Mathematical modeling of HIV-HCV co-infection model: Impact of parameters on reproduction number

Introduction

Emerging and reemerging infectious illnesses are of public health importance, and mathematics has traditionally been employed to acquire a realistic understanding into the transmission dynamics and control of these diseases. Both Hepatitis C Virus (HCV) and Human Immunodeficiency Virus (HIV) are considered blood-borne viruses because they are spread through contact with the blood of an infected individual.¹ In 2017, 2.3 million people living with HIV were simultaneously infected with HCV, according to the World Health Organization (WHO, 2016). Infectious diseases like HIV and HCV have become critical problems in public health around the world. Africa and South and East Asia bear the heaviest brunt of these co-infections (WHO, 2017). Co-infection with HIV and another disease usually poses greater dangers and has more dire outcomes for individuals. When HIV is present alongside HCV, the viral infection advances much more quickly in the latter. If the CD4 cells is less than 200 cells/mm³, the risk of severe liver injury increases.² Hepatocellular carcinoma, liver cirrhosis, and liver-related mortality are also more likely to occur.³ The international community agrees that strong leadership in the form of well-thought-out programs and policies that focus on prevention, early diagnosis, therapies that respects patients' rights, and high-quality, universally accessible health care is needed to stop the spread of HIV. Concerning co-infection, there have been reports of effective HCV drug combinations in treating people who are both HIV positive and HCV positive. Furthermore, HIV can be treated successfully in the majority of people with HCV.⁴ New antiviral medications have the potential to treat HCV in persons who are HIV-positive and infected with HIV, but additional research is needed to prove their effectiveness.

There are about 40 million PLHIV in the world right now. UNAIDS, the United Nations Program on HIV/AIDS, estimates that in 2020, more than one person every minute would die from an AIDS-related illness.⁵ HIV and HCV can be spread in many ways, such as through injections, sexual contact, and being passed down from parent to child.⁶ People with HIV often also have HBV and/or HCV.^7,8 One of the main reasons people with HIV die is because of liver disease.^9,10 There are over 2 million PLHIV on a global scale who are living with HBV or HCV.^1,7,8 Bi-directional effects explain why people who have HIV who also have HBV and/or HCV have a greater risk of becoming sick and die. HIV patients with HBV and/or HCV quickly develop AIDS,¹¹ and antiretroviral drugs are more harmful to the.^12–14 On the other hand, when PLHIV change their immune response, it leads to less HCV viral clearance, reactivation, and replication in co-infected individual. Aspartate aminotransferase, alanine aminotransferase, and alkaline phosphatase levels rise as a result, and chronic liver disease complications like cirrhosis, hepatic decompensation, and hepatocellular carcinoma as well as a higher death rate^15–17 progress more quickly. People living with HIV who also have HBV, HCV, or both have a greater risk of infection transmission. However, there has been relatively little deterministic study of HCV chronic infection co-infected with HIV. For instance, Ref. 18, introduced and analyzed a deterministic model for HCV and HIV co-infection. Focusing on HCV and HIV co-infection, they hope to better understand the long- and short-term dynamics of both diseases and develop methods for forecasting whether HCV and HIV will eventually become extinct or remain a persistent problem. To ascertain the effect of treatment on the dynamics of each disease, in¹⁹ built and investigated a mathematical model of the co-dynamics of the HCV and HIV/AIDS. The equilibria (disease-free and endemic) are described under which they are both locally and globally asymptotically stable. Similarly, in Ref. 20, investigated mathematical model of co-infection with HIV and HCV. In the case of HIV, the innovation of their strategy is the incorporation of therapy for both infections as well as how it is passed from mother to kid.²¹ Constructed a mathematical model of HCV/HIV co-infection within the host by modifying a model of HCV mono-infection that had previously been published to include an immune system component in infection clearance. They then combined a decline in immunological function with an increase in HIV viral load to examine the impact of HIV co-infection on spontaneous HCV clearance and sustained virologic response (SVR). Also, Ref. 22, through mathematical, created a new co-infection model for the hepatitis C virus (HCV) and human immunodeficiency virus (HIV) (HIV). Examining therapy for both diseases, Additionally, using mathematics,²² developed a new co-infection model for the human immunodeficiency virus (HIV) and hepatitis C virus (HCV) (HIV). Examining prevention, diagnosis, screening, HIV knowledge and awareness, condom use, and largely using numerical simulations, ignorance and awareness, and condom use and mostly employs numerical simulations. In, Ref. 23, constructed two ODE models at the population level to mimic the progression of the HCV and HIV among PWID. Both deterministic and stochastic solutions were used to solve the models describing HCV and HIV parenteral transmission. Additionally, several deterministic models that are relevant to our work have been suggested and examined in Refs. 24–29.

