Mathematical Model Analysis of Substance Abuse and Hepatitis B Co-Existence with Control Interventions

Abstract

Substance abuse addictions and hepatitis B infections are two major public health problems facing humanity globally, especially in areas where the two problems co-exist. A mathematical model was used in this work to study the co-dynamics of substance abuse addictions and hepatitis B infections and investigate their possible control strategies. The mathematical features of the model, such as the disease-free equilibrium, endemic equilibrium, and basic reproduction number, were computed. The stability analysis of the disease-free equilibrium and endemic equilibrium was conducted analytically. The impact of multiple control measures, including public enlightenment, rehabilitation of individuals with substance abuse disorders, treatment of persons infected with hepatitis B, and vaccination of susceptible individuals, was examined numerically. The study reveals how co-existence fundamentally alters system behavior and control effectiveness and offers new insights for designing effective control management strategies.

1. Introduction

Hepatitis B is a liver infection caused by the Hepatitis B Virus (HBV) [1]. It is a major public health problem, with approximately 254 million people living with chronic hepatitis B infection as of 2022, and about 1.2 million new infections each year [1]. In 2022, hepatitis B resulted in an estimated 1.1 million deaths, mostly from liver cancer [1]. Transmission of hepatitis B is mainly through contact with infected blood, semen, or other body fluids, shared needles, or unprotected sex but is preventable with a vaccine [1]. Symptoms include fatigue, fever, jaundice, nausea, vomiting, dark urine, and abdominal pain, though many people have no symptoms, making testing crucial. There is no cure for chronic HBV, but antiviral medications can manage the virus, reduce liver damage, and prevent serious complications. Management of the infection include avoiding alcohol, getting rest, and a mild diet, which can help during acute infection. Public enlightenment on the dangers of the disease is very crucial as a preventive strategy. This agrees with the global health sector strategies whereby the World Heath Organization (WHO) organizes annual World Hepatitis Day campaigns every 28 July to increase awareness and understanding of viral hepatitis [1].

Substance abuse can be understood as the harmful or hazardous use of psychoactive substances, including alcohol and illicit drugs [2]. This is a major public health problem that affects a significant number of individuals globally. For instance, approximately 15.3 million persons have drug use disorders, and the harmful use of alcohol results in 3.3 million deaths each year [3]. The number of cases of substance abuse addiction is currently increasing. For instance, an increase in prevalence and use of Cannabis (the most widely used illicit substance in the African region) is being reported in West and Central Africa with rates between 5.2% and 13.5% [3]. Substance abuse addictions can be properly managed in a rehabilitation facility, while public enlightenment to increase awareness of the dangers of substance abuse addiction can also be helpful as a preventive strategy. There is a strong link between substance abuse and criminal activities such as theft, violence, rape, unprotected sex, and drug-related offenses like trafficking. Individuals with substance use disorders are more likely to be involved in violence, rape, and unprotected sexual activity, which can lead to infections like hepatitis B. This could explain a possible connection between substance abuse addiction and hepatitis B infections in a population where both infections co-exist. This study used a mathematical model to investigate the dynamics and control interventions for the co-existence of substance abuse addictions and hepatitis B infections in a population where both infections are endemic. The study is expected to aid policy-makers and health workers in the proper management of substance abuse addiction and hepatitis B infections in endemic areas.

Mathematical models are very important tools that have been successfully used to study several real world problems such as infectious disease, food insecurity, and climate change [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. A literature review of some of the important research findings on the dynamics and control interventions of substance abuse and hepatitis B infections using mathematical model is presented here. For instance, the potential impact of preventive measures and punishment control strategies on the dynamics of drug addiction, with particular emphasis on how to reduce the addicted population and promote recovery, was explored using mathematical modeling [22]. The dynamics and control of substance abuse in women have been investigated using mathematical modeling [23]. It was discovered that social media awareness is significantly more effective than the fixed control in optimizing drug addiction among women [23]. The dynamics of alcohol–marijuana co-abuse have been studied theoretically using mathematical modeling [24]. Findings from the research revealed that even though alcohol and marijuana are both legal, they can be of great harm to the brain of the individual when combined and consequently lead to significant problems for society as a whole [24]. A review study highlighted the importance of incorporating economic cost, social behavior, and intervention effectiveness into mathematical modeling of drug abuse [25]. Individuals that recovered from the use of illicit drugs may not develop lifelong resistance to drugs. This is according to research findings by [26] using a mathematical model that captures the dynamics of illicit drug use. The interconnectedness between drug abuse, unemployment, mental stress and mental illness on population dynamics has been investigated using mathematical modeling [27]. Several strategies to reduce drug abuse, taking into consideration the treatment type and risk level, have been investigated mathematically [28]. Model analysis results show that the anti-drug campaign has a substantial influence in decreasing the population of drug abusers who received treatment [28]. The influence of policing and rehabilitation on the dynamics of drug and substance abuse in the community has been examined theoretically using a mathematical model [29]. More research findings on the dynamics and control interventions of substance abuse can be found in [15,29,30,31,32,33].

Important research findings from studies in the literature that used mathematical model to study the dynamics and control interventions of hepatitis B are presented here. For instance, research that used a mathematical model in the form of an updated Atangana–Baleanu fractional difference operator for hepatitis B transmission emphasizes the significance of applied mathematics in comprehending how disease spreads [34]. Findings from the study suggested the model’s ability to predict outbreaks of infectious diseases such as hepatitis B and AIDS, which is crucial in developing preventive strategies [34]. The effects of migrant screening and public sensitization on the transmission dynamics of HBV in a population have been explored using a mathematical model [35]. The influence of combined therapy on the dynamics of hepatitis B has been investigated theoretically using a mathematical model [36]. Mathematical models have also been used to investigate the impact of the adaptive immune response during acute HBV infection [37]. The impact of universal immunization of the infants at birth, screening, and treatment of both the acute and chronic cases of hepatitis B infections in the presence of passive immunity has been investigated mathematically [12]. Mathematical models have also been considered to investigate the impact of vaccination, treatment, migration, and screening on the dynamics of hepatitis B infections [13]. A mathematical model was considered to evaluate the long-term effectiveness of the hepatitis B vaccination program [14]. The impact of human immune responses on the dynamics and control interventions of hepatitis B has also been investigated mathematically [38]. Other important research findings on the dynamics and control interventions of hepatitis B infections using mathematical models can be found in [16,17,18,39,40,41,42,43].

The above research findings have made a substantial contribution to the dynamics and control interventions of substance abuse and hepatitis B using mathematical modeling. However, none of these studies considered a situation where both substance abuse addictions and hepatitis B infections co-exist in their model formulations. The aim of the study is to fill this critical gap in the literature by developing a mathematical model that take into consideration the co-existence of substance abuse addictions and hepatitis B infections in a population. The results of this paper are expected to reveal how the co-existence of substance abuse addictions and hepatitis B infections fundamentally alter system behavior and control effectiveness and consequently offer new insights for designing effective control management strategies for such scenarios.

2. Model Development

A mathematical model for the co-dynamics of substance abuse and hepatitis B was formulated based on the following assumptions and reasoning. A total human population where both substance abuse addictions and hepatitis B infections co-exist is considered. The total human population N is partitioned into nine sub-populations, namely, susceptible human population (S), human population vaccinated with hepatitis B vaccine (V), human population addicted to substance abuse ($I_a$), human population infected with hepatitis B ($I_b$), human population co-infected with substance abuse and hepatitis B ($I_{ab}$), individuals under rehabilitation due to substance abuse ($T_a$), individuals under treatment due to hepatitis B infection ($T_b$), individuals under rehabilitation/treatment due to substance abuse and hepatitis B ($T_{ab}$), and recovered individuals (R).

