importance of immunization in reducing the occurrence of influenza. Baba et al. [25] established a mathematical model that analyzes the coexistence of two influenza strains and examines their method of transmission, as well as treatment options. Alharbi et al. [26] studied the transmission dynamics of influenza strains among pilgrims from different hemispheres and the effectiveness of vaccination in preventing infections. Ackerman et al. [27] investigated mathematical models to identify the differences in immune processes between infections caused by different strains of the virus.

The study [28] examines the impact of contact with symptomatic and asymptomatic individuals on infectious disease spread using the SEIR model. It reveals that relative infectiousness and asymptomatic cases significantly influence disease transmission, which reinforces the importance of isolation measures. Ref. [29] involves studying the spread of swine flu, differentiating between symptomatic and asymptomatic individuals, and applying optimal control strategies to manage the infection effectively. It typically involves mathematical modeling, using differential equations to analyze disease dynamics and control interventions. Another study explores bifurcation analysis, examining stability conditions and control measures like vaccination and treatment [30]. These methods help policymakers design effective interventions to mitigate outbreaks [31]. Recent research [30] explores the bifurcation analysis of influenza A (H1N1) models with treatment and vaccination models, using numerical simulations and mathematical theories. They investigate transcritical, Hopf, and backward bifurcations to understand disease transmission dynamics and the basic reproduction number’s role in disease persistence or death. One study focuses on the mathematical modeling of H1N1 transmission and control, using real-world data from Mexico, Italy, and South Africa to determine critical illness factors and forecast trends [32]. Another research paper introduces a susceptible–exposed–infectious–quarantined–recovered (SEIQR) model, incorporating quarantine as a key intervention and applying optimal control theory to balance epidemiological impact with cost-effectiveness [33]. Ref. [34] examines an influenza model with vaccination and treatment, applying Pontryagin’s maximum principle to identify the best strategies for disease control. Here, we propose an SEIVR (susceptible–exposed–infected–vaccinated–recovered)-type mathematical model for H1N1 dynamics with a view of optimally controlling the infection using vaccination and treatment. The infected population is divided into symptomatic ($I_S$) and asymptomatic individuals ($I_A$). Also, we assumed that vaccinated humans may lose immunity and become susceptible again.

Sensitivity analysis is a crucial tool in disease modeling, assessing how variations in model parameters affect predictions [35]. It helps identify key factors affecting disease spread and control strategies, improves model accuracy, enhances decision-making, optimizes resource allocation, and supports uncertainty quantification [36]. It is widely used in infectious disease modeling, epidemiology, and biomedical research to refine strategies and minimize risks. Sensitivity analysis is particularly useful for evaluating the impact of different assumptions on disease forecasts, improving preparedness [35]. In the case of the H1N1 dynamics study, sensitivity analysis helps identify which parameters most influence the model outcomes, allowing researchers to refine predictions and optimize control strategies. Studies have used techniques like the partial rank correlation coefficient (PRCC) [37] and Latin hypercube sampling to assess the impact of different variables on disease transmission [33]. One approach involves analyzing the basic reproduction number to determine how changes in parameters affect the spread of the virus. Here, our focus is on achieving a disease-free system. We analyze the sensitivity of model parameters with respect to the basic reproduction number.

In this research, as a result, we derived an epidemiological population model for H1N1 transmission dynamics. To achieve this, we first calculate the disease-free and endemic steady states. By finding the Jacobian matrix, we studied the stability of the equilibrium points in terms of the basic reproduction number of the system under consideration. Finally, we applied the optimal control theory and formulated an optimal control problem. The optimal control problem is solved using the Hamiltonian function and numerically solved using bvp4c solver in Matlab version R2016a. We determined the optimal vaccination and self-monitoring activities to minimize the disease.

The structure of the article is as follows: we develop the mathematical model in Section 2. Section 3 contains an analysis of fundamental qualities such as non-negativity, boundedness, etc. In Section 4, equilibrium and stability analysis are covered. The optimum control issue is presented in Section 5. We present the numerical simulations in Section 7. Finally, the discussion in Section 8 brings the article to an end.

Table 1. List of recent influenza pandemics throughout the past century [21,38,39].

Pandemic	Year	Strain	No. of Deaths
Spanish flu	1918–1920	H1N1	40–100 million
Asian flu	1957–1958	H2N2	1–2 million
Hong Kong flu	1968–1970	H3N2	0.5–2 million
Swine flu	2009–2010	H1N1	Up to 575,000

Table 1. List of recent influenza pandemics throughout the past century [21,38,39].

Pandemic	Year	Strain	No. of Deaths
Spanish flu	1918–1920	H1N1	40–100 million
Asian flu	1957–1958	H2N2	1–2 million
Hong Kong flu	1968–1970	H3N2	0.5–2 million
Swine flu	2009–2010	H1N1	Up to 575,000

2. Derivation of the Mathematical Model

The assumptions of the mathematical model are discussed here to derive the mathematical model. To capture the transmission dynamics of the influenza virus, the total human population at any time t, denoted by $N\left(t\right)$, is stratified into six epidemiological compartments, each representing a distinct health status or disease progression phase:

(i)

Susceptible population ($S\left(t\right)$): Individuals who have not yet contracted the virus and are at risk of infection through contact with infectious persons.

(ii)

Exposed population ($E\left(t\right)$): Individuals who have been exposed to the virus and are in the incubation period. Although they do not show symptoms, they remain capable of transmitting the virus, thus contributing to its spread in the population.

(iii)

Symptomatic infected population ($I_S\left(t\right)$): Individuals who have developed clinical symptoms of influenza and are actively contagious. This group represents the primary source of detectable and severe transmission cases.

(iv)

Asymptomatic infected population ($I_A\left(t\right)$): Individuals who are infected but do not exhibit clinical symptoms. Despite their asymptomatic status, they play a critical role in the silent transmission of the virus.

(v)

Recovered population ($R\left(t\right)$): Individuals who have overcome the infection and have acquired immunity. They no longer participate in the transmission chain.

(vi)

Vaccinated population ($V\left(t\right)$): Individuals who have received a vaccine and possess varying levels of protection depending on vaccine efficacy.

It is important to note that the exposed ($E\left(t\right)$), symptomatic infected ($I_S\left(t\right)$), and asymptomatic infected ($I_A\left(t\right)$) compartments collectively represent the infected individuals, differentiated based on the presence or absence of clinical symptoms and the stage of infection. This stratification allows for a more nuanced depiction of viral transmission, particularly in accounting for asymptomatic carriers and pre-symptomatic transmission—a key challenge in managing influenza outbreaks.