The HIV and HCV co-infection model

The paradigm of co-infection between HIV and HCV is described in this section.

We determine overall and submodel reproduction rates (HIV only and HCV only models). We investigate global and full model disease-free equilibrium local stability. We determine the reproduction number's sensitivity indices to important model parameters. Simulation diagrams created using Runge-kutta order four embedded in maple 2020.1 software and contour plots created using maple 2020.1, help understand the model's dynamics.

Full model description

The mathematical model that will be considered and investigated is divided into (15) different groups, namely, the susceptible populace for both HIV and HCV $S\left(t\right)$, the HIV-infected unaware $H_u\left(t\right)$, the HIV-infected aware, $H_A\left(t\right)$, HIV on treatment $H_T\left(t\right)$, the AIDS populace aware and on treatment $A_A\left(t\right)$, acutely infected $I_c\left(t\right)$ and chronically infected $C_c\left(t\right)$ infected HCV, HIV-unaware co-infected with acute and chronic HCV ($H_{uI}\left(t\right)\mathit{\text{and}}{H}_{\mathit{uC}}\left(t\right)),$ HIV-aware co-infected with acute and chronic HCV ($H_{\mathit{AI}}\left(t\right)\mathit{\text{and}}{H}_{\mathit{AC}}\left(t\right)),$ HIV-positive individuals receiving treatment for HIV who are co-infected with acute and chronic HCV ($H_{\mathit{TI}}\left(t\right)\mathit{\text{and}}{H}_{\mathit{TC}}\left(t\right)),$ HIV-positive individuals in stage-IV co-infected with acute and chronic HCV $\left(A_{\mathit{AI}}\left(t\right)\mathit{\text{and}}{A}_{\mathit{AC}}\left(t\right)\right)$.

The overall population at time t, represented by $N\left(t\right),$is classified into the 15 classes/subgroups listed in Tables of Nomenclature, each of which corresponds to a different epidemiological status.

(1)

$$N\left(t\right)=S\left(t\right)+H_u\left(t\right)+H_A\left(t\right)+H_T\left(t\right)+A_a\left(t\right)+I_c\left(t\right)+C_c\left(t\right)+H_{UI}\left(t\right)+H_{\mathit{AI}}\left(t\right)+H_{\mathit{TI}}\left(t\right)+H_{\mathit{UC}}\left(t\right)+H_{\mathit{AC}}\left(t\right)+H_{\mathit{TC}}\left(t\right)+A_{\mathit{AI}}\left(t\right)+A_{\mathit{aC}}\left(t\right)$$

In Figure 1, the epidemiology of co-infection with HIV and HCV is depicted schematically. The many compartments (circles) symbolize the various disease phases, and the arrows depict how people progress from one phase to the next. At time t, susceptible individuals S are assumed to enter the population at a constant rate, $\left(1-\phi H_u\right)\Lambda .$ Some newborns acquire HIV at parturition and are subsequently enrolled directly into the infectious class, $H_u$ where $\phi$, is the rate of newborn HIV infection and $\Lambda$ is the rate of recruitment through immigration or emigration. Individuals in all classes die at a consistent natural mortality rate, $\mu .$ Individuals with AIDS $\left(A_a,A_{\mathit{aI}},A_{\mathit{aC}}\right)$ have an extra death rate owing to AIDS, ${\mathrm{d}}_a.$We assume that HIV-infected people who are receiving treatment do not spread the virus.^30,31 Despite the complexity of disease co-dynamics, we will make the simple assumption that co-infected and mono-infected people can only transmit one of the two diseases—HIV or HCV—at a time. Individual $S$, who is susceptible to HIV infection, is at risk of acquiring HIV infection at a rate of ${\lambda }_H$, (force of infection related to HIV) when in contact with the $H_U,H_A,\mathit{\text{and}}{A}_a$ populations, where

(2)

$${\lambda }_H=c_h\left(1-\mathit{\psi \xi }\right)b_h\frac{H_U\left(t\right)+A_A\left(t\right)+{\kappa }_1\left(H_{UI}\left(t\right)+H_{\mathit{uC}}\left(t\right)\right)}{N}$$

Figure 1. The compartmental flow diagram of the HIV-HCV co-infection.

