Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy BIOMATH https://biomath.math.bas.bg/biomath/index.php/biomath B f Biomath Forum ORIGINAL ARTICLE Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy Edison Mayanja1,2,∗, Livingstone S. Luboobi3, Juma Kasozi1, Rebecca N. Nsubuga4 1Department of Mathematics, Makerere University, Kampala, Uganda edisonmayanja.phd@gmail.com 0000-0002-8077-3083 kasozi@cns.mak.ac.ug 0000-0002-0941-9604 2Department of Economics and Statistics, Kabale University, Kabale, Uganda 3Independent Researcher, C/O Department of Mathematics, Makerere University, Kampala, Uganda luboobi@gmail.com 0000-0003-0438-9845 4Independent Researcher, Kampala, Uganda nrebeccansubuga@gmail.com 0000-0001-8527-6222 Received: November 27, 2021, Accepted: July 15, 2022, Published: August 11, 2022 Abstract: In this work, we formulated and analysed a deterministic model to study the HIV-HCV co-infection dynamics in presence of HIV therapy. The HCV chronic stage was split into two periods: the period before and the period after onset of cirrhosis. This was done because the HCV chronic stage of infection is long, asymptomatic and infectious. The effective reproduction numbers, one of our outcome measures, were computed using the next generation matrix method. Numerical simulations were performed to support the analytical results from the model. The different parameters in the model were sub- jected to a sensitivity analysis to determine their relative importance on the HIV-HCV co-infection dynamics. The results indicated that both HIV and HCV infections enhance each other; and in the long run, increasing the rates at which people are put on HIV treatment reduces the prevalence of HCV in the community; however, it increases the prevalence of HIV. Therefore, there should be increased safer sexual behaviour campaigns among individuals on HIV treatment. Keywords: HIV/AIDS, HCV, co-infection, reproduction number, sensitivity analysis, therapy I. INTRODUCTION The human immunodeficiency virus (HIV) and hep- atitis C virus (HCV) are a global challenge. These two viruses have a considerable impact on the global morbidity and mortality. HIV is a virus that weakens the immune system by destroying the CD4+ T-cells and hence making it harder for the body to fight off other infections. Hepatitis C infection is a liver disease caused by hepatitis C virus (HCV). Globally, chronic HCV infection is the leading cause of chronic liver disease, and is associated with cirrhosis, hepatocellular carcinoma, and liver failure related mortality [1]. Both HCV and HIV are blood borne viruses caught through exposure to HCV and HIV infected blood, respectively; having common routes of transmission [2, 3], namely by: injection drug use, sexual contact, mother to child transmission during pregnancy or birth, blood and blood products transfusion, organ transplants from infected Copyright: © 2022 Edison Mayanja, Livingstone S. Luboobi, Juma Kasozi, Rebecca N. Nsubuga. This article is distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *Corresponding author Citation: Edison Mayanja, Livingstone S. Luboobi, Juma Kasozi, Rebecca N. Nsubuga, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy, Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 1/15 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:edisonmayanja.phd@gmail.com https://orcid.org/0000-0002-8077-3083 mailto:kasozi@cns.mak.ac.ug https://orcid.org/0000-0002-0941-9604 mailto:luboobi@gmail.com https://orcid.org/0000-0003-0438-9845 mailto:nrebeccansubuga@gmail.com https://orcid.org/0000-0001-8527-6222 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy donors, and exposure to blood by health care profes- sionals [3]. Several treatment options of HCV exist, however, infection with HCV takes long to manifest unlike infection with HIV. Therefore, most of the individuals infected with HCV may not be aware that they are infected until many years later [4], and thus would not seek for treatment. Screening, diagnosis and treatment of HCV infected individuals remains a global challenge. Globally, in most countries there is: lack of prioritiza- tion and hence lack of dedicated budgets or programmes for hepatitis C [5]. According to the Global hepatitis report of 2017 [6], it was revealed that globally, in 2015 of the 71 million people living with HCV: only 20% had been tested for HCV and knew their status (Africa had the lowest proportion diagnosed of 6%); 7% started treatment; and cumulatively, 7.7% had ever received HCV treatment; most of these treatments were older, less effective interferon-based regimens. Unlike in industrialized countries, where it is recommended that all HIV infected patients should be screened for HCV on entry into the healthcare system; in most resource limited countries like Uganda there is no mandatory HCV screening even among HIV infected individuals who are under health care. This is due to the high HCV screening and treatment costs [7, 8]. However, HCV screening is only recommended when investigating whether HCV is the possible cause of liver disease among HIV infected patients [7, 9]. This is still the situation in Uganda [9]. HIV and HCV share transmission routes, as such HIV-HCV coinfection is common [10]. In 2015, it was estimated that, globally close to 71 million peo- ple were chronically infected with HCV; and of the 36.7 million people that were living with HIV, 2.3 million (6.3%) had been co-infected with HCV [6]. Co- infection with HIV completely changes the dynamics of HCV infection. HIV infection reduces the chance of spontaneous clearance of acute HCV [11], with 80% of the acute HCV infected individuals developing chronic HCV infection [10]. There is rapid progression to cirrhosis and higher rates of liver failure in HIV-HCV co-infected individuals than they are in individuals who are infected with HCV only [11,12]. Cirrhosis has been observed to occur 12 to 16 years earlier in persons co- infected with HCV and HIV [10] compared to the 20 to 30 years in individuals infected with only HCV [7]. In this era of antiretroviral therapy (ART) and in efforts to achieve the 95-95-95 (HIV testing, treatment, and viral suppression) target, numbers of AIDS-related deaths have greatly reduced, hence making HIV infec- tion a chronic illness among individuals infected with only HIV. However, co-infection with HCV complicates the management of HIV by increasing the risk of death among HIV infected individuals. For persons living with HIV and co-infected with HCV, liver- related morbidity and mortality has become the leading cause of non-AIDS-related morbidity and deaths [13]. Furthermore, HIV infected individuals on highly active antiretroviral therapy (HAART) and co-infected with HCV, have slower CD4+ T-cells recovery than those infected with only HIV [14]. Over the years, mathematical models have been applied in the modelling of HIV and its common co- infections such as, Tuberculosis, Malaria, and hepatitis viruses to understand the dynamics of HIV and its co-infections. For example, [2, 3, 11, 12, 15] and [16] developed mathematical models to study the dynamics of HIV-HCV co-infection. Some of these models have considered treatment for both HIV and HCV. These models have either considered HCV infection in stages, namely: acute and chronic or simply HCV infection without considering stages of infection. However, the chronic stage of HCV infection needs to be given a special attention because is very long, asymptomatic and infectious. Recently, Mayanja et al. (2020) [17], formulated and analysed a deterministic model to study the HIV and HCV co-infection dynamics in absence of therapy; but with the HCV chronic stage split into two stages i.e. before and after onset of cirrhosis and its complications. Findings revealed that, in the long run, the number of individuals co-infected with HIV and latent HCV was far greater than that in any other class of individuals. The dynamics of HIV-HCV co-infection in