The interaction among these sub-populations that results in co-dynamics of substance abuse and hepatitis B is described as follows. The S increases through birth or migration at a recruitment rate $\mathrm{\Lambda }$ and gets vaccinated with the hepatitis vaccine at a rate $\varphi$ and vaccine efficacy $ϵ$. An individual in S becomes an addictive substance abuse user and/or gets infected with hepatitis B at a rate ${\beta }_a$, ${\beta }_b$, respectively. The $I_a$ can get treatment at a rehabilitation facility at a rate ${\sigma }_a$ or die due to complications from substance abuse at a rate ${\delta }_a$. Similarly, $I_b$ can get treatment at a rate ${\sigma }_b$ or die due to complications from hepatitis B at a rate ${\delta }_b$. Those that are co-infected $I_{ab}$ need special treatment (simultaneously treating substance abuse and hepatitis B) at a rate ${\sigma }_{ab}$ or die due to complications from substance abuse and hepatitis B at a rate ${\delta }_{ab}$. The treated individuals from each of these categories $T_a,T_b,T_{ab}$ recover at a rate ${\gamma }_a,{\gamma }_b,{\gamma }_{ab}$ respectively. Public enlightenment reduces transmission due to substance abuse and hepatitis B by a quantity ${\kappa }_a$ and ${\kappa }_b$, respectively. Natural death occurs in each class at a rate $\mu$. Those who recover from substance abuse through a proper rehabilitation process are not likely to become addicted again. Based on this, we assumed that recovered humans do not become susceptible to substance abuse again throughout the study period. Based on these assumptions and formulations, the following mathematical model for substance abuse and hepatitis B co-dynamics is obtained

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \mathrm{\Lambda }-(1-{\kappa }_a){\beta }_{a}S{I}_a-(1-{\kappa }_b){\beta }_{b}S{I}_b-(\varphi +\mu )S,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& \varphi S-(1-{\kappa }_b)(1-\epsilon ){\beta }_{b}V{I}_b-\mu V,\hfill \\ \hfill {\displaystyle \frac{d{I}_a}{dt}}& =& (1-{\kappa }_a){\beta }_{a}S{I}_a-(1-{\kappa }_b){\beta }_{ab}{I}_a{I}_b-({\sigma }_a+\mu +{\delta }_a)I_a,\hfill \\ \hfill {\displaystyle \frac{d{I}_b}{dt}}& =& (1-{\kappa }_b){\beta }_{b}S{I}_b+(1-{\kappa }_b)(1-\epsilon ){\beta }_{b}V{I}_b-(1-{\kappa }_a){\beta }_{ba}{I}_b{I}_a-({\sigma }_b+\mu +{\delta }_b)I_b,\hfill \\ \hfill {\displaystyle \frac{d{I}_{ab}}{dt}}& =& (1-{\kappa }_b){\beta }_{ab}{I}_a{I}_b+(1-{\kappa }_a){\beta }_{ba}{I}_b{I}_a-({\sigma }_{ab}+\mu +{\delta }_{ab})I_{ab},\hfill \\ \hfill {\displaystyle \frac{d{T}_a}{dt}}& =& {\sigma }_a{I}_a-({\gamma }_a+\mu )T_a,\hfill \\ \hfill {\displaystyle \frac{d{T}_b}{dt}}& =& {\sigma }_b{I}_b-({\gamma }_b+\mu )T_b,\hfill \\ \hfill {\displaystyle \frac{d{T}_{ab}}{dt}}& =& {\sigma }_{ab}{I}_{ab}-({\gamma }_{ab}+\mu )T_{ab},\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =& {\gamma }_a{T}_a+{\gamma }_b{T}_b+{\gamma }_{ab}{T}_{ab}-\mu R,\hfill \end{array}$$

(1)

with initial conditions, $S\left(0\right)\ge 0$, $V\left(0\right)\ge 0$, $I_a\left(0\right)\ge 0$, $I_b\left(0\right)\ge 0$, $I_{ab}\left(0\right)\ge 0$, $T_a\left(0\right)\ge 0$, $T_b\left(0\right)\ge 0$, $T_{ab}\left(0\right)\ge 0$, and $R\left(0\right)\ge 0$. The description of variables and parameters is presented in

Table 1. Description for variables for model (1).

Variables	Description	Unit
N	Total population of humans	Humans km−2
S	Susceptible population of humans	Humans km−2
V	Humans vaccinated with hepatitis B vaccine	Humans km−2
$I_a$	Individuals addicted to using substance abuse	Humans km−2
$I_b$	Individuals infected with hepatitis B	Humans km−2
$I_{ab}$	Humans co-infected/addicted to substance abuse and hepatitis B	Humans km−2
$T_a$	$I_a$ on rehabilitation treatment	Humans km−2
$T_b$	$I_b$ on medical treatment	Humans km−2
$T_{ab}$	$I_{ab}$ on treatment for both substance abuse and hepatitis B	Humans km−2
R	Recovered individuals	Humans km−2

and

Table 2. Description of parameters for model (1).

Parameters	Description	Unit
$\mathrm{\Lambda }$	Recruitment rate of humans into S	Humans km−2 Year−1
${\beta }_a$	Rate of addiction to substance abuse	km2 Humans−1 Year−1
${\beta }_b$	Rate of transmission of hepatitis B	km2 Humans−1 Year−1
${\beta }_{ab}$	Transmission rate from $I_a$ to $I_{ab}$	km2 Humans−1 Year−1
${\beta }_{ba}$	Addiction rate from $I_b$ to $I_{ab}$	km2 Humans−1 Year−1
$\mu$	Natural mortality rate of humans	Year−1
${\delta }_a$	Addiction induce mortality rate of $I_a$	Year−1
${\delta }_b$	Disease induces mortality rate of $I_b$	Year−1
${\delta }_{ab}$	Disease/addiction induces mortality rate of $I_{ab}$	Year−1
$\varphi$	Hepatitis B vaccination rate	Year−1
$\epsilon$	Efficacy of the hepatitis B vaccine	Dimensionless
${\sigma }_a$	Treatment rate of $I_a$	Year−1
${\sigma }_b$	Treatment rate of $I_b$	Year−1
${\sigma }_{ab}$	Treatment rate of $I_{ab}$	Year−1
${\gamma }_a$	Recovery rate of $T_a$	Year−1
${\gamma }_b$	Recovery rate of $T_b$	Year−1
${\gamma }_{ab}$	Recovery rate of $T_{ab}$	Year−1
${\kappa }_a$	Reduction in substance abuse addiction due to public enlightenment	Dimensionless
${\kappa }_b$	Reduction in hepatitis B infection due to public enlightenment	Dimensionless

.

To reduce the complexity of equations in model analysis, the following substitutions are made where applicable: ${\alpha }_1=(1-{\kappa }_a){\beta }_a$, ${\alpha }_3={\sigma }_a+\mu +{\delta }_a$, ${\alpha }_4={\gamma }_a+\mu$, $b_1=(1-{\kappa }_b){\beta }_b$, $b_2=(1-{\kappa }_b)(1-\epsilon ){\beta }_b$, $b_3={\sigma }_b+\mu +{\delta }_b$, $b_4={\gamma }_b+\mu$, $c_1=\varphi +\mu$, $c_2=(1-{\kappa }_b){\beta }_{ab}$, $c_3=(1-{\kappa }_a){\beta }_{ba}$, $c_4={\sigma }_{ab}+\mu +{\delta }_{ab}$, $c_5={\gamma }_{ab}+\mu .$

To confirm that model (1) is well posed mathematically and epidemiologically, it suffices to establish the positivity and boundedness of solutions of the model. The proof of positivity and boundedness of solutions of the model (1) are presented in Theorems 1 and 2 respectively.

Theorem 1.

Let $\mathrm{\Omega }=\{(S,V,I_a,I_b,I_{ab},T_a,T_b,T_{ab},R)\in {\mathbb{R}}_{+}^9:N\le {\displaystyle \frac{\mathrm{\Lambda }}{\mu }}$, $S\left(0\right)\ge 0,V\left(0\right)\ge 0$, $I_a\left(0\right)\ge 0$, $I_b\left(0\right)\ge 0$, $I_{ab}\left(0\right)\ge 0$, $T_a\left(0\right)\ge 0$, $T_b\left(0\right)\ge 0$, $T_{ab}\left(0\right)\ge 0$, $R\left(0\right)\ge 0\}$ then, the solutions $S\left(t\right),V\left(t\right),I_a\left(t\right),I_b\left(t\right),I_{ab}\left(t\right),T_a\left(t\right),T_b\left(t\right),T_{ab}\left(t\right),R\left(t\right)$ of the model (1) will remain positive for all time $t>0$.

Proof. 