To describe the progression and spread of influenza within the stratified population and to incorporate the influence of media-driven public health interventions, we introduce a deterministic system of nonlinear ordinary differential equations. This system models the time-dependent interactions among the compartments under the influence of media awareness, which can alter behavioral patterns such as adherence to preventive measures, the rate of vaccination, or a reduction in contact frequency with infected individuals. We proposed the following model:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \lambda -{\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\mu +a)S+\theta V,\hfill \\ \hfill {\displaystyle \frac{dE}{dt}}& =& {\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\kappa +\mu )E,\hfill \\ \hfill {\displaystyle \frac{d{I}_S}{dt}}& =& \rho \kappa E+b{I}_A-({\sigma }_2+\mu )I_S,\hfill \\ \hfill {\displaystyle \frac{d{I}_A}{dt}}& =& (1-\rho )\kappa E-(b+{\sigma }_1+\mu )I_A,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& aS-(\theta +\delta +\mu )V,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =& \delta V+{\sigma }_1{I}_A+{\sigma }_2{I}_S-\mu R,\hfill \end{array}$$

(1)

with the initial values

$$\begin{array}{c}\hfill S\left(0\right)>0,I_S\left(0\right)>0,I_A\left(0\right)>0,V\left(0\right)>0,R\left(0\right)>0.\end{array}$$

(2)

In the formulation of the epidemiological model described by Equation (1) and schematic diagram in Figure 1, each parameter carries specific biological significance. These parameters, which collectively define the dynamics of population transitions under viral infection and vaccination strategies, are elucidated as follows:

• $\lambda$ denotes the constant production rate of viral agents within the infected host population. This parameter captures the continual generation of infectious particles that contribute to disease propagation.
• $\beta$ represents the infection transmission rate, quantifying the probability of successful disease spread upon contact between susceptible and infectious individuals.
• The natural death rate of individuals within the model is denoted by $\mu$, encompassing mortality unrelated to the disease or vaccination process.
• The parameter a characterizes the rate of vaccination administration across the population, effectively indicating the proportion of susceptible individuals receiving immunization over time.
• Vaccination efficacy is described by $\theta$, which is assumed to be low; this reflects the limited protective effect conferred by the vaccine, potentially due to weak immune response or pathogen resistance.
• The rate of progression from the exposed class to the infectious class is captured by $\kappa$, indicating the transition speed from latent infection to active disease.
• $\rho$ corresponds to the proportion of the population subject to disease transmission and progression, influencing the overall dynamics of the epidemic spread.
• The parameters ${\sigma }_1$ and ${\sigma }_2$ represent distinct recovery rates from infection, potentially differentiating recovery outcomes based on treatment status or disease severity.
• The recovery rate of vaccinated individuals who may still undergo infection is denoted by $\delta$, incorporating post-vaccination outcomes within the model framework.
• b is the rate at which an asymptomatic infected human joins the symptomatic infected population.

The numerical values and descriptions of these parameters are comprehensively listed in

Table 2. List of parameters of model (1) with short description and values [18,25,40].

Parameter	Definition	Value
$\lambda$	Constant reproduction rate	0.03–1
$\beta$	Disease transmission rate	0.5∼1
$\mu$	Death rate	0.2
$\theta$	Rate of loss of immunity	0.00833
a	Rate of vaccination	0.0027375
$\kappa$	Rate of infraction from exposed population	0.15∼0.6
$\rho$	Probability of population	0.2∼0.75
${\sigma }_1$	Rate of recovery for the infected population	$0.2$
${\sigma }_2$	Rate of recovery for the affected population	$0.8$
b	Rate at which asymptomatic infected goes	0.0027
	to symptomatic infected human class
$\delta$	Rate at which the vaccination recovers	0.22∼0.75

, which supports the analytical and simulation aspects of the model.

3. Preliminary Knowledge

It is not hard to show that the solutions exist for the system (1) and it is unique for the initial condition (2) [41]. We show that the solutions are non-negative for the positive initial values as in (2).

For the purpose of analyzing the behavior of the influenza virus model (1), we present the characteristics below. We have the following theorem on the non-negativity of solutions:

3.1. Positive Invariance

Theorem 1.

Given the initial conditions in (2), all solutions to system (1) for all $t>0$ are non-negative.

Proof. 

Let

$$\begin{array}{c}\hfill T_1=sup\{t>0:S\left(t\right)>0,E\left(t\right)>0,I_S\left(t\right)>0,I_A\left(t\right)>0,V\left(t\right)>0,R\left(t\right)>0\}.\end{array}$$

Using the initial condition (2), we have $T_1\ge 0$. For $T_1<\infty$, one of $S,E,I_S,I_A,V,$ and R is equal to zero at $T_1$.

Now, from the first equation of (1), we can write

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \lambda -{\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\mu +a)S+\theta V,\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}+\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right]S& =& (\lambda +\theta V)\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & {\displaystyle \frac{d}{dt}}\left\{S\left(t\right)exp\left[{\int }_0^t\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right]\right\}\hfill \\ & & =(\lambda +\theta V)exp\left[{\int }_0^t\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right].\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & S\left(T_1\right)exp\left\{\left[{\int }_0^t\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right]\right\}\hfill \\ & & =S\left(0\right)+{\int }_0^t\left\{(\lambda +\theta V)exp\left[{\int }_0^t\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right]\right\}dz.\hfill \end{array}$$

So that

$$\begin{array}{ccc}\hfill S\left(T_1\right)& =& S\left(0\right)exp\left[-{\int }_0^t\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right]\hfill \\ & & +exp\left[-{\int }_0^t\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right]\hfill \\ & & \times {\int }_0^t\left\{(\lambda +\theta V)exp\left[\left({\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +a)\right)du\right]\right\}dz>0.\hfill \end{array}$$

Again, from system (1), we get

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dE}{dt}}& =& {\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\kappa +\mu )E,\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dE}{dt}}+(\kappa +\mu )E& =& {\displaystyle \frac{\beta S(I_S+I_A)}{N}}\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}\left\{E\left(t\right)exp\left[{\int }_0^t(\kappa +\mu )du\right]\right\}& =& {\displaystyle \frac{\beta S(I_S+I_A)}{N}}exp\left[{\int }_0^t(\kappa +\mu )du\right].\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill E\left(T_1\right)exp\left[{\int }_0^t(\kappa +\mu )du\right]-E\left(0\right)& =& {\int }_0^t\left\{{\displaystyle \frac{\beta S(I_S+I_A)}{N}}exp\left[{\int }_0^t(\kappa +\mu )du\right]\right\}dz.\hfill \end{array}$$

So that

$$\begin{array}{ccc}\hfill E\left(T_1\right)& =& E\left(0\right)exp\left[-{\int }_0^t(\kappa +\mu )du\right]+exp\left[-{\int }_0^t(\kappa +\mu )du\right]\hfill \\ & & \times {\int }_0^t\left\{{\displaystyle \frac{\beta S(I_S+I_A)}{N}}exp\left[{\int }_0^t(\kappa +\mu )du\right]\right\}dz>0.\hfill \end{array}$$

Similarly, from (1), we get

$$\begin{array}{ccc}\hfill I_S\left(T_1\right)& =& I_S\left(0\right)exp[-{\int }_0^t({\sigma }_2+\mu )du]+exp[-{\int }_0^t({\sigma }_2+\mu )du]\hfill \\ & & \times {\int }_0^t\{(\rho \kappa E+b{I}_A)exp[{\int }_0^t({\sigma }_2+\mu )du]\}dz>0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_A\left(T_1\right)& =& I_A\left(0\right)exp\left[-{\int }_0^t(b+{\sigma }_1+\mu )du\right]+exp\left[-{\int }_0^t(b+{\sigma }_1+\mu )du\right]\hfill \\ & & \times {\int }_0^t\left\{(1-\rho )\kappa Eexp\left[(b+{\sigma }_1+\mu )du\right]\right\}dz>0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill V\left(T_1\right)& =& V\left(0\right)exp\left[-{\int }_0^t(\theta +\delta +\mu )du\right]+exp[-{\int }_0^t(\theta +\delta +\mu )du]\hfill \\ & & \times {\int }_0^t\left\{aSexp\left[{\int }_0^t(\theta +\delta +\mu )du\right]\right\}dz>0,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill R\left(T_1\right)& =& R\left(0\right)exp\left[-{\int }_0^t\left(\mu R\right)du\right]\hfill \\ & & \times {\int }_0^t\left\{(\delta V+{\sigma }_1{I}_A+{\sigma }_2{I}_S)exp\left[{\int }_0^t\left(\mu R\right)du\right]\right\}dz>0.\hfill \end{array}$$

Using the same procedure as above, we can show that for all $t>0$, $S\left(t\right)>0,E\left(t\right)>0,I_S\left(t\right)>0,I_A\left(t\right)>0,V\left(t\right)>0$ and $R\left(t\right)>0$. Thus, we conclude that all trajectories of system (1) remain non-negative in ${\mathbb{R}}_{+}^6$ for any all $t>0$. Hence, the proof is completed. □

3.2. Boundedness

We now begin with the theorem that guarantees that the solutions of the system (1) are constrained to have non-negative beginning values.