The parameter $b_h$ is the chance that a person will get HIV from a contact, and the parameter, the average annual number of sexual partners for someone at risk of contracting HIV is $c_h$. To highlight the usage of condoms as a crucial prevention measure, We presume that $\mathit{\psi \xi }\in \left[0,1\right]$indicates the degree of condom protection. If $\xi =0,$ condom use offers no protection, $\xi =1$ denotes perfect protection, where $\psi$ is the use of a condom.

When compared to persons who are only infected with HIV, the relative infectiousness of persons who are acutely infected with HCV and unaware of their HIV infection $\left(H_{\mathit{UI}}\right)$ and individuals who are chronically infected with HCV and AIDS $\left(A_{\mathit{Ac}}\right)$, is accounted for by the parameters ${\kappa }_1>1$. We make the assumption that persons who are co-infected are approximately three times more infectious than individuals who just have one infection.^32,33 HIV unaware class $,H_U,H_{\mathit{UI}},H_{\mathit{UC}}$ singly and dually infected with HCV advances to HIV diagnosed class $H_A,H_{\mathit{AI}},H_{\mathit{AC}}$ after testing at a rate, ${\alpha }_1,{\alpha }_2,{\alpha }_3$and those in aware HIV was enrolled on therapy at the rate ${\theta }_1,{\theta }_3,{\theta }_5$ in class $H_T,H_{\mathit{TI}},H_{\mathit{TC}}.$Nevertheless, some individuals who were placed on HIV treatment default from or drop out of the HAART treatment²⁵ after which they develop AIDS due to drug resistance and progress to class $A_A,A_{\mathit{AI}},A_{\mathit{AC}}$ at a rate ${\upsilon }_1,{\upsilon }_2,{\upsilon }_3.$ People with HIV and HCV who don't know their HIV status, $H_U,H_{\mathit{UI}},H_{\mathit{UC}}$ and didn't get tested move to the AIDS class $A_A,A_{\mathit{AI}},A_{\mathit{AC}}.$at a rate ${\rho }_1,{\rho }_2,{\rho }_3$, People with AIDS symptoms singly and dually infected with HCV are given treatment at a rate of ${\theta }_2,{\theta }_4,{\theta }_6$ respectively. AIDS infected can respond well to treatment and return to $H_T,H_{\mathit{TI}},H_{\mathit{TC}}$^18,20 and die because of AIDS at an incidence $\mathit{da}.$

Susceptible people get HCV infection from people in the $I_c,C_c,H_{\mathit{UI}},H_{\mathit{UC}}$at a rate of ${\lambda }_C$ where ${\lambda }_C$is the risk of getting HCV, which is given by

(3)

$${\lambda }_C=c\left(1-\mathit{\psi \xi }\right)b_c\frac{I_c\left(t\right)+C_c\left(t\right)+{\kappa }_2\left(H_{UI}\left(t\right)+H_{\mathit{uC}}\left(t\right)\right)}{N}$$

To simulate the reality that individuals who are dually infected are more infectious than the mono-infected, we use the notation ${\kappa }_2>1$, where $b_c$the likelihood that contact will result in HCV infection.^19,34,35

People who are only infected with HIV $\left(H_U,H_A,H_T,\mathit{\text{and}}{A}_a\right)$ acquired HCV at a rate $\left({\delta }_1{\lambda }_c,{\delta }_2{\lambda }_c,{\delta }_3{\lambda }_c\right)$ and moved to classes $\left(H_{\mathit{UI}},H_{\mathit{AI}},H_{\mathit{TI}},A_{\mathit{AI}}\right)$, an increased risk of HCV acquisition is accounted for by the modification parameter ${\delta }_1,{\delta }_2,{\delta }_3>1$. HCV-only infected people $\left(\mathit{Ic},C_c\right)$are more likely to obtain HIV $\left(H_{\mathit{uI}},H_{\mathit{uC}}\right)$ than people who are only infected with HCV at a rate ${\mathit{\gamma \lambda }}_H,{\mathit{\tau \lambda }}_H$ where $\gamma ,\tau >1$translates to an increased chance of contracting HIV for people whose immune systems are weakened by HCV.