Here we establish the positivity of solutions of model (1) with initial conditions $S\left(0\right)\ge 0,V\left(0\right)\ge 0,I_a\left(0\right)\ge 0,I_b\left(0\right)\ge 0,I_{ab}\left(0\right)\ge 0,T_a\left(0\right)\ge 0,T_b\left(0\right)\ge 0,T_{ab}\left(0\right)\ge 0$, and $R\left(0\right)\ge 0$. By rearranging the first equation of model (1), ${\displaystyle \frac{dS}{dt}}=\mathrm{\Lambda }-{\alpha }_{1}S{I}_a-b_{1}S{I}_b-c_{1}S$, we obtain

$${\displaystyle \frac{dS}{dt}}+({\alpha }_1{I}_a+b_1{I}_b+c_1)S=\mathrm{\Lambda }.$$

(2)

Multiplying Equation (2) by its integrating factor given by $\mathrm{\Psi }\left(t\right)=e^{\int ({\alpha }_1{I}_a\left(t\right)+b_1{I}_b\left(t\right)+c_1)dt}$ and simplifying gives

$${\displaystyle \frac{d}{dt}}\left(S\left(t\right)\mathrm{\Psi }\left(t\right)\right)=\mathrm{\Lambda }\mathrm{\Psi }\left(t\right).$$

(3)

Integrating both sides of Equation (3) and simplifying gives

$${S\left(\tau \right)\mathrm{\Psi }\left(\tau \right)|}_0^t={\int }_0^t\mathrm{\Lambda }\mathrm{\Psi }\left(\tau \right)d\tau .$$

(4)

Further simplification gives

$$S\left(t\right)={\displaystyle \frac{1}{\mathrm{\Psi }\left(t\right)}}\left(S\left(0\right)+{\int }_0^t\mathrm{\Lambda }\mathrm{\Psi }\left(\tau \right)d\tau \right).$$

(5)

Since $S\left(0\right)\ge 0$ and $\mathrm{\Psi }\left(t\right)=e^{\int ({\alpha }_1{I}_a\left(t\right)+b_1{I}_b\left(t\right)+c_1)dt}>0$, it can be deduced from Equation (5) that $S\left(t\right)\ge 0$ for all $t>0$. Similarly, we can show that $V\left(t\right),I_a\left(t\right),I_b\left(t\right),I_{ab}\left(t\right)$, $T_a\left(t\right),T_b\left(t\right),T_{ab}\left(t\right),R\left(t\right)$ of the model (1) remain positive for all $t>0$. □

Theorem 2.

All feasible solutions of the model (1) are uniformly bounded in a proper subset $\mathrm{\Omega }=\{(S,V,I_a,I_b,I_{ab},T_a,T_b,T_{ab},R)\in {\mathbb{R}}_{+}^9:N\le {\displaystyle \frac{\mathrm{\Lambda }}{\mu }}\}$.

Proof. 

To establish the boundedness of the solutions of model (1), we need to show that a lower bound and upper bound exist for all the solutions of the model. At initial time $t=0$, $S\left(0\right)\ge 0,V\left(0\right)\ge 0,I_a\left(0\right)\ge 0,I_b\left(0\right)\ge 0,I_{ab}\left(0\right)\ge 0,T_a\left(0\right)\ge 0,T_b\left(0\right)\ge 0,T_{ab}\left(0\right)\ge 0$, and $R\left(0\right)\ge 0$.

For the upper bound, we consider the sum of all the equations in model (1) since $N=S+V+I_a+I_b+I_{ab}+T_a+T_b+T_{ab}+R$ to obtain

$${\displaystyle \frac{dN}{dt}}={\displaystyle \frac{dS}{dt}}+{\displaystyle \frac{dV}{dt}}+{\displaystyle \frac{d{I}_a}{dt}}+{\displaystyle \frac{d{I}_b}{dt}}+{\displaystyle \frac{d{I}_{ab}}{dt}}+{\displaystyle \frac{d{T}_a}{dt}}+{\displaystyle \frac{d{T}_b}{dt}}+{\displaystyle \frac{d{T}_{ab}}{dt}}+{\displaystyle \frac{dR}{dt}}.$$

(6)

Simplifying Equation (6) gives

$${\displaystyle \frac{dN}{dt}}=\mathrm{\Lambda }-\mu N-{\delta }_a{I}_a-{\delta }_b{I}_b-{\delta }_{ab}{I}_{ab}\le \mathrm{\Lambda }-\mu N.$$

(7)

Integrating both sides, applying the initial condition $N\left(0\right)=N_0$ and solving the inequality (7) gives

$$N\le {\displaystyle \frac{\mathrm{\Lambda }}{\mu }}.$$

(8)

Therefore, we obtain

$$0\le N\le {\displaystyle \frac{\mathrm{\Lambda }}{\mu }}.$$

(9)

Thus, the feasible solution set of model enters and remains in the region $\mathrm{\Omega }=\{(S,V,I_a,I_b,I_{ab},T_a,$$T_b,T_{ab},R) \in {\mathbb{R}}_{+}^9:N\le {\displaystyle \frac{\mathrm{\Lambda }}{\mu }}\}$. □

3. Analyses and Results

In this section, the qualitative analysis of the substance abuse and hepatitis B co-dynamics model (1) is presented systematically. The analysis is made up of three scenarios: the first scenario is a special case where only substance abuse is endemic in the population, the second scenario is a special case where only hepatitis B is endemic in the population, and the third scenario is the general model (1) where both substance abuse and hepatitis B are endemic in the population. The results of these analyses are expected to improve our understanding of the model dynamics and consequently aid in decision-making of the management of substance abuse addictions and hepatitis B infections for the above-listed scenarios.

3.1. Substance Abuse Sub-Model

The substance abuse sub-model is a special case of model (1) where only substance abuse affects the population. Qualitative analysis of this special case, which is essential for improved understanding of the dynamics and possible control of substance abuse, is presented in this section. The analysis of this substance abuse sub-model will aid in understanding the dynamics of the general model (1). This can be achieved through a mathematical comparison of the analysis of the substance abuse sub-model and general model (1). The substance abuse sub-model is determined by setting $V=I_b=I_{ab}=T_b=T_{ab}=0$ in the model (1), to obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \mathrm{\Lambda }-(1-{\kappa }_a){\beta }_{a}S{I}_a-\mu S,\hfill \\ \hfill {\displaystyle \frac{d{I}_a}{dt}}& =& (1-{\kappa }_a){\beta }_{a}S{I}_a-({\sigma }_a+\mu +{\delta }_a)I_a,\hfill \\ \hfill {\displaystyle \frac{d{T}_a}{dt}}& =& {\sigma }_a{I}_a-({\gamma }_a+\mu )T_a,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =& {\gamma }_a{T}_a-\mu R.\hfill \end{array}$$

(10)

The dynamics and possible control intervention for substance abuse described by the qualitative analysis of sub-model (10) are presented in this sections. The fourth line of the sub-model (10), denoted by ${\displaystyle \frac{dR}{dt}}$, can be excluded in the qualitative analysis since the variable R does not affect the remaining part of model (10).

3.1.1. Equilibrium Points of Substance Abuse Sub-Model (10)

The substance abuse sub-model (10) has two important equilibrium points, namely, the substance abuse free equilibrium (SAFE) and substance abuse endemic equilibrium (SAEE). The SAFE is the equilibrium point of the model where there is no substance-addicted individual in the population, while the SAEE is the equilibrium point of the model where there is a substance-addicted individual in the population. The analytical presentation of the SAFE of the sub-model (10) is given by

$$\begin{array}{c}\hfill (S^0,I_a^0,T_a^0)=\left({\displaystyle \frac{\mathrm{\Lambda }}{\mu }},0,0\right).\end{array}$$

(11)

Similarly, the analytical presentation of the SAEE of the sub-model (10) is given by

$$\begin{array}{c}\hfill (S^{\ast },I_a^{\ast },T_a^{\ast })=\left({\displaystyle \frac{({\sigma }_a+\mu +{\delta }_a)}{(1-{\kappa }_a){\beta }_a}},{\displaystyle \frac{\mu (S^0-S^{\ast })}{(1-{\kappa }_a){\beta }_a{S}^{\ast }}},{\displaystyle \frac{{\sigma }_a{I}_a^{\ast }}{{\gamma }_a+\mu }}\right).\end{array}$$

(12)

3.1.2. The Basic Reproduction Number of Substance Abuse Sub-Model (10)

The basic reproduction number denoted by ${\mathcal{R}}_0^a$ is the average number of new cases caused by one case in a fully susceptible population [5,7]. Its value shows whether an outbreak will persist or be eradicated [5,7]. For the substance abuse sub-model (10), ${\mathcal{R}}_0^a$ is computed using the next-generation matrix method of [7], as follows. The next-generation matrix for the substance abuse sub-model (10) is

$$F{V}^{-1}=\left(\begin{array}{cc}{\displaystyle \frac{(1-{\kappa }_a){\beta }_a{S}^0}{{\sigma }_a+\mu +{\delta }_a}}& 0\\ 0& 0\end{array}\right),$$

(13)

where

$$F=\left(\begin{array}{cc}(1-{\kappa }_a){\beta }_a{S}^0& 0\\ 0& 0\end{array}\right)\mathrm{and}V=\left(\begin{array}{cc}{\sigma }_a+\mu +{\delta }_a& 0\\ -{\sigma }_a& {\gamma }_a+\mu \end{array}\right).$$

The dominant eigenvalue of the next-generation matrix $F{V}^{-1}$ is the basic reproduction number and is determined as

$$\begin{array}{c}\hfill {\mathcal{R}}_0^a={\displaystyle \frac{(1-{\kappa }_a){\beta }_a{S}^0}{{\sigma }_a+\mu +{\delta }_a}}.\end{array}$$

(14)

3.1.3. Stability Analysis of Substance Abuse Sub-Model (10)

The stability analysis of a model about its equilibrium point describes the dynamics of the model about the equilibrium point. The stability analysis of the substance abuse sub-model (10) about the SAFE and SAEE is summarized in the following theorems.

