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 t , the HIV-infected unaware H U t , the HIV-infected aware, H A t , HIV on treatment H T t , the AIDS populace aware and on treatment A A t , acutely infected I c t and chronically infected C c t infected HCV, HIV-unaware co-infected with acute and chronic HCV ( H u I t and H uC t ) , HIV-aware co-infected with acute and chronic HCV ( H AI t and H AC t ) , HIV-positive individuals receiving treatment for HIV who are co-infected with acute and chronic HCV ( H TI t and H TC t ) , HIV-positive individuals in stage-IV co-infected with acute and chronic HCV A AI t and A AC t .

The overall population at time t, represented by N t , is classified into the 15 classes/subgroups listed in Tables of Nomenclature, each of which corresponds to a different epidemiological status. N t = S t + H U t + H A t + H T t + A A t + I c t + C c t + H u I t + H AI t + H TI t + H UC t + H AC t + H TC t + A AI t + A AC t (1)

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, 1 − φ H u Λ . Some newborns acquire HIV at parturition and are subsequently enrolled directly into the infectious class, H U where φ , is the rate of newborn HIV infection and Λ is the rate of recruitment through immigration or emigration. Individuals in all classes die at a consistent natural mortality rate, μ . Individuals with AIDS A A A AI A AC have an extra death rate owing to AIDS, 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 λ H , (force of infection related to HIV) when in contact with the H U , H A , and A A populations, where λ H = c h 1 − ψξ b h H U t + A A t + κ 1 H u I t + H uC t N (2)

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 ψξ ∈ 0 1 indicates the degree of condom protection. If ξ = 0 , condom use offers no protection, ξ = 1 denotes perfect protection, where ψ 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 H UI and individuals who are chronically infected with HCV and AIDS A Ac , is accounted for by the parameters κ 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 UI , H UC singly and dually infected with HCV advances to HIV diagnosed class H A , H AI , H AC after testing at a rate, α 1 , α 2 , α 3 and those in aware HIV was enrolled on therapy at the rate θ 1 , θ 3 , θ 5 in class H T , H TI , H TC . Nevertheless, some individuals who were placed on HIV treatment default from or drop out of the Highly Active Antiretroviral Therapy (HAART) ²⁵ after which they develop AIDS due to drug resistance and progress to class A A , A AI , A AC at a rate υ 1 , υ 2 , υ 3 . People with HIV and HCV who don't know their HIV status, H U , H UI , H UC and didn't get tested move to the AIDS class A A , A AI , A AC , at a rate ρ 1 , ρ 2 , ρ 3 . People with AIDS symptoms singly and dually infected with HCV are given treatment at a rate of θ 2 , θ 4 , θ 6 respectively. AIDS infected can respond well to treatment and return to H T , H TI , H TC ^{18 , 20} and die because of AIDS at an incidence d a .

Susceptible people get HCV infection from people in the I c , C c , H UI , H UC at a rate of λ C where λ C is the risk of getting HCV, which is given by λ C = c 1 − ψξ b c I c t + C c t + κ 2 H u I t + H uC t N (3)

To simulate the reality that individuals who are dually infected are more infectious than the mono-infected, we use the notation κ 2 > 1 , where b c the likelihood that contact will result in HCV infection. ^{19 , 34 , 35}

People who are only infected with HIV H U H A H T and A a acquired HCV at a rate δ 1 λ c δ 2 λ c δ 3 λ c and moved to classes H UI H AI H TI A AI , an increased risk of HCV acquisition is accounted for by the modification parameter δ 1 , δ 2 , δ 3 > 1 . HCV-only infected people Ic C c are more likely to obtain HIV H uI H uC than people who are only infected with HCV at a rate γλ H , τλ H where γ , τ > 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 UI , H TI , A AI at a rate η , becomes chronically infected and are treated for chronic HCV epidemic at r i , i = 1 , 2 , 3 , 4 , 5 while the remaining populace, ω 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 λ C if expose or engage in risk behaviors such as injection drug use, ³³ drinking alcohol, ³⁶ multiple sex partners and sex between two men ³⁷ since the clearance does not confer permanent immunity. ³⁸

An HCV-positive person stays acutely infected for an average of 1 / σ 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 ε 1 to impact spontaneous clearance and the term ε 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.