HIV and AIDS patients, dually infected with the acute HCV $H_{\mathit{UI}},H_{\mathit{TI}},A_{\mathit{AI}};$at a rate $\eta ,$ becomes chronically infected and are treated for chronic HCV epidemic at $\left(r_i,i=1,2,\dots \right)$ while the remaining populace $,\omega$ spontaneously clear the virus to return to susceptible class $S$. We then assume that an individual whose immune system helps in clearing the virus can become re-infected at rate ${\lambda }_C$ if expose or engage in risk behaviors such as injection drug use,³³ drinking alcohol,³⁶ multiple sex partners and sex between two men³⁷ since the clearance does not confer permanent immunity.³⁸

An HCV-positive person stays acutely infected for an average of $1/{\sigma }_c$ days. Since newer combinations of direct-acting antivirals (DAAs) have showed high cure rates of 90%-95% in phase II and III clinical trials, we did not take treatment failure for chronic HCV carriers into account. However, researchers are beginning to report sporadic incidences of treatment failure in HCV.^33,39,40

In people with HIV and HCV co-infection, little is known regarding the relationship between spontaneous HCV clearance and sustained HIV infection control.⁴¹ Co-infection reduces the possibility of the acute HCV virus clearing itself naturally.^33,42,43 Because HIV speeds up the development of HCV, a high viral load for this virus may also indicate a rapid progression of liver disease.^33,37,43 In order to take into account the additional viral load resulting from co-infection, we use the term ${\epsilon }_1$ to impact spontaneous clearance and the term ${\epsilon }_2$to accelerate the disease progression, due to co-infection.³⁰ Due to the fact that HCV and HIV-1 are spread through the same ways, about 10–15 percent of acute HCV infections clear up on their own, but less than 10 percent of HIV-1 infections do. The compartmental flow diagram for the HIV-HCV co-infection model is depicted in (Figure 1).

Table 1. List of nomenclature for HIV-HCV Model (4) are described as follows.

Variable/Parameter	Description
$S\left(t\right)$	Susceptible Individuals
$H_U\left(t\right)$	Unaware HIV individuals
$H_A\left(t\right)$	Aware HIV individuals
$H_T\left(t\right)$	HIV on Treatment Individuals
$A_A\left(t\right)$	AIDS individual
$I_C\left(t\right)$	Acute HCV Individual
$C_C\left(t\right)$	Chronic HCV Individuals
$H_{\mathit{UI}}\left(t\right)$	Unaware HIV individual co-infected with Acute HCV
$H_{\mathit{AI}}\left(t\right)$	Aware HIV individual co-infected with Acute HCV
$H_{\mathit{TI}}\left(t\right)$	HIV individual on treatment co-infected with Acute HCV
$H_{\mathit{UC}}\left(t\right)$	Unaware HIV individual co-infected with Chronic HCV
$H_{\mathit{AC}}\left(t\right)$	Aware HIV individual co-infected with Chronic HCV
$H_{\mathit{TC}}\left(t\right)$	HIV individual on treatment co-infected with Chronic HCV
$A_{\mathit{AI}}\left(t\right)$	AIDS patient with acute HCV
$A_{\mathit{AC}}\left(t\right)$	AIDS patient with chronic HCV
$\Lambda$	Recruitment rate
$\omega$	Spontaneous clearance for Acute HCV
$\eta$	Progression rate from Acute to Chronic HCV/Non-spontaneous clearance rate
$r_i,i=1,2,\dots$	HCV treatment rate for HCV
${\lambda }_H$	Infectiousness connected to HIV infection
${\lambda }_C$	Infectiousness connected to HIV infection
$\gamma$	Modification parameter for Acute HCV
$\tau$	Modification parameter for Chronic HCV
${\alpha }_1,i=1,2,3$	HIV testing rate
${\delta }_{i,}i=1,2,\dots$	Modification parameter
${\epsilon }_1$	Factor that influences spontaneous HCV clearance in the presence of co-infection.
${\epsilon }_2$	Factor that accelerate HCV disease progression in presence of co-infection
$\phi$	HIV infection rate among infants.
${\rho }_i,i=1,2,3$	Progression rate from unaware HIV to AIDS
${\theta }_{i,}i=1,2,3,\dots$	HIV/AIDS treatment rate
${\upsilon }_{i,}i=1,2,3$	HIV defaulters from treatment rate (progression rate from aware HIV to AIDs)
$\mu$	Natural Mortality
$d_a$	Mortality due to AIDS
$d_c$	Mortality due to HCV
$\frac{1}{{\sigma }_c}$	Average time a person infected with HCV remains in an acute infection condition.
$c_C$	HCV contact rate
$c_h$	HIV contact rate
$b_h$	Transmission Coefficient for HIV
$b_c$	Transmission Coefficient for HIV