Theorem 2.

All solutions of system (1) are bounded in the region Ψ, defined by

$$\begin{array}{ccc}\hfill \mathrm{\Psi }& =& \left\{(S,E,I_S,I_A,V,R)\in R^5:0\le S+E+I_S+I_A+V+R\le {\displaystyle \frac{\lambda }{\mu }}\right\}.\hfill \end{array}$$

Proof. 

Here, $N=S+E+I_S+I_A+V+R$. Thus, using system (1), we can write

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dN}{dt}}& =& {\displaystyle \frac{dS}{dt}}+{\displaystyle \frac{dE}{dt}}+{\displaystyle \frac{d{I}_S}{dt}}+{\displaystyle \frac{d{I}_A}{dt}}+{\displaystyle \frac{dV}{dt}}+{\displaystyle \frac{dR}{dt}}\hfill \\ & \le & \lambda -\mu N.\hfill \end{array}$$

This gives ${\displaystyle \frac{dN}{dt}}+\mu N\le \lambda$.

We use the theory of functional differential equation, $N\left(t\right)\le {\displaystyle \frac{\lambda }{\mu }}(1-e^{-\mu t})+N^0{e}^{-\mu t}.$

Thus, we have

$$\begin{array}{c}\hfill \underset{t\to 0}{lim\; sup}N\left(t\right)={\displaystyle \frac{\lambda }{\mu }}.\end{array}$$

(3)

That means $\mathrm{\Psi }$ is an invariant set for system (1) and each solution that starts in $\mathrm{\Psi }$ always remains in $\mathrm{\Psi }$. □

4. Basic Reproduction Number, Equilibria, and Stability Analysis

This section contains the derivation of the basic reproduction number, the existence of steady states, and their stability analysis.

4.1. The Basic Reproduction Number

To calculate $R_0$, we use the next-generation method [42]. We consider two vectors $\mathcal{F}$ and $\mathcal{G}$ defined for system (1).

$$\mathcal{F}=\left(\begin{array}{c}{\displaystyle \frac{\beta S(I_S+I_A)}{N}}\\ 0\\ 0\end{array}\right),\mathcal{G}=\left(\begin{array}{c}(\kappa +\mu )E\\ -\rho \kappa E-b{I}_A+(\mu +{\sigma }_2)I_S\\ -(1-\rho )\kappa E+(b+{\sigma }_1+\mu )I_A\end{array}\right).$$

(4)

Next, we ensure that the entry-wise non-negative new infection matrix is F, and G is the non-singular Metzler matrix [43] that defines the transitions of influenza infection between the infection compartments and the matrices, which are given as follows:

$$F=\left(\begin{array}{ccc}0& {\displaystyle \frac{\beta S}{N}}& {\displaystyle \frac{\beta S}{N}}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),G=\left(\begin{array}{ccc}(\kappa +\mu )& 0& 0\\ -\rho \kappa & (\mu +{\sigma }_2)& -b\\ (\rho -1)\kappa & 0& (b+{\sigma }_1+\mu )\end{array}\right),$$

$$\begin{array}{ccc}\hfill G^{-1}& =& \left(\begin{array}{ccc}{\displaystyle \frac{1}{(\kappa +\mu )}}& 0& 0\\ {\displaystyle \frac{\kappa (\rho {\sigma }_1+\mu \rho +b)}{(\kappa +\mu )(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}}& {\displaystyle \frac{1}{(\mu +{\sigma }_2)}}& {\displaystyle \frac{b}{(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}}\\ {\displaystyle \frac{(1-\rho )\kappa }{(\kappa +\mu )(b+{\sigma }_1+\mu )}}& 0& {\displaystyle \frac{1}{(b+{\sigma }_1+\mu )}}\end{array}\right).\hfill \end{array}$$

Therefore, we have

$$\begin{array}{ccc}\hfill F{G}^{-1}& =& \left[\begin{array}{ccc}0& {\displaystyle \frac{\beta S}{N}}& {\displaystyle \frac{\beta S}{N}}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\times \left[\begin{array}{ccc}{\displaystyle \frac{1}{(\kappa +\mu )}}& 0& 0\\ {\displaystyle \frac{\kappa (\rho {\sigma }_1+\mu \rho +b)}{(\kappa +\mu )(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}}& {\displaystyle \frac{1}{(\mu +{\sigma }_2)}}& {\displaystyle \frac{b}{(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}}\\ {\displaystyle \frac{(1-\rho )\kappa }{(\kappa +\mu )(b+{\sigma }_1+\mu )}}& 0& {\displaystyle \frac{1}{(b+{\sigma }_1+\mu )}}\end{array}\right],\hfill \\ & =& \left[\begin{array}{ccc}{\displaystyle \frac{\beta S^0\kappa [(b+\mu +{\sigma }_2)+\rho ({\sigma }_1-{\sigma }_2)]}{N(\kappa +\mu )(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}}& {\displaystyle \frac{\beta S^0}{N(\mu +{\sigma }_2)}}& {\displaystyle \frac{\beta S^0(b+{\sigma }_2+\mu )}{N(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

(5)

With the help of the next-generation matrix, we have $R_0$, the largest eigenvalue of $F{G}^{-1}$ at $E_0(S^0,0,0,0,V^0,R^0)$

$$R_0={\displaystyle \frac{\beta S^0\kappa [b+\mu +\rho {\sigma }_1+{\sigma }_2\left(1-\rho \right)]}{N^0(\kappa +\mu )(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )}},$$

(6)

where $S^0={\displaystyle \frac{\lambda (\theta +\delta +\mu )}{(\mu +a)\left(\right(\theta +\delta +\mu )-a\theta )}}$ and $N^0={\displaystyle \frac{\lambda (\theta +\delta +\mu )+\lambda a+a\delta \lambda }{(\mu +a)\left(\right(\theta +\delta +\mu )-a\theta )}}.$ Clearly, $R_0>0.$

Theorem 3.

Model (1) has a basic reproduction number given in (6) at $E_0$. For $R_0>1$, system (1) switches to an endemic state.

4.2. Equilibrium Points

There are two equilibria of the system:

(i).

The infection-free equilibrium is

$$\begin{array}{c}\hfill E_0\left({\displaystyle \frac{\lambda (\theta +\delta +\mu )}{(\mu +a)\left(\right(\theta +\delta +\mu )-a\theta )}},0,0,0,{\displaystyle \frac{\lambda a}{(\mu +a)\left(\right(\theta +\delta +\mu )-a\theta )}},{\displaystyle \frac{a\delta \lambda }{\mu \left(\right(\theta +\delta +\mu )-a\theta )}}\right).\end{array}$$

The disease-free equilibrium $E_0$ is always feasible, as $\theta +\delta +\mu >a\theta$ always holds.