absence of therapy were dominated by HIV. In this work, we extend the work in [17] by introducing HIV treatment and model the dynamics of HIV-HCV co- infection in presence of HIV treatment. Though HIV and HCV may share other transmission routes, in this work we only considered sexual transmission among sexually active individuals as done in [17]. II. MODEL FORMULATION A. Description of the HIV-HCV co-infection dynamics We categorized the chronic HCV stage as in [17], that is: latent and advanced HCV, where latent HCV is characterized by a long infectious period during which infection is undiagnosable because there are no or mild symptoms for a long time; and advanced HCV characterized by onset of cirrhosis and its related complications. The total population is divided into eleven distinct compartments, defined as follows: the Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 2/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy susceptible, S(t); infected with HIV without AIDS symptoms and not on HIV treatment, Ih(t); infected with HIV without AIDS symptoms and are on HIV treatment, IT (t); individuals who have developed AIDS symptoms, Ã(t); acutely HCV-infected, Ia(t); latent HCV, Il(t); advanced HCV, B̃(t); co-infected with HIV and acute HCV, not on HIV treatment, Iha(t); co- infected with HIV and acute HCV, on HIV treatment, ITa(t); co-infected with HIV and latent HCV, not on HIV treatment, Ihl(t); and those co-infected with HIV and latent HCV, on HIV treatment, ITl(t). The assumptions regarding the transmission of HIV and HCV are as follows: susceptibles are sexually active individuals at age a and above (mean age of sexual debut in Uganda is 16 years [18], however, 18 years is the legal age of marriage); HCV or HIV transmission is through sexual acts; for simplicity, we assume that HIV and HCV cannot be transmitted simultaneously; individuals infected with HIV are put on HIV treatment immediately they are diagnosed; individuals in the AIDS class do not revert to earlier classes despite successful treatment; AIDS cases on HIV treatment adhere never to have new sexual partners otherwise practice safe sex if they must, whereas those not on HIV treatment have undergone counselling and do not engage in unprotected sexual activities, similarly, for individuals in the advanced HCV class don’t spread diseases. We suppose that there is a constant recruitment rate Λ into the susceptible class through sexual maturing; and a constant natural mortality at a per capita rate µ in all classes. Susceptible individuals are infected with HCV at per capita rate πc, which is given by: πc = c̃βc[Ia + Il + ρ(Iha + ITa + Ihl + ITl)] N , (1) where c̃ is the average number of sexual partners ac- quired per year; βc is the HCV transmission probability per sexual contact; N is the total active population; and ρ > 1 is the enhancement factor for increased risk of being infected with HCV by a dually infected individual. Susceptible individuals are infected with HIV at per capita rate πh, which is given by: πh = c̃βh[Ih + r3IT + ω(Iha + r3ITa + Ihl + r3ITl)] N , (2) where βh is the HIV transmission probability per sexual contact; r3 < 1 is a reduction parameter catering for the reduced risk of being infected with HIV by an HIV infected individual on HIV treatment; and ω > 1 is the enhancement factor for increased risk of being infected with HIV by a dually infected individual. Individuals co-infected with HIV and HCV have higher viral loads of HIV and HCV as compared to those mono infected. This may increase their risk of transmission of each of the viruses [19]. Both ω and ρ model the fact that co-infected individuals are more infectious than their counterparts who are mono infected [12]. When susceptible individuals are infected with HIV, they enter the class of HIV infected not on treatment, Ih(t). Individuals in the class Ih(t), once detected are put on HIV treatment at a rate δ1 to enter the class IT (t). Individuals in IT (t) class, progress to AIDS class (Ã) at a rate r2α, where r2 < 1 is a reduction parameter catering for the reduced risk of progression to AIDS due to HIV treatment. Individuals in class Ih(t), who are not on HIV treatment, progress to AIDS class at a rate α. Apart from natural death, individuals in AIDS class have an additional AIDS-induced death at a per capita rate σÃ. Susceptible individuals once infected with HCV enter the class of acute HCV infected, Ia(t). Some of the Individuals in Ia(t) class, clear acute HCV spontaneously at a rate τ and become susceptible to HCV again. Others who fail to spontaneously clear, progress to latent HCV class, Il(t), at a rate γ. Then, individuals from Il(t) progress to advanced HCV class, B̃(t), at a rate φ. Apart from natural death, individuals in class B̃(t) have an additional advanced HCV-induced death at a per capita rate σB̃. Since HIV weakens the immune system, this leaves the body more vulnerable to other infections and ill- nesses. Hence, individuals infected with HIV are at higher risk of contracting HCV than their counterparts without HIV [20]. HIV also increases HCV RNA, thus making sexually active HIV infected individuals at an increased risk of sexual transmission of HCV [21]. Due to HIV and HCV having similar ways of transmission, it means that individuals who are infected with HIV are at a high risk of being infected with HCV and vice versa. Therefore, we included amplification parameters, ki=1,2 > 1, to cater for the increased risk of getting infected with HCV for those individuals who are already infected with HIV. Amplification parameters, qi=1,2 > 1, were included to cater for the increased risk of getting infected with HIV for those individuals who are already infected with HCV [12]. A sexual encounter between individuals in classes Ih(t) and Ia(t), is likely to result into a co-infection with HIV and acute HCV, where: individuals who are infected with HIV only and not on HIV treatment, Ih(t), get infected with acute HCV at a rate k1πC to enter the class of those individuals co-infected with Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 3/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy TABLE I PARAMETER VALUES USED IN THE NUMERICAL SIMULATIONS OF HIV-HCV CO-INFECTION MODEL. Parameter Description Value Source βh HIV transmission probability 0.0217 yr −1 Assumed βc HCV transmission probability 0.05 yr−1 [15] c̃ Average number of sexual partners acquired 4∗ yr−1 [17] θ Rate of progression from Iha(t) to Ihl(t) 0.52 yr −1 [17] α Rate of progression from HIV to AIDS 0.068∗ yr−1 Assumed φ Rate of progression from Il to B̃ class 0.095 ∗ yr−1 [17] µ Per capita natural mortality rate 0.0158 yr−1 [17] γ Rate of progression from Ia to Il class 2 yr −1 [17] Λ Recruitment rate 602095 yr−1 [17] k1,k2 Amplification factor for individuals in Ih and IT classes, respectively 1.001 [12] q1,q2 Amplification factor for individuals in Ia and Il classes, respectively 1.0001 [12] ω Enhancement factor for increased risk of being infected with HIV by a co-infected individual 1.0002 [15] ρ Enhancement factor for increased risk of being infected with HCV by a co- infected individual 1.0002 [12] τ Rate of spontaneous clearance of acute HCV 0.27 yr−1 [15] r1 Reduction factor for risk of acute HCV spontaneous clearance in presence of co-infection 0.25 [11] r2 Reduction factor for progressing from IT to à 0.2 Assumed r3 Reduction factor of risk of being infected by an HIV infected individual, on HIV treatment 0.5 Assumed δi=1,2,3 Rates at which individuals who are infected with HIV are identified and put on HIV treatment 0.12 yr−1 Assumed ϕ Rate of progression from IT a to IT l class 0.52 yr −1 [17] Parameter values with * have: c̃ ∈ [1, 4], α ∈ [0.066, 0.1], φ ∈ [0.095, 0.1]; yr represents year. HIV and acute HCV but not on HIV treatment, Iha(t); whereas those who are infected with acute HCV, Ia(t), become co-infected with HIV at a rate q1πh. In addi- tion, some of the individuals who are in class Iha(t) can