Mathematically, the flow chart leads to the 15 systems of ordinary differential equations listed below:

(4)

$$\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=\left(1-{\mathit{\phi H}}_u\right)\Lambda +{\omega }_0{\sigma }_c{I}_c+r_1{C}_c-\left({\lambda }_H+{\lambda }_C+\mu \right)S\\ \frac{d{H}_U}{\mathit{dt}}={\lambda }_{H}S+\mathit{\phi \Lambda }{H}_U+\omega {ϵ}_1{\sigma }_c{H}_{\mathit{UI}}+r_2{H}_{\mathit{UC}}-\left({\delta }_1{\lambda }_C+{\alpha }_1+{\rho }_1+\mu \right)H_U\\ \frac{d{H}_A}{\mathit{dt}}={\alpha }_1{H}_U+\omega {ϵ}_1{\sigma }_c{H}_{\mathit{AI}}+r_3{H}_{\mathit{AC}}-\left({\delta }_2{\lambda }_C+{\theta }_1+\mu \right)H_A\\ \frac{d{H}_T}{\mathit{dt}}={\theta }_1{H}_A+\omega {ϵ}_1{\sigma }_c{H}_{\mathit{TI}}+r_4{H}_{\mathit{TC}}+{\theta }_2{A}_A-\left({\delta }_3{\lambda }_C+\mu +{\upsilon }_1\right)H_T\\ \frac{d{A}_A}{\mathit{dt}}={\rho }_1{H}_U\mathbf{+}{\upsilon }_1{H}_T+\omega {ϵ}_1{\sigma }_c{A}_{\mathit{AI}}+r_5{A}_{\mathit{Ac}}-\left({\delta }_4{\lambda }_C+\mu +d_a+{\theta }_2\right)A_A\\ \frac{d{I}_C}{\mathit{dt}}={\lambda }_{C}S-\left({\omega }_0+{\eta }_0\right){\sigma }_c{I}_C-\left(\gamma {\lambda }_H+\mu \right)I_C\\ \frac{d{C}_c}{\mathit{dt}}={\eta }_0{\sigma }_c{I}_c-\left(\tau {\lambda }_H+\mu +d_c+r_1\right)C_c\\ \frac{d{H}_{\mathit{UI}}}{\mathit{dt}}={\delta }_1{\lambda }_C{H}_U+\gamma {\lambda }_H{I}_c-\left(\eta {ϵ}_2{\sigma }_c+{\alpha }_2+\omega {ϵ}_1{\sigma }_c+{\rho }_2+\mu \right)H_{\mathit{UI}}\\ \frac{d{H}_{\mathit{AI}}}{\mathit{dt}}={\alpha }_2{H}_{\mathit{AI}}+{\delta }_2{\lambda }_C{H}_A-\left(\eta {ϵ}_2{\sigma }_c+{\theta }_3+\omega {ϵ}_1{\sigma }_c+\mu \right)H_{\mathit{AI}}\\ \frac{d{H}_{\mathit{TI}}}{\mathit{dt}}={\theta }_3{H}_{\mathit{AI}}+{\delta }_3{\lambda }_C{H}_T+{\theta }_4{A}_{\mathit{AI}}-\left(\eta {ϵ}_2{\sigma }_c+{\upsilon }_3+\omega {ϵ}_1{\sigma }_c+\mu \right)H_{\mathit{TI}}\\ \frac{d{H}_{\mathit{UC}}}{\mathit{dt}}=\tau {\lambda }_H{C}_c+\eta {ϵ}_2{\sigma }_c{H}_{\mathit{UI}}-\left(r_2+{\alpha }_3+{\rho }_3+\mu +d_c\right)H_{\mathit{UC}}\\ \frac{d{H}_{\mathit{AC}}}{\mathit{dt}}={\alpha }_3{H}_{\mathit{UC}}+\eta {ϵ}_2{\sigma }_c{H}_{\mathit{AI}}-\left(r_3+{\theta }_5+\mu +d_c\right)H_{\mathit{AC}}\\ \frac{d{H}_{\mathit{TC}}}{\mathit{dt}}={\theta }_5{H}_{\mathit{AC}}+\eta {ϵ}_2{\sigma }_c{H}_{\mathit{TI}}+{\theta }_6{A}_{\mathit{AC}}-\left(r_4+{\upsilon }_3+\mu +d_c\right)H_{\mathit{TC}}\\ \frac{d{A}_{\mathit{AI}}}{\mathrm{d}t}={\delta }_4{\lambda }_C{A}_A+{\rho }_2{H}_{\mathit{UI}}+{\upsilon }_2{H}_{\mathit{TI}}-\left(\eta {ϵ}_2{\sigma }_c+{\theta }_4+\omega {ϵ}_1{\sigma }_c+\mu +d_a\right)A_{\mathit{AC}}\\ \frac{d{A}_{\mathit{AC}}}{\mathit{dt}}=\eta {ϵ}_2{\sigma }_c{A}_{\mathit{AI}}+{\rho }_3{H}_{\mathit{UC}}+{\upsilon }_3{H}_{\mathit{TC}}-\left(r_5+{\theta }_6+\mu +d_a+d_c\right)A_{\mathit{AC}}\end{array}$$