Theorem 3.

The substance abuse sub-model (10) is locally asymptotically stable about the SAFE (11) provided ${\mathcal{R}}_0^a<1$.

Proof. 

To prove Theorem 3, it suffices to show that all the eigenvalues of the Jacobian of the sub-model (10) about the SAFE (11) have a negative real part. The Jacobian of sub-model (10) about the SAFE (11) is

$$J_0=\left(\begin{array}{ccc}-\mu & (1-{\kappa }_a){\beta }_a{S}^0& 0\\ 0& ({\sigma }_a+\mu +{\delta }_a)({\mathcal{R}}_0^a-1)& 0\\ 0& 0& -({\gamma }_a+\mu )\end{array}\right).$$

(15)

The eigenvalues of the Jacobian are

$$\begin{array}{ccc}\hfill {\lambda }_1& =& -\mu ,\hfill \\ \hfill {\lambda }_2& =& ({\sigma }_a+\mu +{\delta }_a)({\mathcal{R}}_0^a-1),\hfill \\ \hfill {\lambda }_3& =& -({\gamma }_a+\mu ).\hfill \end{array}$$

Obviously, ${\lambda }_1,{\lambda }_2$ are negative. However, ${\lambda }_3<0$ if ${\mathcal{R}}_0^a<1$. Thus, we conclude that the sub-model (10) is locally asymptotically stable about the SAFE (11) provided ${\mathcal{R}}_0^a<1$. □

The implication of Theorem 3 is that substance abuse can be eradicated if the initial number of addicted users is sufficiently small (within the neighborhood of the SAFE) and ${\mathcal{R}}_0^a<1$. However, if the initial addicted population is large, the dynamics of the substance abuse sub-model about the SAFE require the investigation of the global stability of the substance abuse sub-model (10) about the SAFE (11). The global stability analysis of the substance abuse sub-model (10) about the SAFE (11) is presented in Theorem 4.

Theorem 4.

The substance abuse sub-model (10) is globally asymptotically stable about the SAFE (11) provided ${\mathcal{R}}_0^a<1$.

The proof of Theorem 4 will be established using a stability result in [44], which is stated in Lemma 1.

Lemma 1.

Consider a model system written in the form

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{Z}_1}{dt}}& =& F(Z_1,Z_2)\hfill \\ \hfill {\displaystyle \frac{d{Z}_2}{dt}}& =& G(Z_1,Z_2),G(Z_1,0)=0,\hfill \end{array}$$

(16)

where $Z_1\in R^m$ and $Z_2\in R^n$. $Z_0=(Z_1^{\ast },0)$ denotes the disease-free equilibrium of the system. Assume that

(H1) 

For ${\displaystyle \frac{d{Z}_1}{dt}}=F(Z_1,0)$, $Z_1^{\ast }$ is globally asymptotically stable;

(H2) 

$G(Z_1,Z_2)=A{Z}_2-\widehat{G}(Z_1,Z_2)$, $\widehat{G}(Z_1,Z_2)\ge 0$ for $(Z_1,Z_2)\in \mathrm{\Omega },$ where the Jacobian $A={\displaystyle \frac{\partial G}{\partial Z_2}}(Z_1,0)$ is an M-matrix (the off diagonal elements of A are non-negative) and Ω is the region where the model makes biological sense.

Then the disease-free equilibrium $Z_0$ is globally asymptotically stable provided that ${\mathcal{R}}_0<1$ [44].

Proof. 

To apply Lemma 1, it suffices to show that the substance abuse sub-model (10) satisfies conditions H1 and H2 of the Lemma. From the sub-model (10), let $Z_1=S$, $Z_2=(I_a,T_a)$. So, we obtain

$${\displaystyle \frac{d{Z}_1}{d\tau }}=F(Z_1,0)=\left(\begin{array}{c}\lambda -\mu S\end{array}\right),$$

(17)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{d{Z}_2}{dt}}=G(Z_1,Z_2)=\left(\begin{array}{c}(1-{\kappa }_a){\beta }_{a}S{I}_a-({\sigma }_a+\mu +{\delta }_a)I_a\\ {\sigma }_a{I}_a-({\gamma }_a+\mu )T_a\end{array}\right).\end{array}$$

(18)

Clearly, $F(Z_1,0)$ is a linear ordinary differential equation, and its exact solution can be obtained as $S\left(t\right)={\displaystyle \frac{\lambda }{\mu }}+A_1{e}^{-\mu t}$, where $A_1$ is a constant of integration. As $t⟶\infty$, $S\left(t\right)⟶S^0={\displaystyle \frac{\mathrm{\Lambda }}{\mu }}$, confirming the global stability of $F(Z_1,0)$.

For the $G(Z_1,Z_2)$, the Jacobian of (18) about the substance abuse free equilibrium is determined as

$$\begin{array}{c}\hfill A=\left(\begin{array}{cc}(1-{\kappa }_a){\beta }_a{S}^0-({\sigma }_a+\mu +{\delta }_a)& 0\\ {\sigma }_a& -({\gamma }_a+\mu )\end{array}\right),\end{array}$$

(19)

where A is an M-matrix with all its off-diagonal elements non-negative. From Equations (18) and (19), the term $\widehat{G}(Z_1,Z_2)$ is determined as

$$\begin{array}{c}\hfill \widehat{G}(Z_1,Z_2)=\left(\begin{array}{c}(1-{\kappa }_a){\beta }_a{I}_a(S^0-S)\\ 0\end{array}\right).\end{array}$$

(20)

Clearly, $\widehat{G}(Z_1,Z_2)\ge 0$ since $S^0\ge S$. This confirms that condition $\left(H_2\right)$ holds, and thus the proof is complete. □

The implication of these results is that the substance abuse epidemic could be eradicated irrespective of the number of addicted humans at the initial stage of the epidemic, provided that the control measures are sufficient to keep the ${\mathcal{R}}_0^a$ below unity. However, if the control measures are not effective enough to keep the ${\mathcal{R}}_0^a$ below unity, substance abuse addictions are likely to become endemic in the population. The endemic dynamics of substance abuse sub-model (10) when ${\mathcal{R}}_0^a>1$ are presented in Theorem 5.

Theorem 5.

The substance abuse sub-model (10) is locally asymptotically stable about the SAEE (12) provided ${\mathcal{R}}_0^a>1$.

Proof. 

To prove Theorem 5, it suffices to show that the characteristic polynomial of the Jacobian of the sub-model (10) about the SAEE (12) satisfies the Routh–Hurwitz stability criterion. The Jacobian of sub-model (10) about the SAFE (12) is computed as

$$J_0=\left(\begin{array}{ccc}-\mu -(1-{\kappa }_a){\beta }_a{I}_a^{\ast }& (1-{\kappa }_a){\beta }_a{S}^{\ast }& 0\\ (1-{\kappa }_a){\beta }_a{I}_a^{\ast }& 0& 0\\ 0& {\sigma }_a& -({\gamma }_a+\mu )\end{array}\right).$$

(21)

The characteristic polynomial of the Jacobian about the equilibrium point (12) is given by

$$\begin{array}{c}\hfill a_3{\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0=0,\end{array}$$

where

$$\begin{array}{ccc}\hfill a_3& =& 1,\hfill \\ \hfill a_2& =& 2\mu +(1-{\kappa }_a){\beta }_a{I}_a^{\ast }+{\gamma }_a,\hfill \\ \hfill a_1& =& {(1-{\kappa }_a)}^2{\beta }_a^2{I}_a^{\ast }{S}^{\ast }+(\mu +(1-{\kappa }_a){\beta }_a{I}_a^{\ast })({\gamma }_a+\mu ),\hfill \\ \hfill a_0& =& {(1-{\kappa }_a)}^2{\beta }_a^2{I}_a^{\ast }{S}^{\ast }({\gamma }_a+\mu ).\hfill \end{array}$$

Algebraically, it can be establish that $a_3>1$, $a_2>0$, $a_1>0$, $a_0>0$ and $a_2{a}_1>a_3{a}_0$, confirming the Routh–Hurwitz stability criterion. Thus, we conclude that the sub-model (10) is locally asymptotically stable about the SAEE (12) provided ${\mathcal{R}}_0^a>1$. □

The implication of this result is that substance-addicted individuals will persist in the population provided that ${\mathcal{R}}_0^a>1$. This situation has several negative consequences to society, and thus every effort should be made to reduce ${\mathcal{R}}_0^a$ below unity to reduce the number of substance-addicted individuals in society.