spontaneously clear acute HCV at a rate r1τ and return to the Ih(t) class. These individuals are susceptible to HCV infection again. Due to the fact that the probability of spontaneous clearance of the HCV virus is reduced in case of co-infection [11], a reduction parameter r1 < 1 was introduced to cater for the reduced risk of spontaneous clearance of acute HCV due to the co- infection of acute HCV and HIV. Some of the individuals in class Iha(t) who fail to spontaneously clear acute HCV, are detected to be co- infected with HIV and are immediately put on HIV treatment at a rate δ3 to enter the class of dually infected with HIV and acute HCV but on HIV treatment, ITa(t); whereas those undetected to be co-infected with HIV, progress to dually infected with latent HCV and HIV but not on HIV treatment, Ihl(t), at a rate θ. When individuals in classes Ih(t) and Il(t) sexually interact, individuals who are in the class Ih(t) are likely to become co-infected with acute HCV at a rate k1πc to enter Iha(t) class whereas individuals in the class Il(t) are likely to become co-infected with HIV at a rate q2πh to enter Ihl(t) class. When individuals in classes IT (t) and Ia(t) sexually interact, individuals who are in the class IT (t) are likely to become co-infected with acute HCV at a rate k2πc to enter class ITa(t); whereas individuals in the class Ia(t) are likely to become co-infected with HIV at a rate q1πh to enter class Iha(t). When individuals in classes IT (t) and Il(t) sexually interact, individuals who are in the class IT (t) are likely to become co-infected with acute HCV at a rate k2πc to enter class ITa(t) whereas individuals in the class Il(t) are likely to become co- infected with HIV at a rate q2πh to enter class Ihl(t). Individuals who are dually infected with latent HCV and HIV but not on HIV treatment, Ihl(t), are detected and put on HIV treatment at a rate δ2 to enter the class ITl(t). Some of the individuals in class ITa(t) spontaneously clear acute HCV at a rate r1τ and return to IT (t) whereas those who fail to spontaneously clear acute HCV, progress to class ITl(t) at a rate of ϕ. As we summarize the description of the HIV-HCV co-infection dynamics, the parameters presented in the description of HIV-HCV co-infection dynamics in Sub- section II-A are summarized in Table I. Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 4/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy B. Compartmental model for the HIV-HCV co-infection dynamics The HIV-HCV co-infection dynamics are presented in a compartmental diagram (Figure 1). C. Mathematical model From the compartmental diagram in Figure 1, the associated mathematical model is shown as a system of differential Equations (3a)–(3k): dS dt = Λ + τIa −µS −πcS −πhS, (3a) dIh dt = πhS + r1τIha −k1πcIh −δ1Ih −µIh −αIh, (3b) dIT dt = δ1Ih + r1τITa −k2πcIT −r2αIT −µIT , (3c) dIa dt = πcS − τIa −q1πhIa −γIa −µIa, (3d) dIl dt = γIa −q2πhIl −φIl −µIl, (3e) dIha dt = k1πcIh + q1πhIa −r1τIha −θIha − δ3Iha −µIha, (3f) dIhl dt = θIha + q2πhIl −δ2Ihl −µIhl, (3g) dITl dt = δ2Ihl + ϕITa −µITl, (3h) dITa dt = k2πcIT + δ3Iha −ϕITa −µITa −r1τITa, (3i) dà dt = r2αIT + αIh −σÃÃ−µÃ, (3j) dB̃ dt = φIl −σB̃B̃ −µB̃, (3k) where πc and πh are as defined in (1) and (2), respec- tively. The initial values of the variables of the system (3a)–(3k) are as follows: S(0) > 0,Ih(0) ≥ 0,IT (0) ≥ 0, Ã(0) ≥ 0,Ia(0) ≥ 0,Il(0) ≥ 0, B̃(0) ≥ 0,Iha(0) ≥ 0,Ihl(0) ≥ 0, ITl(0) ≥ 0 and ITa(0) ≥ 0. Since we assume that due to diminished immunity, AIDS cases, Ã(t), and advanced HCV infected cases, B̃(t), do not engage in sexual activity. As a result, Equations (3a)–(3i) are independent of Ã(t) and B̃(t). Therefore, compartments à and B̃ do not feed into any other compartments. Thus, the total active population at time t, N(t), is given by (4): N(t) = S(t) + Ih(t) + IT (t) + Ia(t) + Il(t) + Iha(t) + Ihl(t) + ITl(t) + ITa(t). (4) III. MODEL ANALYSIS AND RESULTS A. Positivity and boundedness of solutions of HIV-HCV co-infection model It is important to establish whether system (3a)–(3i) is well posed and biologically meaningful. A study of the non-negativity and boundedness properties of the solutions of the HIV-HCV co-infection model (3a)–(3i) is made in this Subsection. Lemma 1. The solutions S(t), Ih(t), IT (t), Ia(t), Il(t), Iha(t), Ihl(t), ITl(t), ITa(t) of the system (3a)– (3i) are non-negative for t ≥ 0. Proof: Let the initial values of the variables of the system (3a)–(3i) be non-negative with S(0) > 0. We prove that the solution component of S(t) remains posi- tive. Assume that there exists a first time t1 : S(t1) = 0, S ′ (t1) < 0 and S(t) > 0,Ih(t) > 0,IT (t) > 0,Ia(t) > 0,Il(t) > 0,Iha(t) > 0,Ihl(t) > 0,ITl(t) > 0, ITa(t) > 0 for 0 < t < t1. From (3a) of the system, we have: dS(t1) dt = Λ + τIa(t1) > 0, which is a contradiction and consequently S(t) remains positive. The non-negativity of the other variables can similarly be proved. Therefore, the solutions of the system are non- negative for t ≥ 0. Lemma 2 (Invariant region). The region Ω = { (S(t),Ih(t),IT (t),Ia(t),Il(t),Iha(t),Ihl(t), ITl(t),ITa(t)) ∈ R9+ : N(t) ≤ max { N0, Λ µ }} , is positively invariant and attracting with respect to the model. Proof: Let (S(t),Ih(t),IT (t),Ia(t),Il(t),Iha(t),Ihl(t), ITl(t),ITa(t)) ∈ R9+ be any solution of the system with non-negative initial condition given by (S(0),Ih(0),IT (0),Ia(0),Il(0),Iha(0),Ihl(0), ITl(0),ITa(0)). Adding Equations (3a)–(3i), gives: dN dt = Λ −µN −αIh(t) −r2αIT −φIl. (5) Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 5/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy 𝑰𝒉 𝑰𝑻 𝑰𝒉𝒍 𝑰𝒉𝒂 𝑩 𝑰𝒍 𝑰𝒂 𝑺 𝜇𝐼ℎ 𝜋ℎ𝑆 𝜎𝐵 𝐵 ∅𝐼𝑙 𝛾𝐼𝑎 𝜇𝑩 𝜇𝐼𝑙 𝜇𝐼𝑎 𝜗𝐼ℎ𝑎 𝛿2𝐼ℎ𝑙 𝜇𝑆 ∧ (𝜎𝐴 + 𝜇)𝑨 𝛿1𝐼ℎ 𝑨 𝑰𝑻𝒂 𝑰𝑻𝒍 𝜇𝐼𝑇 𝛼𝐼ℎ 𝜇𝐼ℎ𝑙 𝜇𝐼𝑇𝑎 𝑟2𝛼𝐼𝑇 Fig. 1. Compartmental diagram for HIV-HCV co-infection dynamics. Solid arrows indicate movement from one compartment to another whereas dashed connections indicate the interaction between the connected compartments. For Ih(t),IT (t),Il(t) ≥ 0 for t ≥ 0, (5) reduces to: dN(t) dt ≤ Λ −µN, (6) from which N(t) ≤ Λ µ + ( N0 − Λ µ ) e−µt (7) where N0 ≥ 0 is the initial total population size. Two observations are made: I: If N0 > Λ µ , then (7) implies N(t) ≤ N0 for all values of t. II: If N0 < Λ µ , then (7) implies N(t) ≤ Λ µ for all values of t. Therefore, N(t) ≤ max{N0, Λµ} for all values of t ≥ 0. Hence, every feasible solution of the model system that starts in the region Ω remains in the region for all values of t. Thus, the region Ω is biologically feasible and positively invariant. Therefore, the model is epidemiologically and math- ematically well posed. Before analysing the dynamics of the HIV-HCV co- infection model (3a)–(3i), it is instructive to first analyse the HIV-only and HCV-only submodels. This is done in the Subsections III-B, III-C and III-D. Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 6/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy B. The HIV-only submodel We set Ia(t) = Il(t) = Iha(t) = Ihl(t) = ITl(t) = ITa(t) = 0 in system (3a)–(3i), thus the HIV-only submodel is as in (8a)–(8c): dSHIV dt = Λ −πhSHIV −µSHIV , (8a) dIh dt = πhSHIV −αIh − δ1Ih −µIh, (8b) dIT dt = δ1Ih −r2αIT −µIT , (8c) where πh = c̃βh[Ih + r3IT ] NHIV (9) and NHIV = SHIV + Ih + IT . (10) 1) The HIV-free equilibrium and effective reproduc- tion number: The HIV-free equilibrium point, ε0HIV , for system (8a)–(8c) is obtained by setting the right- hand side of system (8a)–(8c) equal to zero, and hence is found to be: ε0HIV = (S 0 HIV ,I 0 h,I 0 T ) = ( Λ µ , 0, 0). The effective reproduction number is defined as the spectral radius of the next generation matrix [23]. As interpreted in [12], it is the “expected number of secondary infections produced by a single infectious individual during his/her entire infectious period in the presence of an intervention strategy”. Using the next generation matrix method [23], we obtain the Jacobian matrices of new HIV infections, FHIV , and for the rate of transfer into and out of compartment i by all other processes, VHIV , evaluated at HIV-free equilibrium as: FHIV = [ c̃βh c̃βhr3 0 0 ] and VHIV = [ (δ1 + α + µ) 0 −δ1 (r2α + µ) ] . The effective reproduction number of the HIV-only submodel, RHIV , is given by the spectral radius of the next generation matrix, FHIV V −1 HIV . That is, RHIV = c̃βh (δ1 + α + µ) ( 1 + r3δ1 (r2α + µ) ) . (11) Expressing (11) as: RHIV = c̃βh (r2α + µ) ( r2α + µ + r3δ1 δ1 + µ + α ) = c̃βh (r2α + µ) f(δ1), (12) in which f(δ1) = ( r2α + µ + r3δ1 δ1 + µ + α ) = i + r3δ1 g + δ1 , (13) where i = r2α + µ and g = α + µ. Now, lim δ1→+∞ f(δ1) = r3 and lim δ1→0 f(δ1) = i g . (14) From (12) and (14), we deduce that, varying δ1 alone while other parameters are kept fixed, RHIV is bounded, that is: RHIV (δ1) ≤ c̃βh (r2α + µ) max { i g ,r3 } . (15) Using Theorem 2 in [23], the following result fol- lows. Theorem 1. The HIV-free equilibrium ε0HIV is locally asymptotically stable if RHIV < 1 and unstable other- wise. We can establish the following results using, for instance, the techniques in [17]: Theorem 2. The HIV-free equilibrium ε0HIV is glo- bally-asymptotically stable whenever RHIV < 1. Theorem 3. The HIV-only submodel has a unique endemic equilibrium if and only if RHIV > 1. The global stability property of the endemic equi- librium of the HIV-only submodel can be investigated using the techniques in [17]. C. The HCV-only submodel To obtain the HCV-only submodel, we set Ih(t) = IT (t) = Iha(t) = Ihl(t) = ITl(t) = ITa(t) = 0 in (3a)–(3i). Thus, we obtain: dSHCV dt = Λ + τIa −πcSHCV −µSHCV , (16a) dIa dt = πcSHCV −γIa − τIa −µIa, (16b) dIl dt = γIa −µIl −φIl, (16c) where πc = c̃βc[Ia + Il] NHCV and NHCV = SHCV + Ia + Il. (17) Basic reproduction number: The HCV-only sub- model has a basic reproduction number, RHCV , as derived in [17], is given by: RHCV = c̃βc (γ + τ + µ) ( 1 + γ (µ + φ) ) . (18) Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 7/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy In [17], it is deduced that, keeping other parameters constant and varying γ alone, RHCV is bounded. That is: RHCV (γ) ≤ c̃βc (µ + φ) max {d e , 1 } , (19) where d = µ + φ and e = µ + τ. Existence and stability of HCV-free and HCV en- demic equilibria: The existence of HCV-free and HCV endemic equilibria and their stability is as studied in [17]. D. The disease-free equilibrium and effective reproduc- tion number for the HIV-HCV co-infection model The disease-free equilibrium of the HIV-HCV co- infection model (3a)–(3i) is given by: ε0 = (S0f,I 0f h ,I 0f T ,I 0f a ,I 0f l ,I 0f ha,I 0f hl ,I 0f Tl ,I 0f Ta) = (Λ µ , 0, 0, 0, 0, 0, 0, 0, 0 ) . Using the next generation matrix method [23] on model Equations (3a)–(3i), we obtain Jacobian of new infections matrix at disease free-equilibrium F̃ as:  c̃βh c̃βhr3 0 0 n1 n1 n1r3 n1r3 0 0 0 0 0 0 0 0 0 0 c̃βc c̃βc c̃βcρ c̃βcρ c̃βcρ c̃βcρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   where n1 = c̃βhω; and the Jacobian matrix for the rate of transfer from one compartment to another by all other processes at disease-free equilibrium V as: V =   a1 0 0 0 −r1τ 0 0 0 −δ1 a2 0 0 0 0 0 −r1τ 0 0 a3 0 0 0 0 0 0 0 −γ a4 0 0 0 0 0 0 0 0 a5 0 0 0 0 0 0 0 −θ a6 0 0 0 0 0 0 0 −δ2 a7 −ϕ 0 0 0 0 −δ3 0 0 a8   where a1 = (δ1 + α + µ), a2 = (r2α + µ), a3 = (γ + τ + µ), a4 = (φ + µ), a5 = (r1τ + θ + δ3 + µ), a6 = (δ2 + µ), a7 = µ, a8 = (ϕ + µ + r1τ). The effective reproduction number, R0, for the HIV- HCV co-infection model is the maximum of eigenval- ues of the next generation matrix F̃V −1: λ1 = c̃βh (δ1 + µ + α) ( 1 + r3δ1 (r2α + µ) ) , λ2 = c̃βc (γ + τ + µ) ( 1 + γ (µ + φ) ) , and λ3 = λ4 = λ5 = λ6 = λ7 = λ8 = 0. That is, R0 = max { c̃βh (δ1 + µ + α) ( 1 + r3δ1 (r2α + µ) ) , c̃βc (γ + τ + µ) ( 1 + γ (µ + φ) ) , 0, 0, 0, 0, 0, 0 } . (20) Thus R0 = max{RHIV ,RHCV} (21) where RHIV and RHCV are the reproduction numbers of HIV-only and HCV-only submodels as indicated in Equations (11) and (18), respectively. From (21), it is deduced that the disease with the bigger effective reproduction number will dominate the dynamics of the HIV-HCV co-infection. Using Theorem 2 in [23], the following result fol- lows. Theorem 4. The disease-free equilibrium ε0 of the HIV-HCV co-infection model is locally asymptotically stable if R0 < 1 and unstable otherwise. 1) Global stability of disease-free equilibrium for HIV-HCV co-infection model: To study the global be- haviour of system (3a)–(3i), we use the Theorem in [24] as shown in Appendix A. Re-writing model (3a)–(3i) in the form of Equation (A.1) and using the same notation as used in [24], we have: X = (S), Z = (Ih,IT ,Ia,Il,Iha,Ihl,ITl,ITa), F(X, 0) = [Λ −µS], and A =   g1 h1 0 0 h2 c̃βhω ωh1 ωh1 δ1 g2 0 0 0 0 0 r1τ 0 0 g3 c̃βc c̃βcρ c̃βcρ c̃βcρ c̃βcρ 0 0 γ g4 0 0 0 0 0 0 0 0 g5 0 0 0 0 0 0 0 θ g6 0 0 0 0 0 0 0 δ2 g7 ϕ 0 0 0 0 δ3 0 0 g8   Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 8/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy where g1 = c̃βh − (δ1 + α + µ), g2 = −(r2α + µ), g3 = c̃βc − (γ + τ + µ), g4 = −(φ + µ), g5 = −(r1τ + θ + µ + δ3), g6 = −(δ2 + µ), g7 = −µ, g8 = −(ϕ + µ + r1τ), h1 = c̃βhr3, h2 = c̃βhω + r1τ. Ĝ defined as Ĝ(X,Z) = AZ−G(X,Z) is given by: Ĝ(X,Z) =   Ĝ1(X,Z) Ĝ2(X,Z) Ĝ3(X,Z) Ĝ4(X,Z) Ĝ5(X,Z) Ĝ6(X,Z) Ĝ7(X,Z) Ĝ8(X,Z)   =   c̃βh(1 − SN )(Ih + r3IT + ωIha + ωr3ITa +ωIhl + ωr3ITl) + k1πcIh k2πcIT c̃βc(1 − SN )(Ia + Il + ρIha + ρITa + ρIhl +ρITl) + q1πhIa q2πhIl −k1πcIh −q1πhIa −q2πhIl 0 −k2πcIT   . It can be seen that since Ĝ5(X,Z) < 0, Ĝ6(X,Z) < 0, and Ĝ8(X,Z) < 0, then Ĝ(X,Z) < 0. This implies that the second condition (H2) in Theorem by [24] is not fulfilled. Thus, the disease-free equilibrium of system (3a)–(3i) may not be globally asymptotically stable for R0 < 1. 2) HIV-HCV co-infection endemic equilibrium: Es- tablishment of the expressions for the endemic equi- librium for the HIV-HCV co-infection model (3a)– (3i) analytically is laborious, therefore, its existence and stability could be numerically investigated as, for instance, done in [17], by varying the initial values of the variables to determine whether they would level off to the same non-zero values in the long run, irrespective of the different initial values of the variables. 3) Impact of HIV infection on HCV infection and vice versa: The impact of HCV infection on HIV infection and vice versa is analysed by expressing the reproduction number of one infection in terms of the other. The impact of HIV infection on HCV infection is analysed by expressing the reproduction number of HCV infection in terms of that of HIV infection, that is, RHCV = RHIV βc(δ1 + α + µ)(r2α + µ)(µ + φ + γ) βh(r2α + µ + r3δ1)(γ + τ + µ)(µ + φ) . (22) Now, taking the partial derivative of RHCV in (22) with respect to RHIV , gives: ∂RHCV ∂RHIV = βc(δ1 + α + µ)(r2α + µ)(µ + φ + γ) βh(r2α + µ + r3δ1)(γ + τ + µ)(µ + φ) . (23) Since, according to (23), ∂RHCV ∂RHIV > 0, epidemiolog- ically this is an implication that an increase in HIV cases would result in an increase in HCV cases in the population. That is, HIV prevalence enhance HCV infections in the population. Thus, HIV control has a positive effect in controlling the HCV transmission dynamics. Similarly, it can be shown that ∂RHIV ∂RHCV = βh(γ + τ + µ)(µ + φ)(r2α + µ + r3δ1) βc(µ + φ + γ)(δ1 + α + µ)(r2α + µ) . (24) Since, from (24), ∂RHIV ∂RHCV > 0, this implies that an increase in HCV cases would result in an increase in HIV cases in the population. That is, HCV prevalence enhance HIV infections in the population. Thus, HIV and HCV infections enhance each other. This is in concurrence with the findings in [2] and [12]. E. Sensitivity analysis 1) Derivation of parameter values: In Uganda’s efforts to achieve the 90-90-90 targets, in 2018, of the 1.4 million Ugandans that were living with HIV, 84% knew their HIV status, 72% were on treatment and 64% were virally suppressed [22]. In this work, the rate at which HIV infected individuals are identified and put on HIV treatment has been assumed to be 0.12, that is: δi=1,2,3 = 0.12. We make a further assumption that there is no difference in the duration in acute HCV stage between individuals who are dually infected with HIV and on HIV treatment, and those not on HIV treatment. Hence, θ = ϕ = 0.52. Some of the parameter values are cited from the respective studies with literature similar to this work, and others have been assumed only to illustrate numerical results. All the parameter input values are summarised in Table I indicating the sources of parameter values. 