(ii).

The endemic equilibrium is $E_1(S^{*},E^{*},I_S^{*},I_A^{*},V^{*},R^{*}),$ where

$$\begin{array}{ccc}\hfill S^{*}& =& {\displaystyle \frac{(\kappa +\mu )({\sigma }_2+\mu )(b+\mu +{\sigma }_1)N}{[(b+{\sigma }_2)+\mu (1+\sigma )+\rho ({\sigma }_1-{\sigma }_2-\mu )]\beta \kappa }},\hfill \\ \hfill I_S^{*}& =& {\displaystyle \frac{(b+\rho {\sigma }_1+\rho \mu )\kappa E^{*}}{({\sigma }_2+\mu )(b+\mu +{\sigma }_1)}},I_A^{*}={\displaystyle \frac{(1-\rho )\kappa E^{*}}{(b+\mu +{\sigma }_1)}},\hfill \\ \hfill V^{*}& =& {\displaystyle \frac{(\kappa +\mu )({\sigma }_2+\mu )(b+\mu +{\sigma }_1)Na}{(\theta +\delta +\mu )[(b+{\sigma }_2)+\mu (1+\sigma )+\rho ({\sigma }_1-{\sigma }_2-\mu )]\beta \kappa }},\hfill \\ \hfill R^{*}& =& {\displaystyle \frac{\left[(\kappa +\mu ){(\mu +{\sigma }_2)}^2{(b+\mu +{\sigma }_1)}^2\delta Na\right]+\left[(1-\rho )({\sigma }_2+\mu )(b+\mu +{\sigma }_2+\rho {\sigma }_1-\rho {\sigma }_2)\beta {\sigma }_1\kappa E^{*}\right]}{(b+\mu +{\sigma }_2+\rho {\sigma }_1-\rho {\sigma }_2)(b+{\sigma }_1+\mu )({\sigma }_2+\mu )\beta }}\hfill \\ & & +{\displaystyle \frac{[b+\rho {\sigma }_1+\rho \mu )(b+\mu +{\sigma }_2+\rho {\sigma }_1-\rho {\sigma }_2)\beta ]\kappa {\sigma }_2{E}^{*}}{(b+\mu +{\sigma }_2+\rho {\sigma }_1-\rho {\sigma }_2)(b+{\sigma }_1+\mu )({\sigma }_2+\mu )\beta }},\hfill \end{array}$$

and $E^{*}$ is the positive root of the following equation:

$$\lambda N-\beta S^{*}(I_S^{*}+I_A^{*})-(\mu +a)S^{*}N+\theta V^{*}N=0.$$

(7)

Remark 1.

The calculation of determining $E^{*}$ from (7) is completed. Thus, we determine the positive value of $E^{*}$ using numerical calculations. Detailed numerical simulations show that the endemic equilibrium $E_1$ exists when the basic reproduction number $R_0>1$.

4.3. Local Stability

Theorem 4.

The system is stable at $E_0$ when $R_0<1$ and unstable when $R_0>1$.

Proof. 

At $E_0$, we have

$$\begin{array}{ccc}\hfill J_{E_0}& =\left[a_{ij}\right]=& \left(\begin{array}{cccccc}-(\mu +\theta )& 0& -{\displaystyle \frac{\beta S^0}{N}}& -{\displaystyle \frac{\beta S^0}{N}}& \theta & 0\\ 0& -(\kappa +\mu )& {\displaystyle \frac{\beta S^0}{N}}& {\displaystyle \frac{\beta S^0}{N}}& 0& 0\\ 0& \rho \kappa & -(\mu +{\sigma }_2)& b& 0& 0\\ 0& (1-\rho )\kappa & 0& -(b+{\sigma }_1+\mu )& 0& 0\\ a& 0& 0& 0& -(\theta +\delta +\mu )& 0\\ 0& 0& {\sigma }_1& {\sigma }_2& \delta & -\mu \end{array}\right).\hfill \end{array}$$

Here the eigenvalues are $\left[\xi =-\mu \right]$ and the other five eigenvalues are calculated from

$$\begin{array}{ccc}\hfill \left|\begin{array}{ccccc}-(\mu +\theta )-\xi & 0& -{\displaystyle \frac{\beta S^0}{N}}& -{\displaystyle \frac{\beta S^0}{N}}& \theta \\ 0& -(\kappa +\mu )-\xi & {\displaystyle \frac{\beta S^0}{N}}& {\displaystyle \frac{\beta S^0}{N}}& 0\\ 0& \rho \kappa & -(\mu +{\sigma }_2)-\xi & b& 0\\ 0& (1-\rho )\kappa & 0& -(b+{\sigma }_1+\mu )-\xi & 0\\ a& 0& 0& 0& -(\theta +\delta +\mu )-\xi \end{array}\right|& =& 0,\hfill \end{array}$$

and the characteristic equation at $E_0$ is

$$\begin{array}{c}\hfill \left\{x^2+B_{1}x+B_2\right\}\times \left\{x^3+B_3{x}^2+B_{4}x+B_5\right\}=0,\end{array}$$

(8)

where

$$\begin{array}{ccc}\hfill B_1& =& -a_{11}-a_{55},\hfill \\ \hfill B_2& =& a_{11}{a}_{55}-a_{15}{a}_{51},\hfill \\ \hfill B_3& =& -a_{22}-a_{33}-a_{44},\hfill \\ \hfill B_4& =& -a_{23}{a}_{32}+a_{22}{a}_{33}-a_{24}{a}_{42}+a_{22}{a}_{44}+a_{33}{a}_{44},\hfill \\ \hfill B_5& =& a_{24}{a}_{33}{a}_{42}-a_{23}{a}_{34}{a}_{42}+a_{23}{a}_{32}{a}_{44}-a_{22}{a}_{33}{a}_{44},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill a_{22}& =& -(\kappa +\mu ),a_{33}=-(\mu +{\sigma }_2),a_{44}=-(b+{\sigma }_1+\mu ),\hfill \\ \hfill a_{24}& =& a_{23}={\displaystyle \frac{\beta S^0}{N^0}},a_{42}=(1-\rho )\kappa ,a_{32}=\rho \kappa ,a_{34}=b,a_{51}=a,a_{55}=-(\theta +\delta +\mu ).\hfill \end{array}$$

If $B_i>0,i=1,2,...,5$, then all the roots must be negative. For $B_2>0$, we obtain the threshold condition for the stability of $E_0$. We get the stability condition as $\beta S^0\kappa [b+\mu +\rho {\sigma }_1+{\sigma }_2\left(1-\rho \right)]-N^0(\kappa +\mu )(\mu +{\sigma }_2)(b+{\sigma }_1+\mu )<0$, which implies that $R_0<1$, resulting in the removal of infection. □

4.4. Local Equilibrium Stability of the Endemic System

At $E^{*}$, we have the charismatic equation

$$\begin{array}{ccc}\hfill |J_{E^{*}}-\xi |& =& \left(\begin{array}{ccccc}{Y}_{11}-\xi & 0& Y_{13}& Y_{14}& Y_{15}\\ Y_{21}& Y_{22}-\xi & Y_{23}& Y_{24}& 0\\ 0& Y_{32}& Y_{33}-\xi & Y_{34}& 0\\ 0& Y_{42}& 0& Y_{44}-\xi & 0\\ Y_{51}& 0& 0& 0& Y_{55}-\xi \end{array}\right),\hfill \end{array}$$