Points of equilibrium, reproduction numbers and the stability analyses

In this section, computation of disease-free equilibrium (DFE) and the endemic equilibrium (EE) will be carried out, and their stability will be examined using associative reproduction number.

Disease-free equilibrium and the effective reproduction number

In this part, we calculate model $R_0^{\prime }s$ RN.

The effective reproduction number $R_{\mathit{eHC}}$, is known as the spectral radius of the next generation matrix,⁴⁶ governs ${E_{\mathit{oHC}}}^{\prime }s$ linear stability. In the presence of a strategic intervention, the effective reproduction number is frequently understood as the estimated number of secondary infections produced by a single infectious individual during his/her entire infectious phase. Nevertheless, in the suggested model, the infectious persons can be classified into any of these fourteen classes $H_U,H_A,H_T,A_A,I_C,C_C,H_{\mathit{UI}},H_{\mathit{UC}},$ $H_{\mathit{AI}},H_{\mathit{AC}},H_{\mathit{TI}},H_{\mathit{TC}},$ $A_{\mathit{AI}},A_{\mathit{AC}}$with the estimated count of secondary infections varying according to the class. Model's effective reproduction number (the total sum of secondary infections caused by HIV or HCV infected individual throughout the full contagious period in the context of treatment) is given using the latter technique as.

We will now estimate the reproduction number, $R_{\mathit{eHC}}$, of the entire model (4). Model (4)'s infection-free equilibrium state $E_{0\mathit{HC}}$ is given by:

On system (4), we evaluate the matrices for the new transmittable terms $F$, the terms $V$, and matrix ${\mathit{FV}}^{-1}$, based on submission in (1) – (4) above. The reproduction number is then the spectral radius of ${\mathit{FV}}^{-1}$. $R_{0\mathit{HC}}$ is given after some mathematical manipulation (please see the Appendix for a complete proof):

Where $k_1=\mu +{\alpha }_1+{\rho }_1,k_2=\mu +{\theta }_1,k_3=\mu +{\nu }_1,k_4=\mu +d_a+{\theta }_2,k_5=\mu +d_c+r_1$

The following lemma is derived from Theorem 2 of Ref. 46.