2) Computation of R0: Substituting for the param- eter values in Table I in (11) and (18), we obtain RHIV = 1.295 and RHCV = 1.667. From (21), R0 = max{1.295, 1.667} = 1.667 which is RHCV . Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 9/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy Hence, the dynamics of HIV-HCV co-infection in pres- ence of HIV therapy is dominated by HCV. Introduction of HIV treatment changes the dynamics of HIV-HCV co-infection. For the HIV-HCV co-infection dynamics in absence of treatment, the dynamics were dominated by HIV [17]. Therefore, there is need to investigate the dynamics of HIV-HCV co-infection in presence of therapies for both infections. 3) Computation of sensitivity indices of the effective reproduction numbers with respect to the parameters of the HIV-HCV co-infection model: The sensitivity anal- ysis of the effective reproduction number of the HIV- HCV co-infection model, R0, to each of the param- eter values has been performed using the normalized forward sensitivity index method [25]. This helps to determine the relative contribution of each parameter on R0 such that appropriate intervention strategies can be taken. The normalized forward sensitivity index of Re with respect to parameter, x, is defined as the ratio of the relative change in Re to the relative change in parameter, x, that is: rRex = ∂Re ∂x × x Re . (25) Sensitivity analysis of R0 to each of the parameter values has been computed separately for RHIV and RHCV , since R0 = max{RHIV ,RHCV}. Sensitivity indices of RHIV and RHCV have been calculated analytically using formulas rRHIVx = ∂RHIV ∂x × x RHIV (26) and rRHCVx = ∂RHCV ∂x × x RHCV , (27) respectively. Table II presents sensitivity indices of both RHIV and RHCV . TABLE II SENSITIVITY INDICES OF RHIV AND RHCV WITH RESPECT TO PARAMETERS. Parameter Index of RHIV Parameter Index of RHCV βh +1.0000 βc +1.0000 c̃ +1.0000 c̃ +1.0000 r2 −0.3105 φ −0.8123 r3 +0.6712 µ −0.14201 α −0.6442 τ −0.1181 µ −0.4382 γ +0.0725 δ1 +0.0823 The interpretation of the sensitivity indices presented in Table II is as follows: for a parameter with a negative index, it implies that the corresponding ba- sic reproduction number decreases (increases) with an increase (decrease) in the value of that parameter while keeping the values of other parameters fixed. More still, a positive index implies that the corresponding basic reproduction number increases (decreases) with an increase (decrease) in the value of that parameter. For example, increasing (decreasing) the value of average number of sexual partners acquired per year, c̃, by 10% while other parameter values are kept fixed, increases (decreases) the value of RHIV by 10%. In addition, a 10% increase (decrease) in the value of the rate of progression of individuals infected with HIV to AIDS, α, while other parameter values are kept fixed, decreases (increases) the value of RHIV by 6.4%. Similarly, the sensitivity indices of other parameters can be interpreted. From Table II, we deduce that endemicity of HIV infection increases when the values of βh, c̃, δ1, and r3 are increased and or those of r2, α, and µ are decreased. The most sensitive parameters in HIV infection are c̃ and βh (which are equally sensitive) followed by r3, α and r2. Therefore, interventions should target and concentrate on reducing the values of c̃ and βh. Parameters c̃, βh, δ1, r2, r3 and α are related in a way that: an increase in the rate at which individuals who are infected with only HIV are identified and put on HIV treatment, δ1, reduces the value of reduction factor of HIV individuals on treatment progressing to AIDS, r2, hence increasing the duration in the HIV stage (reduction in α). An increase in δ1 results into increased r3 which in the long run leads to increased HIV infections. This is because HIV infected individuals on treatment have a prolonged life span, look healthy and can easily get many sexual partners as they would wish like any HIV negative individual. They have a prolonged time of infecting other individuals with HIV. Therefore, for reduced HIV infections, HIV infected individuals on HIV treatment need to be sensitised on how to live positively (for example, having safe sex using condoms and having few sexual partners). The sensitivity indices of RHCV in Table II are similar to those in [17], since the HCV-only submodel and the corresponding parameter values in this work are the same as those in [17]. Thus, from Table II, we deduce that endemicity of HCV infection increases when the values of βc, c̃, and γ are increased and or those of φ, τ, and µ are decreased. The most sensitive parameters in HCV infection are c̃ and βc (which are equally sensitive) followed by φ. In Subsection III-E2, it is revealed that the dynamics of HIV-HCV co-infection Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 10/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy in presence of HIV therapy is dominated by HCV. Therefore, R0 will be more sensitive to βc, c̃, and φ just like RHCV . From sensitivity analysis, we deduce that, βh (or βc) and c̃ are equally likely to increase HIV (or HCV) infections. Increment in the values of these parameters is leading other parameters in increasing the HIV (or HCV) infection. This is in agreement with the findings in [17] in which they investigated the dynamics of HIV-HCV co-infection in absence of treatment for the two infections. Therefore, for reduced HIV (or HCV) infections: individuals need to greatly reduce the rate of sexual partner acquisition, c̃; and transmit-ability probabilities βh (or βc); that is, by having safe sex which doesn’t expose them to infected blood, like using condoms as recommended in [17]. IV. NUMERICAL SIMULATIONS We use the ode45 solver in Matlab to perform a nu- merical simulation of the HIV-HCV co-infection model using parameter values presented in Table I. The initial values for the variables are set as follows: in Uganda, the total population in 2014 was 34, 634, 650 of which 49.2% was aged 15-64 years [26]. In this study, the population aged 15-64 years is taken to be sexually active. This implies that sexually active population, P = 49.2 100 × 34, 634, 650 = 17, 040, 248. In 2015, the HIV prevalence in Uganda was 6.2% [22]. This implies that sexually active population that was living with HIV in 2015 = 6.2 100 × 17, 040, 248 = 1, 056, 496. In Uganda, HCV prevalence is 2.7% [1]. This implies that sexually active population infected with HCV= 2.7 100 × 17, 040, 248 = 460, 087. Sexually active population co-infected with HIV and HCV= 2.7 100 × 1, 056, 496 = 28, 526. In 2018, 72% of HIV infected Ugandans were on HIV treatment [22]. In [3, 4] and [12] it is revealed that approximately 85% of the people infected with acute HCV develop chronic HCV. Basing on these facts, the following initial values for the variables are derived: IT (0) = 740, 138,Ih(0) = 287, 832,ITl(0) = 17, 458,ITa(0) = 3, 081,Il(0) = 366, 827,Ia(0) = 64, 734,Ihl(0) = 6, 789 and Iha(0) = 1, 198. There- fore, S(0) = P − [IT (0) + Ih(0) + Il(0) + Ia(0) + ITl(0) + ITa(0) + Iha(0) + Ihl(0)] = 15, 552, 191. In Figures 2, 3, 4, and 5, values for the rates at which individuals who are infected with only HIV, dually infected with HIV and latent HCV, and dually infected with HIV and acute HCV are identified and put on HIV treatment, δ1, δ2, and δ3, respectively, are varied to investigate the effect of increasing HIV treatment on 0 5 10 15 20 0.7 0.75 0.8 0.85 0.9 0.95 T re a tm e