(9)

where,

$$\begin{array}{ccc}\hfill Y_{11}& =& -{\displaystyle \frac{\beta (I_S+I_A)}{N}}-(\mu +\theta ),Y_{13}=-{\displaystyle \frac{\beta S}{\left(N\right)}},Y_{14}=-{\displaystyle \frac{\beta S}{\left(N\right)}},\hfill \\ \hfill Y_{15}& =& \theta ,Y_{21}={\displaystyle \frac{\beta (I_S+I_A)}{N}},Y_{22}=-(\kappa +\mu ),Y_{23}={\displaystyle \frac{\beta S}{N}},\hfill \\ \hfill Y_{24}& =& {\displaystyle \frac{\beta S}{N}},Y_{32}=\rho \kappa ,Y_{33}=-(\mu +{\sigma }_2),Y_{34}=b,\hfill \\ \hfill Y_{42}& =& (1-\rho )\kappa ,Y_{44}=-(b+{\sigma }_1+\mu ),Y_{51}=a,Y_{55}=-(\theta +\delta +\mu ).\hfill \end{array}$$

At $E_1$, the characteristic equation is

$$\begin{array}{ccc}\hfill {\xi }^5+{\mathrm{\Phi }}_1{\xi }^4+{\mathrm{\Phi }}_2{\xi }^3+{\mathrm{\Phi }}_3{\xi }^2+{\mathrm{\Phi }}_4\xi +{\mathrm{\Phi }}_5\xi & =& 0,\hfill \end{array}$$

(10)

where

$$\begin{array}{ccc}\hfill {\mathrm{\Phi }}_1& =& -(Y_{11}+Y_{22}+Y_{33}+Y_{44}+Y_{55}),\hfill \\ \hfill {\mathrm{\Phi }}_2& =& -Y_{23}{Y}_{32}+Y_{22}{Y}_{33}-Y_{24}{Y}_{42}+Y_{22}{Y}_{44}+Y_{33}{Y}_{44}-Y_{15}{Y}_{51}+Y_{22}{Y}_{55}+Y_{33}{Y}_{55}\hfill \\ & & +Y_{44}{Y}_{55}+Y_{11}(Y_{22}+Y_{33}+Y_{44}+Y_{55}),\hfill \\ \hfill {\mathrm{\Phi }}_3& =& -Y_{11}(Y_{13}{Y}_{32}-Y_{23}{Y}_{32}+Y_{22}{Y}_{33}+Y_{14}{Y}_{42}-Y_{24}{Y}_{42}+Y_{22}{Y}_{44}+Y_{33}{Y}_{44}\hfill \\ & & +Y_{22}{Y}_{55}+Y_{33}{Y}_{55}+Y_{44}{Y}_{55})-(Y_{23}{Y}_{34}{Y}_{42}-Y_{23}{Y}_{32}{Y}_{44}-(Y_{23}{Y}_{34}{Y}_{42}\hfill \\ & & -Y_{23}{Y}_{32}{Y}_{44}+Y_{22}{Y}_{33}{Y}_{44}-Y_{15}{Y}_{22}{Y}_{51}-Y_{15}{Y}_{33}{Y}_{51}-Y_{15}{Y}_{44}{Y}_{51}\hfill \\ & & -Y_{23}{Y}_{32}{Y}_{55}+Y_{22}{Y}_{33}{Y}_{55}+Y_{22}{Y}_{44}{Y}_{55}+Y_{33}{Y}_{44}{Y}_{55}-Y_{24}{Y}_{42}(Y_{33}+Y_{55})),\hfill \\ \hfill {\mathrm{\Phi }}_4& =& Y_{15}(Y_{23}{Y}_{32}+Y_{24}{Y}_{42}-Y_{33}{Y}_{44}-Y_{22}(Y_{33}+Y_{44}))Y_{51}+(-Y_{24}{Y}_{33}{Y}_{42}+Y_{23}{Y}_{34}{Y}_{42}\hfill \\ & & -Y_{23}{Y}_{32}{Y}_{44}+Y_{22}{Y}_{33}{Y}_{44})Y_{55}+Y_{11}(-Y_{13}{Y}_{34}{Y}_{42}+Y_{23}{Y}_{34}{Y}_{42}+Y_{13}{Y}_{32}{Y}_{44}\hfill \\ & & -Y_{23}{Y}_{32}{Y}_{44}+Y_{22}{Y}_{33}{Y}_{44}+Y_{13}{Y}_{32}{Y}_{55}-Y_{23}{Y}_{32}{Y}_{55}+Y_{22}{Y}_{33}{Y}_{55}\hfill \\ & & +Y_{22}{Y}_{44}{Y}_{55}+Y_{33}{Y}_{44}{Y}_{55}+Y_{14}{Y}_{42}(Y_{33}+Y_{55})-Y_{24}{Y}_{42}(Y_{33}+Y_{55})),\hfill \\ \hfill {\mathrm{\Phi }}_5& =& -Y_{51}{Y}_{15}(Y_{24}{Y}_{33}{Y}_{42}-Y_{23}{Y}_{34}{Y}_{42}+Y_{23}{Y}_{32}{Y}_{44}-Y_{22}{Y}_{33}{Y}_{44})-Y_{11}(Y_{14}{Y}_{33}{Y}_{42}\hfill \\ & & -Y_{24}{Y}_{33}{Y}_{42}-Y_{13}{Y}_{34}{Y}_{42}+Y_{23}{Y}_{34}{Y}_{42}+Y_{13}{Y}_{32}{Y}_{44}-Y_{23}{Y}_{32}{Y}_{44}+Y_{22}{Y}_{33}{Y}_{44})Y_{55}.\hfill \end{array}$$

According to Routh–Hurwitz conditions, the characteristic Equation (10) has roots with negative real parts if the conditions

$$\begin{array}{ccc}& i.& {\mathrm{\Phi }}_5>0,{\mathrm{\Phi }}_1{\mathrm{\Phi }}_2-{\mathrm{\Phi }}_3>0,\hfill \\ & ii.& {\mathrm{\Phi }}_3({\mathrm{\Phi }}_1{\mathrm{\Phi }}_2-{\mathrm{\Phi }}_3)-{\mathrm{\Phi }}_1({\mathrm{\Phi }}_1{\mathrm{\Phi }}_4-{\mathrm{\Phi }}_5)>0,\hfill \\ & iii.& {\mathrm{\Phi }}_4\left[{\mathrm{\Phi }}_3({\mathrm{\Phi }}_1{\mathrm{\Phi }}_2-{\mathrm{\Phi }}_3)-{\mathrm{\Phi }}_1({\mathrm{\Phi }}_1{\mathrm{\Phi }}_4-{\mathrm{\Phi }}_5)\right]\hfill \\ & & -{\mathrm{\Phi }}_5\left[{\mathrm{\Phi }}_2({\mathrm{\Phi }}_1{\mathrm{\Phi }}_2-{\mathrm{\Phi }}_3)-({\mathrm{\Phi }}_1{\mathrm{\Phi }}_4-{\mathrm{\Phi }}_5)\right]>0,\hfill \end{array}$$

are satisfied.

5. Optimal Control Problem Formulation

In this section, we have considered two controls, namely the awareness through governance and self-learning and vaccination to control the disease. Here, our main focus is to minimize the number of infected and exposed people. For this purpose, we have introduced the following control strategies $(u_v,u_i)$:

• The control function $u_v\left(t\right)$ represents the role of vaccination.
• The control $u_i\left(t\right)$ plays the role in controlling the infected class.