3.2. Hepatitis B Sub-Model

The hepatitis B sub-model is another special case of the co-dynamics model (1) where hepatitis B is the only disease affecting the population. Qualitative analysis of this special case is crucial for understanding the dynamics and control interventions of hepatitis B and is presented in this section. The hepatitis sub-model is obtained by setting $I_a=I_{ab}=T_a=T_{ab}=0$ in the model (1) to obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& \mathrm{\Lambda }-(1-{\kappa }_b){\beta }_{b}S{I}_b-(\varphi +\mu )S,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =& \varphi S-(1-{\kappa }_b)(1-\epsilon ){\beta }_{b}V{I}_b-\mu V,\hfill \\ \hfill {\displaystyle \frac{d{I}_b}{dt}}& =& (1-{\kappa }_b){\beta }_{b}S{I}_b+(1-{\kappa }_b)(1-\epsilon ){\beta }_{b}V{I}_b-({\sigma }_b+\mu +{\delta }_b)I_b,\hfill \\ \hfill {\displaystyle \frac{d{T}_b}{dt}}& =& {\sigma }_b{I}_b-({\gamma }_b+\mu )T_b,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =& {\gamma }_b{T}_b-\mu R.\hfill \end{array}$$

(22)

The analysis of sub-model (22) that will reveal the qualitative dynamics and control of hepatitis B disease outbreak is presented in this section.

3.2.1. Equilibrium Points of the Hepatitis B Sub-Model (22)

The hepatitis B sub-model (22) has two important equilibrium points, namely, the disease-free equilibrium (DFE) and endemic equilibrium (EE). The analytical presentation of the DFE of the hepatitis B sub-model (22) is given by

$$\begin{array}{c}\hfill (S^0,V^0,I_b^0,T_b^0)=\left({\displaystyle \frac{\mathrm{\Lambda }}{\mu +\varphi }},{\displaystyle \frac{\varphi S^0}{\mu }},0,0\right).\end{array}$$

(23)

3.2.2. The Basic Reproduction Number of the Hepatitis B Sub-Model (22)

The basic reproduction number for the hepatitis B sub-model (22) denoted by ${\mathcal{R}}_0^b$ is computed using the next-generation matrix method [7] and is determined as

$$\begin{array}{c}\hfill {\mathcal{R}}_0^b={\displaystyle \frac{(1-{\kappa }_b){\beta }_b{S}^0+(1-{\kappa }_a)(1-ϵ){\beta }_b{V}^0}{{\sigma }_b+\mu +{\delta }_b}}.\end{array}$$

(24)

3.2.3. Stability Analysis of the Hepatitis B Sub-Model (22)

The results of the stability analysis of the hepatitis B sub-model (22) are summarized in the subsequent theorems.

Theorem 6.

The hepatitis B sub-model (22) is locally asymptotically stable about the DFE (23) provided ${\mathcal{R}}_0^b<1$.

Proof. 

To prove Theorem 6, it suffices to show that all the eigenvalues of the Jacobian of the sub-model (22) about the equilibrium point (23) have a negative real part. The Jacobian of sub-model (22) about the DFE (23) is

$$J_0=\left(\begin{array}{cccc}-(\mu +\varphi )& 0& -(1-{\kappa }_b){\beta }_b{S}^0& 0\\ -\varphi & -\mu & -(1-{\kappa }_b)(1-ϵ){\beta }_b{V}^0& 0\\ 0& 0& ({\sigma }_b+\mu +{\delta }_b)({\mathcal{R}}_0^b-1)& 0\\ 0& 0& 0& -({\gamma }_b+\mu )\end{array}\right).$$

(25)

The eigenvalues of the Jacobian are

$$\begin{array}{ccc}\hfill {\lambda }_1& =& -(\mu +\varphi ),\hfill \\ \hfill {\lambda }_2& =& \mu ,\hfill \\ \hfill {\lambda }_3& =& -({\gamma }_b+\mu ),\hfill \\ \hfill {\lambda }_4& =& ({\sigma }_b+\mu +{\delta }_b)({\mathcal{R}}_0^b-1).\hfill \end{array}$$

Clearly, ${\lambda }_1,{\lambda }_2,{\lambda }_3$ are negative. However, ${\lambda }_4<0$ if ${\mathcal{R}}_0^b<1$. Thus, we conclude that the hepatitis B sub-model (22) is locally asymptotically stable about the DFE (23) provided ${\mathcal{R}}_0^b<1$. □

The implication of Theorem 6 is that hepatitis B disease can be eradicated if the initial infection is small (within the neighborhood of the DFE) and provided the control measures are effective enough to keep the ${\mathcal{R}}_0^b$ below unity. However, if the initial infection is large, the actual dynamics of the hepatitis B sub-model about the DFE required the investigation of the global stability analysis of the sub-model (22) about the DFE (23). The result of global stability analysis of the hepatitis B sub-model (22) about the DFE (23) is presented in Theorem 7.

Theorem 7.

The hepatitis B sub-model (22) is globally asymptotically stable about the DFE (23) provided ${\mathcal{R}}_0^b<1$.

The proof of Theorem 7 can be established using a similar approach to that used in the proof of Theorem 4. The implication of this result is that hepatitis B can be eradicated regardless of the initial infections, provided the control measures are effective enough to keep the ${\mathcal{R}}_0^b$ below unity.

In the absence of hepatitis B vaccination, an endemic equilibrium exists for the hepatitis B sub-model (22) and is given by

$$\begin{array}{c}\hfill (S^{\ast },I_b^{\ast },T_b^{\ast })=\left({\displaystyle \frac{({\sigma }_b+\mu +{\delta }_b)}{(1-{\kappa }_b){\beta }_b}},{\displaystyle \frac{\mu (S^0-S^{\ast })}{(1-{\kappa }_b){\beta }_b{S}^{\ast }}},{\displaystyle \frac{{\sigma }_b{I}_b^{\ast }}{{\gamma }_b+\mu }}\right).\end{array}$$

(26)

This illustrates the impact of hepatitis B vaccination. The stability analysis of the endemic equilibrium for the hepatitis B sub-model (22) in the absence of vaccination is presented in Theorem 8.

Theorem 8.

For ${\mathcal{R}}_0^b>1$, the hepatitis B sub-model (22) is locally asymptotically stable about the endemic equilibrium (26) in the absence of vaccination.

Theorem 8 can be established using a similar approach to that used in the proof of Theorem 5. The implication of this result is that in the absence of vaccination, hepatitis B will become endemic in the population if ${\mathcal{R}}_0^b>1$. This situation has several negative consequences, such as an increase in deaths due to the disease, and thus every effort should be made to reduce ${\mathcal{R}}_0^b$ below unity so that hepatitis B will be eradicated from the population.

3.3. Analysis of the Substance Abuse and Hepatitis B Co-Dynamics Model (1)

The dynamical system analysis of the substance abuse and hepatitis B co-dynamics model (1) is presented in this section. The results of the analysis are crucial for improving understanding of the dynamics and control interventions of substance abuse and hepatitis B co-dynamics in the community where the two infections co-exist.

3.3.1. Equilibrium Points of the Substance Abuse and Hepatitis B Co-Dynamics Model (1)

The substance abuse and hepatitis B co-dynamics model (1) has equilibrium points. The analytical presentation of one of the equilibrium points of the substance abuse and hepatitis B co-dynamics model (1) is given by

$$\begin{array}{c}\hfill (S^0,V^0,I_a^0,I_b^0,I_{ab}^0,T_a^0,T_b^0,T_{ab}^0)=\left({\displaystyle \frac{\mathrm{\Lambda }}{\mu +\varphi }},{\displaystyle \frac{\varphi S^0}{\mu }},0,0,0,0,0,0\right).\end{array}$$

(27)

3.3.2. Basic Reproduction Number of the Substance Abuse and Hepatitis B Co-Dynamics Model (1)

The basic reproduction number of the substance abuse and hepatitis B co-dynamics model (1) denoted by ${\mathcal{R}}_0$ is computed using the next-generation matrix method [7] and is determined as

$$\begin{array}{c}\hfill {\mathcal{R}}_0=\max \{{\mathcal{R}}_0^a,{\mathcal{R}}_0^b\},\end{array}$$

(28)

where

$$\begin{array}{ccc}\hfill {\mathcal{R}}_0^a& =& {\displaystyle \frac{(1-{\kappa }_a){\beta }_a{S}^0}{{\sigma }_a+\mu +{\delta }_a}},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\mathcal{R}}_0^b& =& {\displaystyle \frac{(1-{\kappa }_b){\beta }_b{S}^0+(1-{\kappa }_a)(1-ϵ){\beta }_b{V}^0}{{\sigma }_b+\mu +{\delta }_b}}.\hfill \end{array}$$

(30)

This shows that the basic reproduction number of the full co-dynamics model exceeds that of the individual substance abuse and hepatitis B models. Mathematically, an increase in the basic reproduction number implies an increase in the severity of the infections and addictions to substance abuse. Thus, the co-existence of substance abuse addictions and hepatitis B infections implies greater severity of infections and addictions.