n t p ro p o rt io n Time (Years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 Fig. 2. Proportion on HIV treatment under varying rates of initiation on HIV treatment. the number of individuals co-infected with HIV and HCV. Starting with δ1 = δ2 = δ3 = 0.12, these values were doubled, tripled, multiplied four times, and lastly multiplied five times. Figure 2 shows that increasing the rate at which individuals are identified and put on HIV treatment, in the long run increases the proportion of individuals on HIV treatment. A country with such increasing treatment proportions, will nearly achieve the second “95” of the 95-95-95 targets. Figure 3 shows the effect of increasing the values of δ1, δ2, and δ3 on HCV and HIV prevalences. Figure 3(b) is a magnification of Figure 3(a). From Figures 3(a) and 3(b), it is revealed that increasing the values of δ1, δ2, and δ3, eventually, leads to the decrease in the prevalence of HCV. This implies that HIV control has a positive effect in controlling the HCV transmis- sion dynamics as mentioned in subsection III-D3. HIV treatment boosts the immunity of the body to fight off opportunistic infections such as HCV. Figure 3(c) reveals that increasing the rate at which individuals are identified and put on HIV treatment, in the long run increases the prevalence of HIV. This is due to HIV treatment improving the health of these people and prolonging their life span. They get many sexual partners and infect them with HIV. This is in concurrence with the findings in [27] and [28] in which it is inferred that treatment of HIV/AIDS patients would prolong the patients’ lives, and if the treatment does not reduce the infectiousness of such people, they become key agents in the spread of the HIV/AIDS. Therefore, for reduced HIV infections, HIV positively Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 11/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy infected people need continuous sensitisation on how to live positively as mentioned in this work in subsection III-E3. Figure 4(a) shows that increasing the rate at which people are identified and put on HIV treatment, de- creases the number of individuals who are co-infected with HIV and acute HCV not on HIV treatment in the long run. On the other hand, increasing the rate at which people are identified and put on HIV treatment, increases the number of individuals who are co-infected with HIV and acute HCV on treatment in the long run as shown in Figure 4(b). Figure 5(a) reveals that with increasing values of δ1, δ2, and δ3, in the long, there are decreasing numbers of people co-infected with HIV and latent HCV not on HIV treatment, however, the population co-infected with HIV and latent HCV on HIV treatment increases as shown in Figure 5(b). Figures 6 and 7 show the effect of varying amplifi- cation parameters catering for increased risk of getting infected with HCV (or HIV) for those individuals who are already infected with HIV (or HCV), k1 and k2 (or q1 and q2), on the size of individuals co-infected with HIV and HCV. Figure 6(a) shows that when the value of k1, is increased, eventually, there will be an increase in the number of individuals infected with HIV but on treatment becoming co-infected with acute HCV. More still, Figure 6(b) shows that when the value of k2, is increased, in the long run, there will be an increase in the number of individuals infected with HIV but on treatment becoming co-infected with acute HCV. Figure 7(a) reveals that when the value of q1 is increased, eventually, there will be an increase in the number of individuals acutely infected with HCV get- ting co-infected with HIV. In addition, Figure 7(b) shows that when the value of q2 is increased, there will be an increase in the number of individuals latently infected with HCV becoming co-infected with HIV, in the long run. Therefore, Figures 7 and 6 confirm that individuals who are already infected with HIV (or HCV) are at an increased risk of being infected with HCV (or HIV). This is in agreement with the literature in [20] and [12] which reveals that individuals who are already infected with HIV have an increased risk of contracting HCV and vice versa. V. CONCLUSION Building on the earlier work in [17], we have added the aspect of HIV treatment. Thus, we formulated and analysed a mathematical model for the HIV-HCV co-infection dynamics in presence of HIV therapy. 0 2 4 6 8 10 12 14 16 18 20 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 H C V p re va le n ce Time (Years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (a) HCV prevalence against time 18.8 19 19.2 19.4 19.6 19.8 20 0.09 0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 H C V p re va le n ce Time (Years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (b) Magnified graph of HCV prevalence against time 0 2 4 6 8 10 12 14 16 18 20 0.0595 0.06 0.0605 0.061 0.0615 0.062 0.0625 0.063 0.0635 0.064 H IV p re va le n ce Time (Years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (c) HIV prevalence against time Fig. 3. Simulation results showing HIV-HCV co-infection dynamics under varying rates of initiation on HIV treatment. Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 12/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy 0 2 4 6 8 10 12 14 16 18 20 0 1000 2000 3000 4000 5000 6000 7000 8000 Time (Years) H IV − a cu te H C V c o − in fe ct e d n o t o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (a) Population co-infected with HIV and acute HCV not on HIV treatment against time 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 Time (Years) H IV − a cu te H C V c o − in fe ct e d o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (b) Population co-infected with HIV and acute HCV on HIV treat- ment against time Fig. 4. HIV-acute HCV co-infection dynamics under varying rates of initiation on HIV treatment. Analytical analysis revealed that both HIV and HCV infections enhance each other. The different parameters in the HIV-HCV co-infection model were subjected to a sensitivity analysis and it was deduced that HIV (or HCV) transmission probability per sexual contact and average number of sexual partners acquired per year are not only equally likely to result into increased HIV (or HCV) infections, but also increment in the values of these parameters is leading other parameters in increasing the HIV (or HCV) infections, just like the case in the work in [17] when there was no treatment for both infections. Through numerical simulations it was revealed that in the long run, increasing the rates 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 Time (Years) H IV − la te n t H C V c o − in fe ct e d n o t o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (a) Population co-infected with HIV and latent HCV not on HIV treatment against time 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 x 10 5 Time (Years) H IV − la te n t H C V c o − in fe ct e d o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (b) Population co-infected with HIV and latent HCV on HIV treat- ment against time Fig. 5. HIV-latent HCV co-infection dynamics under varying rates of initiation on HIV treatment. at which people are put on HIV treatment reduces the prevalence of HCV in the community, however, it increases the prevalence of HIV. We recommend that there should be increased safer sexual behaviour campaigns among individuals on HIV treatment. This work can be extended by including treatment for HCV in the model. ACKNOWLEDGMENTS We are very grateful to the financial support extended by Sida Phase-IV bilateral program with Makerere University [2015-2020, project 316] “Capacity building in mathematics and its applications”. Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 13/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 Time (Years) H IV − a cu te H C V c o − in fe ct e d n o t o n t re a tm e n t k 1 =1.001 k 1 =2.001 k 1 =3.001 k 1 =4.001 k 1 =5.001 (a) Iha under varying values of k1 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 7 8 9 10 x 10 4 Time (Years) H IV − a cu te H C V c o − in fe ct e d o n t re a tm e n t k 2 =1.001 k 2 =2.001 k 2 =3.001 k 2 =4.001 k 2 =5.001 (b) IT a under varying values of k2 Fig. 6. HIV-HCV co-infection dynamics under varying values of amplification parameters for the risk of getting co-infected with HCV for HIV infected individuals, k1 and k2. APPENDIX A GLOBAL STABILITY CONDITIONS FOR THE DISEASE-FREE EQUILIBRIUM WHEN R0 < 1 According to Castillo-Chavez et al. [24], for the system written in the form:  dX dt = F(X,Z), dZ dt = G(X,Z), G(X, 0) = 0, (A.1) where the components of X ∈ Rm denotes the num- ber of uninfected individuals and the components of Z ∈ Rn denotes the number of infected individuals. 0 2 4 6 8 10 12 14 16 18 20 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Years) H IV − a cu te H C V c o − in fe ct e d n o t o n t re a tm e n t q 1 =1.0001 q 1 =2.0001 q 1 =3.0001 q 1 =4.0001 q 1 =5.0001 (a) Iha under varying values of q1 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 x 10 4 Time (Years) H IV − la te n t H C V c o − in fe ct e d n o t o n t re a tm e n t q 2 =1.0001 q 2 =2.0001 q 2 =3.0001 q 2 =4.0001 q 2 =5.0001 (b) Ihl under varying values of q2 Fig. 7. HIV-HCV co-infection dynamics under varying values of amplification parameters for the risk of getting co-infected with HIV for HCV infected individuals, q1 and q2. Let U0 = (X∗, 0) denote the disease-free equilibrium of this system. The fixed point U0 = (X∗, 0) is a globally asymptotically stable equilibrium of system (A.1) pro- vided that R0 < 1 and the following conditions (H1) and (H2) are satisfied: (H1): For dX dt = F(X, 0), X∗ is globally asymptot- ically stable, (H2): G(X,Z) = AZ−Ĝ(X,Z), Ĝ(X,Z) ≥ 0 for (X,Z) ∈ Ω, where A = DZG(X∗, 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. If system (A.1) satisfies conditions (H1) and (H2), then the Theorem below holds: Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 14/15 https://doi.org/10.55630/j.biomath.2022.07.158 Mayanja et al, Mathematical Modelling of HIV-HCV Co-infection Dynamics in Presence of HIV Therapy Theorem ([24]). The fixed point U0 = (X∗, 0) is a globally asymptotic stable equilibrium of system (A.1) provided R0 < 1 (locally asymptotically stable) and that conditions (H1) and (H2) are satisfied. REFERENCES [1] M.W. Sonderup, M. Afihene, R. Ally, B. Apica, Y. Awuku Y, L. Cunha, et al., “Hepatitis C in sub-Saharan Africa: the current status and recommendations for achieving elimination by 2030”, The Lancet Gastroenterology & Hepatology, 2(12):910-919, 2017, DOI: 10.1016/S2468-1253(17)30249-2. [2] D.P. Moualeu, J. Mbang, R. Ndoundam, S. Bowong, “Mod- eling and analysis of HIV and hepatitis C co-infections”, Journal of Biological Systems, 19(4):683-723, 2011, DOI: 10.1142/S0218339011004159. [3] Ana R.M. Carvalho, Carla M.A. Pinto, “A coinfection model for HIV and HCV”, Biosystems, 124:46-60, 2014, DOI: 10.1016/j.biosystems.2014.08.004. [4] S. Zhang, Y. Zhou, “Dynamics and application of an epidemiological model for hepatitis C”, Mathematical and Computer Modelling, 56(1-2):36-42, 2012, DOI: 10.1016/j.mcm.2011.11.081. [5] S.E. Schröeder, A. Pedrana, N. Scott, D. Wilson, C. Kuschel, L. Aufegger, et al., “Innovative strategies for the elimination of vi- ral hepatitis at a national level: A country case series”, Liver In- ternational, 39(10):1818-1836, 2019, DOI: 10.1111/liv.14222. [6] World Health Organization, Global Hepatitis Report, 2017. [7] V. Walusansa, M. Kagimu, “Screening for hepatitis C among HIV positive patients at Mulago Hospital in Uganda”, African Health Sciences, 9(3):143-146, 2009. [8] J.I. O’Reilly, P. Ocama, C.K. Opio, A. Alfred, E. Paintsil, E. Seremba, A.N. Sofair, “Risk Factors and Seroprevalence of Hepatitis C among Patients Hospitalized at Mulago Hospital, Uganda”, Journal of Tropical Medicine, Vol. 2011:598341, 2011, DOI: 10.1155/2011/598341. [9] Ministry of Health. Uganda Guidelines for Prevention, Testing, Care and Treatment of Hepatitis B and C Virus Infection, 2019. Accessed: August 2nd, 2022. Available at: http://library.health.go.ug/publications/hepatitis-b/uganda- guidelines-prevention-testing-care-and-treatment-hepatitis-b- and-c. [10] D. Joshi, J. O’Grady, D. Dieterich, B. Gazzard, K. Agarwal, “In- creasing burden of liver disease in patients with HIV infection”, The Lancet, 377(9772):1198-1209, 2011, DOI: 10.1016/S0140- 6736(10)62001-6. [11] A.Y.C. Sanchez, M. Aerts, Z. Shkedy, P. Vickerman, F. Fag- giano, G. Salamina, N. Hens, “A mathematical model for HIV and Hepatitis C co-infection and its assessment from a statistical perspective”, Epidemics, 5(1):56-66, 2013, DOI: 10.1016/j.epidem.2013.01.002. [12] C. P. Bhunu, S. Mushayabasa, “Modelling the transmis- sion dynamics of HIV/AIDS and Hepatitis C virus co- infection”, HIV & AIDS Review, 12(2):37-42, 2013, DOI: 10.1016/j.hivar.2013.03.001. [13] E. Rosenthal, C. Roussillon, D. Salmon-Céron, A. Georget, S. Hénard, T. Huleux, et al., “Liver-related deaths in HIV-infected patients between 1995 and 2010 in France: the Mortavic 2010 study in collaboration with the Agence Nationale de Recherche sur le SIDA (ANRS) EN 20 Mortalité 2010 survey”, HIV Medicine, 16(4):230-239, 2015, DOI: 10.1111/hiv.12204. [14] M.F. Miller, C. Haley, M.J. Koziel, C.F. Rowley, “Impact of Hepatitis C Virus on Immune Restoration in HIV-Infected Pa- tients Who Start Highly Active Antiretroviral Therapy: A Meta- analysis”, Clinical Infectious Diseases, 41(5):713-720, 2005, DOI: 10.1086/432618. [15] Carla M. A. Pinto, Ana Carvalho, “Effects of treatment, aware- ness and condom use in a coinfection model for HIV and HCV in MSM”, Journal of Biological Systems, 23(02):165-193, 2015, DOI: 10.1142/S0218339015500096. [16] M. Zerehpoush, H. Kheiri, “Global analysis of a mathematical model for HIV and Hepatitis C virus co-infection”, Journal of Mathematical Extension, 11(2):19-31, 2017. [17] E. Mayanja, L.S. Luboobi, J. Kasozi, R.N. Nsubuga, “Math- ematical Modelling of HIV-HCV Coinfection Dynamics in Absence of Therapy”, Computational and Mathemati- cal Methods in Medicine, Vol. 2020:2106570, 2020, DOI: 10.1155/2020/2106570. [18] A.M.N. Renzaho, J.K. Kamara, N. Georgeou, G. Kamanga, “Sexual, Reproductive Health Needs, and Rights of Young People in Slum Areas of Kampala, Uganda: A Cross Sectional Study”, PLoS ONE, 12(1):e0169721, 2017, DOI: 10.1371/jour- nal.pone.0169721. [19] D.E. Karageorgopoulos, J. Allen, S. Bhagani, “Hepatitis C in human immunodeficiency virus co-infected individuals: Is this still a “special population”?”, World Journal of Hepatology, 7(15):1936-1952, 2015, DOI: 10.4254/wjh.v7.i15.1936. [20] I. Maier, G.Y. Wu, “Hepatitis C and HIV co-infection: A re- view”, World Journal of Gastroenterology, 8(4):577-579, 2002, DOI: 10.3748/wjg.v8.i4.577. [21] J. Baseke, M. Musenero, H. Mayanja-Kizza, “Prevalence of Hepatitis B and C and relationship to liver damage in HIV infected patients attending Joint Clinical Research Centre Clinic (JCRC), Kampala, Uganda”, African Health Sciences, 15(2):322-327, 2015, DOI: 10.4314/ahs.v15i2.3. [22] UNAIDS Data 2019, Accessed: August 2nd, 2022. Available at: https://www.unaids.org/en/resources/documents/2019/2019- UNAIDS-data. [23] P. van den Driessche, J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Mathematical Biosciences, 180(1-2):29- 48, 2002, DOI: 10.1016/S0025-5564(02)00108-6. [24] C. Castillo-Chavez, Z. Feng, W. Huang, “On the computation of R0 and its role on global stability”, in Mathematical Ap- proaches for Emerging and Reemerging Infectious Diseases: An Introduction, Vol. 1:229-250, 2002, download link. [25] N. Chitnis, J.M. Hyman, J.M. Cushing, “Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model”, Bulletin of Mathematical Biology, 70(5):1272-1296, 2008, DOI: 10.1007/s11538-008- 9299-0. [26] Uganda Bureau of Statistics, The National Population and Housing Census 2014 – Main Report, Kampala, Uganda, 2016. [27] L.S. Luboobi, “A three age-groups