Thus, our state system of the optimal control problem becomes

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \lambda -{\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\mu +u_{v}a)S+\theta V,\hfill \\ \hfill {\displaystyle \frac{dE}{dt}}& =& {\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(u_i\kappa +\mu )E,\hfill \\ \hfill {\displaystyle \frac{d{I}_S}{dt}}& =& u_i\rho \kappa E+b{I}_A-({\sigma }_2+\mu )I_S,\hfill \\ \hfill {\displaystyle \frac{d{I}_A}{dt}}& =& u_i(1-\rho )\kappa E-(b+{\sigma }_1+\mu )I_A,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& u_{v}aS-(\theta +\delta +\mu )V,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =& \delta V+{\sigma }_1{I}_A+{\sigma }_2{I}_S-\mu R.\hfill \end{array}$$

(11)

We propose the objective functional as

$$\begin{array}{ccc}\hfill J(u_v,u_i)& =& {\int }_0^T[P_{1}E+P_2{I}_S+P_3{I}_A+Q_1{u}_v^2+Q_2{u}_i^2]dt.\hfill \end{array}$$

(12)

In the objective function, ${\int }_0^T{P}_{1}Edt$ represents the cost induced by the exposed class, ${\int }_0^T{P}_2{I}_{S}dt$ represents the cost induced by the infected class, ${\int }_0^T{P}_3{I}_{A}dt$ is the cost induced by any phase of the infected class, ${\int }_0^T{Q}_1{u}_v^{2}dt$ is the cost induced by the optimal stage, and ${\int }_0^T{Q}_2{u}_i^{2}dt$ is the cost at the stage.

Here the optimal controls are $u_v^{*},u_i^{*}\in \mathcal{U}$, where

$$\begin{array}{ccc}\hfill J(u_v^{*},u_i^{*})& =& minJ(u_v,u_i),\hfill \end{array}$$

and $\mathcal{U}=\{(u_v,u_i)$|$u_v,u_i$ are Lebesgue measurable on $[0,T]$ with $0\le u_v\le 1,0\le u_i\le 1$}.

5.1. Existence of an Optimal Control

We will determine the existence of the optimal control model (11).

Theorem 5.

$\mathcal{U}=(u_v,u_i)$ and $\mathcal{M}=(S,E,I_S,I_A,V,R)$ exist for the state initial value problem (11), and the functional (12) attains its minimum $J(u_v,u_i)$ over $\mathcal{U}$.

Proof. 

To verify the existence condition, we use the study of Fleming–Rishel’s theorem under certain conditions:

i.

Systems (11) and (12) with the control fraction V admit at least one non-empty solution set.

ii.

The state system should be a linear function dependent on time and time state variables.

iii.

The integral J in (12) is convex on $\mathcal{U}$ and $\alpha >1$, and there exist positive constants $\chi$, $c_1$, and $c_2$ such that

$$\begin{array}{ccc}\hfill J(t,\mathcal{M},u)& \ge & c_1{|u_v,u_i|}^{\chi }-c_2.\hfill \end{array}$$

For every admissible control $\mathcal{U}$, there is a unique solution satisfying the Lipsehiiz condition. Also, we know that the total population is bounded, and it belongs to ${\displaystyle \frac{\delta }{\mu }}$. Also, all the state variables are bounded. Thus, we have a partial derivation of the state system (11) with respect to the state variable system bondedness, which satisfies condition $\left(i\right)$. We can also verify condition $\left(ii\right)$ by checking the linear dependence of the state equation and control $(u_v,u_i)$ for verification of the function condition (iii). We note that linear and quarantine functions are convex [44]. To prove the bondedness on J, we have $Q_1{u}_v^2\le Q_1$, $Q_2{u}_i^2\le Q_2$, $0\le u_v\le 1$, and $0\le u_i\le 1$. As such,

$$\begin{array}{ccc}\hfill J(t,\mathcal{M},u)& =& P_{1}E+P_2{I}_S+P_3{I}_A+Q_1{u}_v^2+Q_2{u}_i^2\hfill \\ & \ge & (Q_1{u}_v^2+Q_2{u}_1^2-u_1^2),\hfill \end{array}$$

(13)

from (13), we get

$$\begin{array}{ccc}\hfill J(t,\mathcal{M},u)& \ge & min(Q_1,Q_2)(u_v^2+u_i^2)-u_i\hfill \\ & =& min\left(Q_1{Q}_2\right)\left|\right|u_v,u_i{\left|\right|}^2-u_i.\hfill \end{array}$$

(14)

Thus,

$$\begin{array}{ccc}\hfill J(t,\mathcal{M},u)& \ge & c_1\left|\right|(u_v,u_i){\left|\right|}^{\chi }-c_2,\hfill \end{array}$$

where $c_1=min(u_v,u_i),c_2=u_i$, and $\chi =2$. □

5.2. Characterization of the Optimal Control Pair

Let $u^{*}\left(t\right)$ represent the amount of medicine as the control input. The essential criteria for an optimum control and related states to meet Pontryagin’s maximum principle are represented by the cost function (12) for system (11). We employ Pontryagin’s maximal principle [45] to choose the best controls for $u_v^{*}\left(t\right)$ and $u_i^{*}\left(t\right)$. By using this technique, we create the Hamiltonian function L with regard to $(u_v\left(t\right),u_i\left(t\right))$, which transforms system (11) along with (12).

According to the control system provided by (11), we determine the optimal control pair for the optimality of the system. To define the Hamiltonian ($\mathcal{H}$) associated with the optimal control problem, the Lagrangian ($\mathcal{L}$) as generated below is also included:

$$\begin{array}{ccc}\hfill \mathcal{L}(u_v,u_i)& =& P_{1}E+P_2{I}_S+P_3{I}_A+Q_1{u}_v^2+Q_2{u}_i^2.\hfill \end{array}$$

(15)

The Hamiltonian is built by combining the state variables and adjoint variables as follows:

$$\begin{array}{ccc}\hfill \mathcal{H}& =& \mathcal{L}(u_v,u_i)\hfill \\ & & +{\xi }_1\left[\lambda -{\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\mu +u_{v}a)S+\theta V\right]\hfill \\ & & +{\xi }_2\left[{\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(u_i\kappa +\mu )E\right]\hfill \\ & & +{\xi }_3\left[u_i\rho \kappa E+b{I}_A-({\sigma }_2+\mu )I_S\right]\hfill \\ & & +{\xi }_4\left[u_i(1-\rho )\kappa E-(b+{\sigma }_1+\mu )I_A\right]\hfill \\ & & +{\xi }_5\left[u_{v}aS-(\theta +\delta +\mu )V\right]\hfill \\ & & +{\xi }_6\left[\delta V+{\sigma }_1{I}_A+{\sigma }_2{I}_S-\mu R\right].\hfill \end{array}$$

(16)