3.3.3. Stability Analysis of the Substance Abuse and Hepatitis B Co-Dynamics (1)

The result of stability analysis of the co-dynamics model (1) is summarized in the theorem below.

Theorem 9.

Substance abuse and hepatitis B co-dynamics model (1) is locally asymptotically stable about the equilibrium point (27) provided ${\mathcal{R}}_0<1$.

Proof. 

To prove Theorem 9, it suffices to show that all the eigenvalues of the Jacobian of the co-dynamics model (1) about the equilibrium point (27) have a negative real part. The Jacobian of co-dynamics model (1) about the equilibrium point (27) is

$$J_0=\left(\begin{array}{cccccccc}-c_1& 0& -{\alpha }_1{S}^0& b_1{S}^0& 0& 0& 0& 0\\ -\varphi & -\mu & 0& b_2{V}^0& 0& 0& 0& 0\\ 0& 0& {\alpha }_3({\mathcal{R}}_0^a-1)& 0& 0& 0& 0& 0\\ 0& 0& 0& b_3({\mathcal{R}}_0^b-1)& 0& 0& 0& 0\\ 0& 0& 0& 0& -c_4& 0& 0& 0\\ 0& 0& 0& 0& 0& -{\alpha }_4& 0& 0\\ 0& 0& 0& 0& 0& 0& -b_4& 0\\ 0& 0& 0& 0& 0& 0& 0& -c_5\end{array}\right).$$

(31)

The eigenvalues of the Jacobian are

$$\begin{array}{ccc}\hfill {\lambda }_1& =& -(\varphi +\mu ),\hfill \\ \hfill {\lambda }_2& =& -\mu ,\hfill \\ \hfill {\lambda }_3& =& ({\sigma }_a+\mu +{\delta }_a)({\mathcal{R}}_0^a-1),\hfill \\ \hfill {\lambda }_4& =& ({\sigma }_b+\mu +{\delta }_b)({\mathcal{R}}_0^b-1),\hfill \\ \hfill {\lambda }_5& =& -({\sigma }_{ab}+\mu +{\delta }_{ab}),\hfill \\ \hfill {\lambda }_6& =& -({\gamma }_a+\mu ),\hfill \\ \hfill {\lambda }_7& =& -({\gamma }_b+\mu ),\hfill \\ \hfill {\lambda }_8& =& -({\gamma }_{ab}+\mu ).\hfill \end{array}$$

Obviously, ${\lambda }_1,{\lambda }_2,{\lambda }_5,{\lambda }_6,{\lambda }_7,{\lambda }_8$ are negative. However, ${\lambda }_3<0$ if ${\mathcal{R}}_0^a<1$ and ${\lambda }_4<0$ if ${\mathcal{R}}_0^b<1$. Thus, we conclude that the substance abuse and hepatitis B co-dynamics model (1) is locally asymptotically stable about the equilibrium point (27) provided ${\mathcal{R}}_0<1$. □

The implication of Theorem 9 is that both substance abuse and hepatitis B can be eradicated if the initial number of addicted users and hepatitis B-infected persons is sufficiently small and ${\mathcal{R}}_0<1$. However, if the initial addicted population and number of hepatitis B-infected persons are large, the dynamics of model (1) about the equilibrium point (27) require the investigation of the global stability of the co-dynamics model (1). The global stability analysis of the co-dynamics model (1) is presented in Theorem 4.

Theorem 10.

For ${\mathcal{R}}_0<1$, the substance abuse and hepatitis B co-dynamics model (1) is not globally asymptotically stable about the equilibrium point (27).

Proof. 

The proof of Theorem 10 will be established using Lemma 1. Based on this, it is sufficient to show that the co-dynamics model (1) fails to satisfy one of the global stability conditions of Lemma 1 [44]. From the co-dynamics model (1), let $Z_1=(S,V),Z_2=(I_a,I_b,I_{ab},T_a,T_b,T_{ab})$. From the co-dynamics model (1), we obtain

$${\displaystyle \frac{d{Z}_1}{dt}}=F(Z_1,0)=\left(\begin{array}{c}\mathrm{\Lambda }-(\varphi +\mu )S\\ \varphi S-\mu V\end{array}\right)$$

(32)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{d{Z}_2}{dt}}=G(Z_1,Z_2)=\left(\begin{array}{c}{\alpha }_{1}S{I}_a-c_2{I}_a{I}_b-{\alpha }_3{I}_a\\ b_{1}S{I}_b+b_{2}V{I}_b-c_3{I}_b{I}_a-b_3{I}_b\\ c_2{I}_a{I}_b+c_3{I}_b{I}_a-c_4{I}_{ab}\\ {\sigma }_a{I}_a-{\alpha }_4{T}_a\\ {\sigma }_b{I}_b-b_4{T}_b\\ {\sigma }_{ab}{I}_{ab}-c_5{T}_{ab}\end{array}\right).\end{array}$$

(33)

The Jacobian of Equation (33) about the equilibrium point (27) is

$$\begin{array}{c}\hfill A=\left(\begin{array}{cccccc}{\alpha }_1{S}^0-{\alpha }_3& 0& 0& 0& 0& 0\\ 0& b_1{S}^0+b_2{V}^0-b_3& 0& 0& 0& 0\\ 0& 0& -c_4& 0& 0& 0\\ {\sigma }_a& 0& 0& -{\alpha }_4& 0& 0\\ 0& {\sigma }_b& 0& 0& -b_4& 0\\ 0& 0& {\sigma }_{ab}& 0& 0& -c_5\end{array}\right),\end{array}$$

(34)

where A is an M-matrix with all its off-diagonal elements non-negative. From Equations (33) and (34), the expression $\widehat{G}(Z_1,Z_2)$ is calculated as

$$\begin{array}{c}\hfill \widehat{G}(Z_1,Z_2)=\left(\begin{array}{c}{\alpha }_1{I}_a(S^0-S)+c_2{I}_a{I}_b\\ b_1{I}_b(S^0-S)+b_2{I}_b(V^0-V)+c_3{I}_a{I}_b\\ -c_2{I}_a{I}_b-c_3{I}_a{I}_b\\ 0\\ 0\\ 0\end{array}\right).\end{array}$$

(35)

Clearly, $\widehat{G}(Z_1,Z_2)$ is not positive. This shows that condition $\left(H_2\right)$ does not hold and hence completes the proof. □

The implication of these results is that it will be difficult to eradicate substance abuse and hepatitis B simultaneously by only lowering ${\mathcal{R}}_0$ below unity. This suggest that eradication is theoretically more difficult in a co-existence scenarios compare to a single scenario.

When the control interventions are not effective and ${\mathcal{R}}_0$ increases above unity, substance abuse addictions and hepatitis B infections are likely to become endemic in the population. The stability analysis about the endemic equilibrium will reveal the endemic dynamics of the co-existence of substance abuse addictions and hepatitis B infections. However, due to the complexity in the equations involved in the stability analysis about the endemic equilibrium of the co-existence model (1), numerical simulations are considered to explore the endemic dynamics of the model as well as the impact of control interventions.

4. Numerical Simulations

Numerical simulations were conducted to explore the dynamics and impact of control measures on the substance abuse and hepatitis B co-dynamics model (1). Insights gained from this study can inform broader strategies aimed at combating substance abuse and hepatitis B in the endemic areas. The values of the parameters used for the numerical simulations are given in

Table 3. Parameters values used in the simulation.