model for the HIV/AIDS epidemic and effects of medical/social intervention”, Mathe- matical and Computer Modelling, 19(9):91-105, 1994, DOI: 10.1016/0895-7177(94)90043-4. [28] M.O. Okongo, J. Kirimi, A.L. Murwayi, D.K. Muriithi, “Math- ematical Analysis of a Comprehensive HIV/AIDS Model: Treatment versus Vaccination”, Applied Mathematical Sciences, 7(54):2687-2707, 2013. Biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 15/15 https://doi.org/10.1016/S2468-1253(17)30249-2 https://doi.org/10.1016/S2468-1253(17)30249-2 https://doi.org/10.1016/S2468-1253(17)30249-2 https://doi.org/10.1016/S2468-1253(17)30249-2 https://doi.org/10.1016/S2468-1253(17)30249-2 https://doi.org/10.1142/S0218339011004159 https://doi.org/10.1142/S0218339011004159 https://doi.org/10.1142/S0218339011004159 https://doi.org/10.1142/S0218339011004159 https://doi.org/10.1016/j.biosystems.2014.08.004 https://doi.org/10.1016/j.biosystems.2014.08.004 https://doi.org/10.1016/j.biosystems.2014.08.004 https://doi.org/10.1016/j.mcm.2011.11.081 https://doi.org/10.1016/j.mcm.2011.11.081 https://doi.org/10.1016/j.mcm.2011.11.081 https://doi.org/10.1016/j.mcm.2011.11.081 https://doi.org/10.1111/liv.14222 https://doi.org/10.1111/liv.14222 https://doi.org/10.1111/liv.14222 https://doi.org/10.1111/liv.14222 https://www.who.int/publications/i/item/9789241565455 https://www.ajol.info/index.php/ahs/article/view/48996 https://www.ajol.info/index.php/ahs/article/view/48996 https://www.ajol.info/index.php/ahs/article/view/48996 https://doi.org/10.1155/2011/598341 https://doi.org/10.1155/2011/598341 https://doi.org/10.1155/2011/598341 https://doi.org/10.1155/2011/598341 https://doi.org/10.1155/2011/598341 http://library.health.go.ug/publications/hepatitis-b/uganda-guidelines-prevention-testing-care-and-treatment-hepatitis-b-and-c http://library.health.go.ug/publications/hepatitis-b/uganda-guidelines-prevention-testing-care-and-treatment-hepatitis-b-and-c http://library.health.go.ug/publications/hepatitis-b/uganda-guidelines-prevention-testing-care-and-treatment-hepatitis-b-and-c http://library.health.go.ug/publications/hepatitis-b/uganda-guidelines-prevention-testing-care-and-treatment-hepatitis-b-and-c http://library.health.go.ug/publications/hepatitis-b/uganda-guidelines-prevention-testing-care-and-treatment-hepatitis-b-and-c http://library.health.go.ug/publications/hepatitis-b/uganda-guidelines-prevention-testing-care-and-treatment-hepatitis-b-and-c https://doi.org/10.1016/S0140-6736(10)62001-6 https://doi.org/10.1016/S0140-6736(10)62001-6 https://doi.org/10.1016/S0140-6736(10)62001-6 https://doi.org/10.1016/S0140-6736(10)62001-6 https://doi.org/10.1016/j.epidem.2013.01.002 https://doi.org/10.1016/j.epidem.2013.01.002 https://doi.org/10.1016/j.epidem.2013.01.002 https://doi.org/10.1016/j.epidem.2013.01.002 https://doi.org/10.1016/j.epidem.2013.01.002 https://doi.org/10.1016/j.hivar.2013.03.001 https://doi.org/10.1016/j.hivar.2013.03.001 https://doi.org/10.1016/j.hivar.2013.03.001 https://doi.org/10.1016/j.hivar.2013.03.001 https://doi.org/10.1111/hiv.12204 https://doi.org/10.1111/hiv.12204 https://doi.org/10.1111/hiv.12204 https://doi.org/10.1111/hiv.12204 https://doi.org/10.1111/hiv.12204 https://doi.org/10.1111/hiv.12204 https://doi.org/10.1086/432618 https://doi.org/10.1086/432618 https://doi.org/10.1086/432618 https://doi.org/10.1086/432618 https://doi.org/10.1086/432618 https://doi.org/10.1142/S0218339015500096 https://doi.org/10.1142/S0218339015500096 https://doi.org/10.1142/S0218339015500096 https://doi.org/10.1142/S0218339015500096 https://www.ijmex.com/index.php/ijmex/article/view/482 https://www.ijmex.com/index.php/ijmex/article/view/482 https://www.ijmex.com/index.php/ijmex/article/view/482 https://doi.org/10.1155/2020/2106570 https://doi.org/10.1155/2020/2106570 https://doi.org/10.1155/2020/2106570 https://doi.org/10.1155/2020/2106570 https://doi.org/10.1155/2020/2106570 https://doi.org/10.1371/journal.pone.0169721 https://doi.org/10.1371/journal.pone.0169721 https://doi.org/10.1371/journal.pone.0169721 https://doi.org/10.1371/journal.pone.0169721 https://doi.org/10.1371/journal.pone.0169721 https://doi.org/10.4254/wjh.v7.i15.1936 https://doi.org/10.4254/wjh.v7.i15.1936 https://doi.org/10.4254/wjh.v7.i15.1936 https://doi.org/10.4254/wjh.v7.i15.1936 https://doi.org/10.3748/wjg.v8.i4.577 https://doi.org/10.3748/wjg.v8.i4.577 https://doi.org/10.3748/wjg.v8.i4.577 https://doi.org/10.4314/ahs.v15i2.3 https://doi.org/10.4314/ahs.v15i2.3 https://doi.org/10.4314/ahs.v15i2.3 https://doi.org/10.4314/ahs.v15i2.3 https://doi.org/10.4314/ahs.v15i2.3 https://www.unaids.org/en/resources/documents/2019/2019-UNAIDS-data https://www.unaids.org/en/resources/documents/2019/2019-UNAIDS-data https://www.unaids.org/en/resources/documents/2019/2019-UNAIDS-data https://doi.org/10.1016/S0025-5564(02)00108-6 https://doi.org/10.1016/S0025-5564(02)00108-6 https://doi.org/10.1016/S0025-5564(02)00108-6 https://doi.org/10.1016/S0025-5564(02)00108-6 https://books.google.com/books?id=pR4CqiTSTMwC https://books.google.com/books?id=pR4CqiTSTMwC https://www.researchgate.net/profile/Carlos-Castillo-Chavez/publication/228915276_ON_THE_COMPUTATION_OF_R_AND_ITS_ROLE_ON_GLOBAL_STABILITY/links/0fcfd50a569d32c708000000/ON-THE-COMPUTATION-OF-R-AND-ITS-ROLE-ON-GLOBAL-STABILITY.pdf https://doi.org/10.1007/s11538-008-9299-0 https://doi.org/10.1007/s11538-008-9299-0 https://doi.org/10.1007/s11538-008-9299-0 https://doi.org/10.1007/s11538-008-9299-0 https://doi.org/10.1007/s11538-008-9299-0 https://ugandajournalistsresourcecentre.com/uganda-national-housing-population-census-2014/ https://ugandajournalistsresourcecentre.com/uganda-national-housing-population-census-2014/ https://doi.org/10.1016/0895-7177(94)90043-4 https://doi.org/10.1016/0895-7177(94)90043-4 https://doi.org/10.1016/0895-7177(94)90043-4 https://doi.org/10.1016/0895-7177(94)90043-4 https://www.researchgate.net/profile/Dennis-Muriithi/publication/283072306_Mathematical_Analysis_of_a_Comprehensive_HIV_AIDS_Model_Treatment_versus_Vaccination/links/5628b57a08ae518e347c6083/Mathematical-Analysis-of-a-Comprehensive-HIV-AIDS-Model-Treatment-versus-Vaccination.pdf https://www.researchgate.net/profile/Dennis-Muriithi/publication/283072306_Mathematical_Analysis_of_a_Comprehensive_HIV_AIDS_Model_Treatment_versus_Vaccination/links/5628b57a08ae518e347c6083/Mathematical-Analysis-of-a-Comprehensive-HIV-AIDS-Model-Treatment-versus-Vaccination.pdf https://www.researchgate.net/profile/Dennis-Muriithi/publication/283072306_Mathematical_Analysis_of_a_Comprehensive_HIV_AIDS_Model_Treatment_versus_Vaccination/links/5628b57a08ae518e347c6083/Mathematical-Analysis-of-a-Comprehensive-HIV-AIDS-Model-Treatment-versus-Vaccination.pdf https://www.researchgate.net/profile/Dennis-Muriithi/publication/283072306_Mathematical_Analysis_of_a_Comprehensive_HIV_AIDS_Model_Treatment_versus_Vaccination/links/5628b57a08ae518e347c6083/Mathematical-Analysis-of-a-Comprehensive-HIV-AIDS-Model-Treatment-versus-Vaccination.pdf https://doi.org/10.55630/j.biomath.2022.07.158 INTRODUCTION MODEL FORMULATION Description of the HIV-HCV co-infection dynamics Compartmental model for the HIV-HCV co-infection dynamics Mathematical model MODEL ANALYSIS AND RESULTS Positivity and boundedness of solutions of HIV-HCV co-infection model The HIV-only submodel The HIV-free equilibrium and effective reproduction number The HCV-only submodel The disease-free equilibrium and effective reproduction number for the HIV-HCV co-infection model Global stability of disease-free equilibrium for HIV-HCV co-infection model HIV-HCV co-infection endemic equilibrium Impact of HIV infection on HCV infection and vice versa Sensitivity analysis Derivation of parameter values Computation of R0 Computation of sensitivity indices of the effective reproduction numbers with respect to the parameters of the HIV-HCV co-infection model NUMERICAL SIMULATIONS CONCLUSION Appendix A: Global stability conditions for the disease-free equilibrium when R0<1 References