By evaluating the two model sub-models listed below.

Model (9) is obtained from model (4) by equating to zero the variables pertaining to HIV dynamics $H_U=H_A=H_T=A_A=H_{\mathit{UI}}=H_{\mathit{UC}}=H_{\mathit{AI}}=H_{\mathit{AC}}=H_{\mathit{TI}}=H_{\mathit{TC}}=A_{\mathit{AI}}=A_{\mathit{AC}}=0$, while model (12) is developed from model (4) by setting to zero the variables pertaining to HCV dynamics ($I_C=C_C=H_{\mathit{UI}}=H_{\mathit{UC}}=H_{\mathit{AI}}=H_{\mathit{AC}}=H_{\mathit{TI}}=H_{\mathit{TC}}=A_{\mathit{AI}}=A_{\mathit{AC}}=0).$ We now compute the system's reproduction number, $R_{\mathit{HIV}}$ (5). We employ the method of the next generation matrix in Ref. 46.

Where ${\lambda }_H=c\left(1-\mathit{\psi \xi }\right)b_h\frac{H_U+A_A}{N_h},$ with total population given as

Disease-free equilibrium (DFE) evaluation of F and V generational matrices is given by

Using Ref. 22, the new infection terms matrices F, and the terms, V, are as follows:

The matrix ${\mathit{FV}}^{-1\prime }s$ eigenvalues are as follows:

The associative basic reproduction number is stated as:

where $\rho$ stands for spectral radius of ${\mathit{FV}}^{-1}$. The following lemma is derived from Theorem 2, Ref. 46.

We then derive the reproduction number, $R_{\mathit{eC}}$, of model (12).

Where ${\lambda }_C=c\left(1-\mathit{\psi \xi }\right)b_c\frac{I_C+C_C}{N_c},$where $N_c$ is the total number of people given as

A state of HCV-free equilibrium for the system of equations in (12) is obtained by:

Using Ref. 22, the new infection terms matrices F, and the terms, V, are thus:

The matrix ${\mathit{FV}}^{-1\prime }s$ eigenvalues are as follows:

The associative basic reproduction number is written as:

The endemic equilibria and stability

The following endemic equilibrium states are available in model system (4):

The global stability of the disease free equilibria

Computation of global stability of the disease-free equilibrium of the whole model (4) is done in this section. To start, we will calculate the stability of the disease-free equilibria of both of the sub-models (9) and (12).

Since the disease-free $H_U=H_A=H_T=A_A=0\to \left(0,0,0,0\right)$ and $S_h\le N_h$, as $t\to \infty$ in ${\Gamma }_h$, $F$ and $V$ are defined as described for system (9) in section 2.2.1. Thus,

If $R_{0H}<1$, then $\rho \left(F-V\right)<1$, which is the same as stating that all eigenvalues of the matrix $F-V$lie in the left-half plane.⁴⁶ Therefore, the linear system described by the equality (23) is stable anytime $R_{0H}<1$ and $H_U=H_A=H_T=A_A=0\to \left(0,0,0,0\right)\mathrm{as}t\to \infty$for this linear ordinary differential equation (ODE) system. As a result of employing a basic comparison theorem,^49–51 we obtain $H_U=H_A=H_T=A_A=0\to \left(0,0,0,0\right)$for the nonlinear system (9) represented by the last four equations of the system. We construct a linear system with $S\left(t\right)=\frac{\Lambda }{\mu }$ by inserting $H_U=H_A=H_T=A_A=0$ into the first equation of model (9). Thus, $\left(S\left(t\right),H_U,H_A,H_T,A_A\right)\to \left(\frac{\Lambda }{\mu },0,0,0,0\right)\mathrm{as}t\to \infty$ for $R_{0H}<1$, so $E_{0H}$is asymptotically stable globally if $R_{0H}<1$.

Now, we follow the same approach to compute the global stability of the disease-free equilibrium of the sub model (9).