Parameters	Value	References
$\mathrm{\Lambda }$	100–380	[22,23]
${\beta }_a$	0.02–0.5	[23]
${\beta }_b$	0.3–0.9	[12]
${\beta }_{ab}$	0.2$({\beta }_a+{\beta }_b)$	Assumed
${\beta }_{ba}$	0.2$({\beta }_a+{\beta }_b)$	Assumed
$\mu$	0.013	[22]
${\delta }_a$	0.167	[23]
${\delta }_b$	0.041	[12]
${\delta }_{ab}$	${\delta }_a+{\delta }_b$	Assumed
$\varphi$	0.52	[45]
$\epsilon$	0.90–1.0	[1]
${\sigma }_a$	0.001–0.1	[22,23]
${\sigma }_b$	0.1	[12]
${\sigma }_{ab}$	0.5$({\sigma }_a+{\sigma }_b)$	Assumed
${\gamma }_a$	0.01	[22]
${\gamma }_b$	0.1	[12]
${\gamma }_{ab}$	0.5$({\gamma }_a+{\gamma }_b)$	Assumed
${\kappa }_a$	0.0–1.0	Assumed
${\kappa }_b$	0.0–1.0	Assumed

. Due to limited access to real data, some of the parameter values were extracted from the literature, while the remainder were assumed within a statistically realistic range. However, future work will consider parameter estimation by fitting the model to real data on substance abuse and hepatitis B as real data become available. The parameter values for co-dynamics are assumed since this is the first time that research has been conducted that involves the co-existence of substance abuse and hepatitis B. For instance, the treatment rate for the co-existence scenario is taken as the average of the treatment rates for substance abuse and hepatitis B.

4.1. Numerical Simulation of Substance Abuse Sub-Model (10)

The numerical simulations for the substance abuse sub-model (10) are presented in this section. This is crucial for understanding the dynamics and possible control intervention for substance abuse in endemic areas.

Figure 1 is a numerical illustration of the dynamics of substance abuse sub-model (10) when ${\mathcal{R}}_0^a<1$ for various initial conditions. The figure shows that when ${\mathcal{R}}_0^a<1$, substance abuse addictions will be totally eradicated irrespective of the initial number of addicted individuals in the population. Consequently, individuals in rehabilitation due to substance abuse and recovered individuals will also be eradicated, while the susceptible individuals in the population increase. This validates Theorems 3 and 4, which state that substance abuse sub-model (10) is locally and globally asymptotically stable about the SAFE point (11) provided ${\mathcal{R}}_0^a<1$. Based on these results, effective control interventions such as public enlightenment and rehabilitation should be implemented such that ${\mathcal{R}}_0^a<1$ to ensure that substance abuse is totally eradicated from the population.

Figure 2 is a numerical illustration of the dynamics of substance abuse sub-model (10) when ${\mathcal{R}}_0^a>1$ for various initial conditions. The figure reveals that when ${\mathcal{R}}_0^a>1$, substance abuse addictions will persist in the population, irrespective of the initial number of addicted individuals. Consequently, individuals in rehabilitation due to substance abuse as well as recovered and susceptible individuals will also persist in the population. This validates Theorem 5, which states that substance abuse sub-model (10) is locally asymptotically stable about the SAEE (12) provided ${\mathcal{R}}_0^a>1$. Based on these results, control interventions such as public enlightenment and rehabilitation of addicted persons should be implemented effectively such that ${\mathcal{R}}_0^a<1$; otherwise, substance abuse addictions will persist in the population irrespective of the initial number of addicted persons.

Figure 3 is a plot illustrating the impact of public enlightenment (${\kappa }_a$) on the dynamics of substance abuse sub-model (10). The figure shows that public enlightenment increases the number of susceptible individuals significantly and consequently leads to a decrease in the population of addicted individuals. Specifically, the figure revealed that the effective implementation of public enlightenment leads to the eradication of substance abuse addictions in the population. Thus, effective implementation of public enlightenment is strongly recommended to achieve a substance abuse-free society.

Figure 4 is a plot illustrating the impact of rehabilitation of addicted individuals (${\sigma }_a$) on the dynamics of substance abuse sub-model (10). The figure shows that rehabilitation of addicted individuals have significant impact on the dynamics of substance abuse. Specifically, an increase in the rate of rehabilitation of addicted individuals leads to a decrease in substance abuse addictions and consequently increases the number of susceptible individuals. The figure also revealed that the effective implementation of rehabilitation of addicted individuals leads to the prompt eradication of substance abuse addictions in the population and is therefore strongly recommended.

4.2. Numerical Simulation of Hepatitis B Sub-Model (22)

The numerical simulations for the hepatitis B sub-model (22) are presented in this section. This is crucial for understanding the dynamics and possible control interventions for a situation where there are only hepatitis B infections in the population.

Figure 5 is a graphical illustration of the dynamics of the hepatitis B sub-model (22) for various initial conditions when ${\mathcal{R}}_0^b<1$. The figure reveals that when ${\mathcal{R}}_0^b<1$, hepatitis B infections are eradicated irrespective of the initial number of infected individuals in the population. Consequently, the treated individuals are also eradicated while the susceptible and vaccinated individuals remain in the population. These results validate Theorems 6 and 7, which state that the hepatitis B sub-model (22) is both locally and globally asymptotically stable about the DFE (23) provided ${\mathcal{R}}_0^b<1$. Therefore, implementing effective control measures such as vaccination, treatment, and public enlightenment that will keep the ${\mathcal{R}}_0^b$ below unity is strongly recommended to eradicate hepatitis B infections from the population irrespective of the initial number of infections.

If the control measures are not effective enough such that ${\mathcal{R}}_0^b>1$, hepatitis B infections are likely to persist in the population. Figure 6 is a graphical illustration of the dynamics of the hepatitis B sub-model (22) for various initial conditions when ${\mathcal{R}}_0^b>1$. The figure reveals that when ${\mathcal{R}}_0^b>1$, hepatitis B infections persist in the population, irrespective of the initial number of infected individuals. Consequently, the treated, recovered and susceptible individuals also remain in the population validating Theorem 8, which states that in the absence of vaccination, hepatitis B sub-model (22) is locally asymptotically stable about the EE (26) provided ${\mathcal{R}}_0^b>1$. Therefore, effective hepatitis B vaccination coverage is strongly recommended to prevent the infections from becoming endemic in the population.

Public enlightenment is one of the methods of preventing the spread of hepatitis B infections. Figure 7 is a plot illustrating the impact of public enlightenment (${\kappa }_b$) on the dynamics of the hepatitis B sub-model (22). The figure reveals that public enlightenment has a significant impact on the overall dynamics (i.e, number of susceptible, vaccinated, infected, and treated individuals). Specifically, the figure reveals that effective implementation of public enlightenment increases the number of susceptible and vaccinated individuals and consequently leads to the eradication of hepatitis B infections. Thus, effective implementation of public enlightenment is strongly recommended for the eradication of hepatitis B infections in endemic areas.

Another method of controlling hepatitis B infection is through the treatment of infected individuals. Figure 8 is a plot illustrating the impact of treatment of infected individuals (${\sigma }_a$) on the dynamics of the hepatitis B sub-model (22). The figure reveals that treatment of infected individuals has a significant impact on hepatitis B infections, especially on the number of infected and treated individuals. Specifically, the figure reveals that effective implementation of treatment of infected individuals will lead to prompt eradication of hepatitis B infections, whereas the disease remains endemic if the treatment is not effectively implemented. Thus, effective treatment of infected individuals is strongly recommended for the possible eradication of hepatitis B infections in endemic regions.

Vaccination is one of the control strategies that is considered for preventing the transmission of hepatitis B infections, especially among the infants to improve their immune system against the infection. Figure 9 is a plot illustrating the impact of the vaccination rate ($\varphi$) on the dynamics of the hepatitis B sub-model (22). The figure shows that vaccination has an impact on hepatitis B infections, especially on the number of vaccinated, infected, and treated individuals. Specifically, the figure shows that the effective vaccination rate increases the number of vaccinated individuals and consequently leads to a decrease and eradication of hepatitis B infections if the vaccination coverage in very high. However, if the vaccination coverage is small, the disease remains endemic in the population. Thus, effective vaccination coverage is strongly recommended for the possible eradication of hepatitis B infections.

4.3. Numerical Simulation of Substance Abuse and Hepatitis B Co-Dynamics Model (1)

The numerical simulations for the substance abuse and hepatitis B co-dynamics model (1) are presented in this section. This is crucial for understanding the dynamics and possible control interventions for substance abuse and hepatitis B co-dynamics in endemic areas.