Here, ${\xi }_i,i=1,2,3,4,5,6$ are adjoint variables. According to the maximum principle, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial S}}& =& -{\xi }_1\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +u_{v}a)\right]+{\xi }_2\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}\right]+{\xi }_5{u}_{v}a,\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial E}}& =& P_1-{\xi }_2(u_i\kappa +\mu )+{\xi }_3{u}_i\rho \kappa +{\xi }_4{u}_i\kappa (1-\rho ),\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial I_S}}& =& P_2-{\xi }_1\left[{\displaystyle \frac{\beta S}{N}}\right]+{\xi }_2\left[{\displaystyle \frac{\beta S}{N}}\right]-{\xi }_3({\sigma }_2+\mu )+{\xi }_6{\sigma }_2,\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial I_A}}& =& P_3-{\xi }_1\left[{\displaystyle \frac{\beta S}{N}}\right]+{\xi }_2\left[{\displaystyle \frac{\beta S}{N}}\right]+{\xi }_{3}b-{\xi }_4\left(b+{\sigma }_1+\mu \right)+{\xi }_6{\sigma }_1,\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial V}}& =& {\xi }_1\theta -{\xi }_5(\theta +\delta +\mu )+{\xi }_6\delta ,\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial R}}& =& -{\xi }_6\mu ,\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial u_v}}& =& 2Q_1{u}_v-aS({\xi }_1-{\xi }_5),\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial u_i}}& =& 2Q_2{u}_i-{\xi }_2\kappa E+{\xi }_3\rho \kappa E+{\xi }_4\kappa E(1-\rho ).\hfill \end{array}$$

(17)

The adjoint system to be estimated for the control input $(u_v\left(t\right),u_i\left(t\right))$ associated with the model state variables $S,E,I_S,I_A,V,$ and R is represented as $(u_v\left(t\right),u_i\left(t\right))$ for the control input, along with $S,E,I_S,I_A,V,R$, which is represented as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{\xi }_1}{dt}}& =& -\left[-{\xi }_1\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +u_{v}a)\right]+{\xi }_2\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}\right]+{\xi }_5{u}_{v}a\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_2}{dt}}& =& -\left[P_1-{\xi }_2(u_i\kappa +\mu )+{\xi }_3{u}_i\rho \kappa +{\xi }_4\kappa u_i(1-\rho )\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_3}{dt}}& =& -\left[P_2-{\xi }_1\left({\displaystyle \frac{\beta S}{N}}\right)+{\xi }_2\left({\displaystyle \frac{\beta S}{N}}\right)-{\xi }_3({\sigma }_2+\mu )+{\xi }_6{\sigma }_2\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_4}{dt}}& =& -\left[P_3-{\xi }_1\left({\displaystyle \frac{\beta S}{N}}\right)+{\xi }_2\left({\displaystyle \frac{\beta S}{N}}\right)+{\xi }_{3}b-{\xi }_4\left(b+{\sigma }_1+\mu \right)+{\xi }_6{\sigma }_1\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_5}{dt}}& =& -\left[{\xi }_1\theta -{\xi }_5(\theta +\delta +\mu )+{\xi }_6\delta \right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_6}{dt}}& =& {\xi }_6\mu .\hfill \end{array}$$

(18)

Here, the transversality conditions are ${\xi }_i\left(T\right)=0,i=1,2,...6.$ Pontryagin’s maximum principle [45] has allowed us to ascertain

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial u^{*}\left(t\right)}}=0.\end{array}$$

(19)

Then,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial u_v}}& =& 2Q_1{u}_v-aS({\xi }_1-{\xi }_5),\hfill \\ \hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial u_i}}& =& 2Q_2{u}_i-{\xi }_2\kappa E+{\xi }_3\rho \kappa E+{\xi }_4\kappa E(1-\rho ).\hfill \end{array}$$

(20)

Solving (20) for $u_v^{*}\left(t\right)$ and $u_i^{*}\left(t\right)$,

$$\begin{array}{ccc}\hfill u_v^{*}\left(t\right)& =& {\displaystyle \frac{aS({\xi }_1-{\xi }_5)}{2Q_1}},\hfill \\ \hfill u_i^{*}\left(t\right)& =& {\displaystyle \frac{\kappa E[{\xi }_2-{\xi }_3\rho -{\xi }_4(1-\rho )]}{2Q_2}}.\hfill \end{array}$$

(21)

The standard form is

$$\begin{array}{ccc}\hfill u_v^{*}\left(t\right)& =& \left\{\begin{array}{cc}0,\hfill & {\displaystyle \frac{aS({\xi }_1-{\xi }_5)}{2Q_1}}<0,\hfill \\ {\displaystyle \frac{aS({\xi }_1-{\xi }_5)}{2Q_1}},\hfill & 0<{\displaystyle \frac{aS({\xi }_1-{\xi }_5)}{2Q_1}}<1,\hfill \\ 1,\hfill & {\displaystyle \frac{aS({\xi }_1-{\xi }_5)}{2Q_1}}>1,\hfill \end{array}\right.\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill u_i^{*}\left(t\right)& =& \left\{\begin{array}{cc}0,\hfill & {\displaystyle \frac{\kappa E[{\xi }_2-{\xi }_3\rho -{\xi }_4(1-\rho )]}{2Q_2}}<0,\hfill \\ {\displaystyle \frac{\kappa E[{\xi }_2-{\xi }_3\rho -{\xi }_4(1-\rho )]}{2Q_2}},\hfill & 0<{\displaystyle \frac{\kappa E[{\xi }_2-{\xi }_3\rho -{\xi }_4(1-\rho )]}{2Q_2}}<1,\hfill \\ 1,\hfill & {\displaystyle \frac{\kappa E[{\xi }_2-{\xi }_3\rho -{\xi }_4(1-\rho )]}{2Q_2}}>1.\hfill \end{array}\right.\hfill \end{array}$$

(23)

Thus, we can write $u_v^{*}\left(t\right)$ as

$$\begin{array}{ccc}\hfill u_v^{*}\left(t\right)& =& max\left(min\left(1,{\displaystyle \frac{aS({\xi }_1-{\xi }_5)}{2Q_1}}\right),0\right).\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill u_i^{*}\left(t\right)& =& max\left(min\left(1,{\displaystyle \frac{\kappa E[{\xi }_2-{\xi }_3\rho -{\xi }_4(1-\rho )]}{2Q_2}}\right),0\right).\hfill \end{array}$$

(25)

Thus, the optimal system becomes

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \lambda -{\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(\mu +u_{v}a)S+\theta V,\hfill \\ \hfill {\displaystyle \frac{dE}{dt}}& =& {\displaystyle \frac{\beta S(I_S+I_A)}{N}}-(u_i\kappa +\mu )E,\hfill \\ \hfill {\displaystyle \frac{d{I}_S}{dt}}& =& u_i\rho \kappa E+b{I}_A-({\sigma }_2+\mu )I_S,\hfill \\ \hfill {\displaystyle \frac{d{I}_A}{dt}}& =& u_i(1-\rho )\kappa E-(b+{\sigma }_1+\mu )I_A,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& u_{v}aS-(\theta +\delta +\mu )V,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =& \delta V+{\sigma }_1{I}_A+{\sigma }_2{I}_S-\mu R,\hfill \\ \hfill {\displaystyle \frac{d{\xi }_1}{dt}}& =& -\left[-{\xi }_1\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}+(\mu +u_{v}a)\right]+{\xi }_2\left[{\displaystyle \frac{\beta (I_S+I_A)}{N}}\right]+{\xi }_5{u}_{v}a\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_2}{dt}}& =& -\left[P_1-{\xi }_2(u_i\kappa +\mu )+{\xi }_3{u}_i\rho \kappa +{\xi }_4{u}_i\kappa (1-\rho )\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_3}{dt}}& =& -\left[P_2-{\xi }_1\left({\displaystyle \frac{\beta S}{N}}\right)+{\xi }_2\left({\displaystyle \frac{\beta S}{N}}\right)-{\xi }_3({\sigma }_2+\mu )+{\xi }_6{\sigma }_2\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_4}{dt}}& =& -\left[P_3-{\xi }_1\left({\displaystyle \frac{\beta S}{N}}\right)+{\xi }_2\left({\displaystyle \frac{\beta S}{N}}\right)+{\xi }_{3}b-{\xi }_4\left(b+{\sigma }_1+\mu \right)+{\xi }_6{\sigma }_1\right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_5}{dt}}& =& -\left[{\xi }_1\theta -{\xi }_5(\theta +\delta +\mu )+{\xi }_6\delta \right],\hfill \\ \hfill {\displaystyle \frac{d{\xi }_6}{dt}}& =& {\xi }_6\mu ,\hfill \end{array}$$

(26)

along with the transversality conditions ${\xi }_i\left(T\right)=0,i=1,2,...,6$.

6. Parameter Sensitivity Analysis

Sensitivity and identifiability studies were conducted by computing the derivative of the clinical score $\mathrm{\Omega }={[S,E,I_S,I_A,V,R]}^T$ with respect to the vector parameter at each data point. The sensitivities are dimensionless because of their scaling with respect to the values of the variables and parameters. The following is a representation of system (1) sensitivity functions with respect to any parameter p:

$$\begin{array}{ccc}\hfill {\mathrm{\Omega }}_i,\omega & =& {\displaystyle \frac{\partial {\mathrm{\Omega }}_i\left(t\right)}{\partial \omega }},i=1,2,...6\hfill \end{array}$$

(27)

Due to these sensitivities, the least important parameter for model output may be found and changed without being used for calibration. The sensitivity functions of DDEs can be found using a variety of techniques [20,46]. However, to determine the sensitivity functions of system (1), we shall employ the so-called direct technique because of its simplicity.

7. Numerical Simulations

In this section, we analyze our model’s behavior and the optimal system dynamics graphically using Matlab version R2016a simulations. All the parameters are taken from Table 2, and some parameters are varied to study their impacts on the system dynamics.

As analyzed in Section 6, we plot the sensitivity of model parameters in Figure 2a,b and also in

Table 3. Sensitivity indices of the parameters.

Parameter	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\mu }$	$\mathit{\kappa }$	${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	a
Sensitivity Index	1	1	−0.86	0.5	−0.48	−0.21	−0.0132
Parameter	$\rho$	$\theta$	$\delta$	b
Sensitivity Index	0.000001	0.00004	−0.00007	−0.000004

. All parameters are sensitive to the study of the H1N1 dynamics. Some parameters, namely $\lambda ,\beta ,\kappa$, etc., are positively sensitive, and some parameters are negatively sensitive. Negatively sensitive parameters can reduce the value of $R_0$; thus, these parameters can minimize the infection.

Figure 3a–f show that when $R_0<1$, the system attains its disease-free state. Also, when $R_0>1$, the system moves to a stable endemic steady state. Figure 4a–c shows the forward bifurcation at the basic reproduction $R_0$. We varied the infection rate $\beta$, and it is observed that when $R_0<1$, the system becomes disease-free, and it shows its stability around $E_0$. It switches to an endemic state when $R_0>1$ and the system attains stability around $E^{*}$.

Figure 5 shows the stability region for the disease-free state and endemic state with respect to parameters $\beta$ and $\kappa$. From this figure, we reveal that with the increasing values of $\beta$ and $\kappa$, the system switches its stability from $E_0$ to $E^{*}$, that is, from a disease-free state to an endemic state. Nonlinear stability of the endemic system is shown in Figure 6. All phase trajectories converge to the same endemic steady state $E^{*}$, showing the global stability of $E^{*}$.

The impact of $\kappa$ is analyzed using Figure 7a–f by plotting the equilibrium values of the model population with respect to $\kappa$. Similarly, in Figure 8, the impact of vaccination is shown. Infection is reduced as the vaccinated population increases.

Also, we consider the control system with control inputs $u_v\left(t\right)$ and $u_i\left(t\right)$ for systems (11) and (12). The time-dependent control functions, along with the state system and adjoint system, are analyzed numerically. Here, the optimal control problem is bounded by $[0,T]$. Here, we consider $T=60$, at which the treatment is stopped. The optimal system is plotted in Figure 9. This figure shows the impacts of controls on the disease. In Figure 10a,b, the optimal trajectories show the effective role of control strategies. Also, we have clearly observed that a low weight factor gives a better result in increasing the susceptible population along with class improvement.

8. Discussion and Conclusions

The H1N1 virus is a major contributor to influenza infections, primarily affecting the upper respiratory tract. Severe manifestations are typically observed in individuals with underlying chronic conditions. To better understand and mitigate the spread of this virus, a mathematical model has been developed and extended to incorporate time-dependent control inputs. The model framework includes the establishment of existence conditions and a thorough stability analysis based on the basic reproduction number $R_0$.

A comprehensive sensitivity analysis reveals that all model parameters significantly influence the dynamics of H1N1 transmission. Notably, parameters such as the transmission rate ($\beta$), recruitment rate ($\lambda$), and contact rate ($\kappa$) exhibit positive sensitivity, indicating that increases in these parameters tend to amplify the spread of infection (Figure 2).

We illustrated the critical role of the basic reproduction number $R_0$ in determining system behavior (Figure 3). When $R_0<1$, the model stabilizes at a disease-free equilibrium ($E_0$), whereas $R_0>1$ leads to a stable endemic equilibrium ($E^{*}$). This transition is further substantiated by the forward bifurcation (Figure 4), which highlights a distinct shift in system dynamics as $R_0$ crosses the threshold value of unity. We have noticed that when the endemic system is feasible, it is globally stable (Figure 6). The impact of the contact rate $\kappa$ is examined, which shows that higher values of $\kappa$ correspond to increased equilibrium infection levels (Figure 7). This finding emphasizes the importance of managing contact rates to control disease spread. Additionally, the positive effect of vaccination is studied, where an increase in vaccinated individuals leads to a marked reduction in infection prevalence (Figure 8).

For the analysis of the control model, Pontryagin’s maximum principle has been employed. The corresponding Hamiltonian function has been constructed, and the adjoint equations, along with the optimal control variables $u_1^{*}$ and $u_2^{*}$, have been derived. To enhance disease mitigation, the model incorporates a control system governed by time-dependent inputs $u_v\left(t\right)$ (vaccination) and $u_i\left(t\right)$ (treatment), analyzed over a finite time horizon $T=60$. We have seen that the optimal control strategies are effective. The simulations reveal that control interventions significantly reduce infection levels. Moreover, a lower weight factor in the cost function yields improved outcomes, including an increase in the susceptible population and enhanced class dynamics, suggesting that more aggressive control measures are beneficial (Figure 9 and Figure 10).

In conclusion, this study presents a rigorous analysis of the H1N1 transmission model, integrating sensitivity analysis, bifurcation theory, and optimal control strategies. The basic reproduction number $R_0$ serves as a pivotal threshold for system stability. Parameters such as $\beta$, $\lambda$, and $\kappa$ play critical roles in shaping disease dynamics. Vaccination and contact reduction are effective in curbing the spread. Optimal control strategies, particularly those with lower cost weights, substantially improve population health outcomes. Thus, the results underscore the importance of targeted interventions and strategic parameter management in controlling H1N1 infection. The integration of mathematical modeling and optimal control provides a valuable framework for public health decision-making and epidemic response planning.