Figure 10 is a graphical illustration of the dynamics of the substance abuse and hepatitis B co-dynamics model (1) when ${\mathcal{R}}_0<1$. The figure reveals that the trajectories of the substance abuse and hepatitis B co-dynamics model (1) when ${\mathcal{R}}_0<1$ will converge at the addiction/disease-free equilibrium. This result validates Theorem 9, which states that the substance abuse and hepatitis B co-dynamics model (1) is locally asymptotically stable about the equilibrium point (27) provided ${\mathcal{R}}_0<1$. Therefore, when ${\mathcal{R}}_0<1$, all the addicted/infected compartments (such as addicted, infected, and treated) will be eradicated, while the susceptible and vaccinated individuals remain in the population. Thus, effective control measures that will keep the ${\mathcal{R}}_0$ below unity are strongly recommended for the possible eradication of substance abuse addiction and hepatitis B infections simultaneously in endemic areas.

Figure 11 is a graphical illustration of the dynamics of the substance abuse and hepatitis B co-dynamics model (1) when ${\mathcal{R}}_0>1$. The figure reveals that when ${\mathcal{R}}_0>1$, the trajectories of the substance abuse and hepatitis B co-dynamics model (1) converge to a possible positive equilibrium where both substance abuse addictions and hepatitis B infections persist. The trajectories in the figure reveal a possible endemic dynamics model (1) when both substance abuse and hepatitis B co-exists in a population. Therefore, if the control measures are not effective enough to keep ${\mathcal{R}}_0$ below unity, both addicted/infected compartments will remain endemic in the population. Thus, to eradicate substance abuse addictions and hepatitis B infections simultaneously in a population, effective control measures are strongly recommended.

Figure 12 is a plot illustrating the impact of public enlightenment ($\kappa ={\kappa }_a={\kappa }_b$) on the dynamics of the substance abuse and hepatitis B co-dynamics model (1). The figure shows that implementing public enlightenment in the entire population increases the number of susceptible and vaccinated individuals but decreases the number of addicted, treated, and recovered individuals. Specifically, effective implementation of public enlightenment leads to eradication of both substance abuse addictions and hepatitis B infections, whereas both substance abuse addictions and hepatitis B infections will remain endemic if control is not effectively implemented. Comparing this general case with the single scenarios revealed that the control measures have more significant impact on the single scenarios, leading to a faster eradication of either substance abuse or hepatitis B infection. Thus, more effort toward implementing the control measures is required for the eradication of substance abuse addictions and hepatitis B infections in the co-existence scenario.

Figure 13 is a plot illustrating the impact of joint rehabilitation of substance-addicted individuals and treatment of hepatitis B-infected persons ($\sigma ={\sigma }_a={\sigma }_b={\sigma }_{ab}$) on the dynamics of the substance abuse and hepatitis B co-dynamics model (1). The figure shows that implementing the simultaneous rehabilitation of substance-addicted individuals and treatment of hepatitis B-infected persons has some impact on both substance abuse addiction and hepatitis B infections. Specifically, effective implementation of joint rehabilitation of addicted individuals and treatment of hepatitis B-infected persons leads to eradication of both substance abuse addictions and hepatitis B infections. Comparing this general case with the single scenarios revealed that rehabilitation or treatment has a more significant impact on the single scenarios, leading to a faster eradication of either substance abuse addictions or hepatitis B infections for the single-epidemic scenario. Thus, more effort toward implementing the joint rehabilitation of substance-addicted individuals and treatment of hepatitis B-infected persons is required for prompt epidemic eradication in the co-dynamics epidemic scenario.

5. Discussion

Substance abuse and hepatitis B are two public health issues that are currently affecting many regions of the world, especially in the areas where both epidemics co-exist. Even though the two problems are distinct, individuals who are addicted to substance abuse have been linked with several abnormal behaviors, including having unprotected sex, which is one of the major routes of transmission of the hepatitis B virus. This study is our attempt to use a mathematical model to study the co-dynamics of substance abuse and hepatitis B and investigate the impact of control strategies.

A mathematical model for the co-dynamics of substance abuse and hepatitis B with their possible control strategies was developed. Special cases of the model where only substance abuse (i.e, substance abuse sub-model) or hepatitis B (i.e, hepatitis B sub-model) are considered to gain insight into the situation where either substance abuse or hepatitis B is endemic in the population. The qualitative analysis of these two special cases was carried out systematically. For the substance abuse sub-model, the important mathematical features of the sub-model were determined and analyzed accordingly. For instance, the existence of a substance abuse-free equilibrium point, the existence of a substance abuse endemic equilibrium point, and the existence of a threshold quantity (${\mathcal{R}}_0^a$) that determines whether the substance abuse will be eradicated or remain endemic were determined. The local and global stability of the substance abuse-free equilibrium point revealed that substance abuse addictions will be eradicated from the population irrespective of the initial number of addicted persons provided that the control measures are effective enough to keep ${\mathcal{R}}_0^a$ below unity. However, if the control measures are not effective enough to keep ${\mathcal{R}}_0^a$ below unity, the local stability of the substance abuse endemic equilibrium shows that substance abuse will persist and remain endemic in the population. These analytical results of the substance abuse sub-model were validated numerically by showing that the trajectories of the substance abuse sub-model converge to the substance abuse free equilibrium point or substance abuse endemic equilibrium point for various initial conditions when ${\mathcal{R}}_0^a<1$ or ${\mathcal{R}}_0^a>1$, respectively. The impact of control measures on this special case revealed that effective implementation of public enlightenment or rehabilitation of addicted individuals decreases and consequently leads to prompt eradication of substance abuse addictions in the population.

Similarly, the analysis of the hepatitis B sub-model is conducted systematically. Some of the important mathematical features of the hepatitis B sub-model includes: existence of disease free equilibrium point, existence of an endemic equilibrium point, existence of the basic reproduction number (${\mathcal{R}}_0^b$). We showed that hepatitis B will be eradicated from the population irrespective of the initial number of infections, provided the control measures are effective enough to keep ${\mathcal{R}}_0^b$ below unity. This was established analytically by proving that the hepatitis B sub-model disease-free equilibrium point is both locally and globally asymptotically stable provided that ${\mathcal{R}}_0^b<1$. However, in the absence of vaccination, we discovered that hepatitis B persists and remains endemic in the population. This was also established analytically by showing that hepatitis B endemic equilibrium is locally asymptotically stable when ${\mathcal{R}}_0^b>1$. The analytical (stability) results of the hepatitis B sub-model were validated numerically. The impact of control measures on this special case revealed that effective implementation of public enlightenment, treatment of infected individuals, or vaccination leads to prompt eradication of hepatitis B infections in the population.

Next, the analysis of the general case of the substance abuse and hepatitis B co-dynamics model was conducted. The important mathematical features of the substance abuse and hepatitis B co-dynamics model, such as the existence of disease free equilibrium point and the basic reproduction number (${\mathcal{R}}_0$), were determined. The local stability of the substance abuse and hepatitis B co-dynamics model free equilibrium point revealed that both substance abuse and hepatitis B infections can be eradicated if the initial number of addicted users and hepatitis B-infected persons is sufficiently small and ${\mathcal{R}}_0<1$. This result was validated numerically by showing that the trajectories of both substance abuse addiction and hepatitis B infection will converge to a substance abuse addiction and hepatitis B infection-free equilibrium when ${\mathcal{R}}_0<1$. On the contrary, if the initial substance-addicted population and number of hepatitis B-infected persons are large, the global instability shows that substance abuse and hepatitis B infections will be difficult to eradicate by just lowering the ${\mathcal{R}}_0$ below unity. When the control measures are not effectively implemented such that ${\mathcal{R}}_0>1$, the trajectories of both substance abuse and hepatitis B infections converge to some possible positive number, indicating the endemicity of substance abuse and hepatitis B. The impact of public enlightenment and joint rehabilitation of addicted individuals/treatment of hepatitis B-infected persons on the dynamics of the substance abuse and hepatitis B co-dynamics model was explored numerically. Effective implementation of public enlightenment has some impact in reducing substance abuse addictions and hepatitis B infections, including their co-existence, and consequently leads to their eradication after a period of continuous implementation. However, using the same number of interventions on the single scenarios of substance abuse or hepatitis B infection has a greater impact on the single scenarios, leading to a faster eradication of either substance abuse or hepatitis B infections. This supports the severity of co-infections of disease which is often more difficult to control compared to single infection. Thus, more control effort is required to eradicate the co-existence of substance abuse addictions and hepatitis B infections in endemic areas.

In conclusion, this research offers a unique perspective in that it presents a mathematical model that explicitly accounts for the co-existence of substance abuse and hepatitis B. The study further integrates multiple control measures, including public enlightenment, rehabilitation of substance-addicted persons, treatment of hepatitis B-infected persons, and vaccination of susceptible individuals. By revealing how co-existence fundamentally alters system behavior and control effectiveness, this research fills a critical gap in the literature and offers new insights for designing effective control management strategies.