Mathematically, the flow chart leads to the system of 15 ordinary differential equations given by: dS dt = 1 − φH u Λ + ω 0 σ c I c + r 1 C c − λ H + λ C + μ S d H U dt = λ H S + φΛ H U + ω ϵ 1 σ c H UI + r 2 H UC − δ 1 λ C + α 1 + ρ 1 + μ H U d H A dt = α 1 H U + ω ϵ 1 σ c H AI + r 3 H AC − δ 2 λ C + θ 1 + μ H A d H T dt = θ 1 H A + ω ϵ 1 σ c H TI + r 4 H TC + θ 2 A A − δ 3 λ C + μ + υ 1 H T d A A dt = ρ 1 H U + υ 1 H T + ω ϵ 1 σ c A AI + r 5 A Ac − δ 4 λ C + μ + d a + θ 2 A A d I C dt = λ C S − ω 0 + η 0 σ c I C − γ λ H + μ I C d C c dt = η 0 σ c I c − τ λ H + μ + d c + r 1 C c d H UI dt = δ 1 λ C H U + γ λ H I c − η ϵ 2 σ c + α 2 + ω ϵ 1 σ c + ρ 2 + μ H UI d H AI dt = α 2 H AI + δ 2 λ C H A − η ϵ 2 σ c + θ 3 + ω ϵ 1 σ c + μ H AI d H TI dt = θ 3 H AI + δ 3 λ C H T + θ 4 A AI − η ϵ 2 σ c + υ 3 + ω ϵ 1 σ c + μ H TI d H UC dt = τ λ H C c + η ϵ 2 σ c H UI − r 2 + α 3 + ρ 3 + μ + d c H UC d H AC dt = α 3 H UC + η ϵ 2 σ c H AI − r 3 + θ 5 + μ + d c H AC d H TC dt = θ 5 H AC + η ϵ 2 σ c H TI + θ 6 A AC − r 4 + υ 3 + μ + d c H TC d A AI d t = δ 4 λ C A A + ρ 2 H UI + υ 2 H TI − η ϵ 2 σ c + θ 4 + ω ϵ 1 σ c + μ + d a A AI d A AC dt = η ϵ 2 σ c A AI + ρ 3 H UC + υ 3 H TC − r 5 + θ 6 + μ + d a + d c A AC (4)

•

People who are being treated for HIV don't spread the virus.

Since equation (4) represents a population of humans, all of the corresponding parameters are positive. The subsequent non-negativity finding is also valid.

HIV and HCV will be analyzed independently. Thereafter, the co-infection analyses will be carried out, with positive initial conditions specified by; S 0 = S 0 , H u 0 = H U 0 , H A 0 = H A 0 , H T 0 = H T 0 , A a 0 = A A 0 , I c 0 = I C 0 , C c 0 = C C 0 , H uI 0 = H uI 0 , H AI 0 = H AI 0 , H TI 0 = H TI 0 , A AI 0 = A AI 0 , H UC 0 = H uC 0 , H AC 0 = H AC 0 , H TC 0 = H TC 0 , A AC 0 = A AC 0 ∈ ℝ + 15 (5)

As a result, the system dynamics ( 4) will be examined in light of the biological elements of the region Φ = { ( S t + H u t + H A t + H T t + A a t + I c t + C c t + H uI t + H AI t + H TI t + A AI t + H UC t + H AC t + H TC t + A aC t ) ∈ ℝ + 15 : N ≤ Λ μ } , (6) Theorem 1:

The system variables ( 4) are positive whenever t > 0. In other words, Solutions of the system ( 4) with a positive initial condition will remain positive for every t > 0.

It follows from the model's first equation (4) that dS dt = 1 − φH u Λ + ω 0 σ c I c + r 1 C c − λ H + λ C + μ S dS dt = 1 − φH u Λ + ω 0 σ c I c + r 1 C c ≥ − λ H + λ C + μ S

which is re-writeable as d dt S t e μt + ∫ 0 t λ H + λ C ( τ ) d τ ≥ Λ e μt + ∫ 0 t λ H + λ C ( τ ) d τ

Hence, S Φ e μ Φ + ∫ 0 Φ λ H + λ C ( τ ) d τ − S 0 ≥ ∫ 0 Φ Λ e μx + ∫ 0 x λ H + λ C τ d τ dx

So that S Φ ≥ S 0 e − μ Φ − ∫ 0 Φ λ H + λ C ( τ ) d τ + e − μ Φ − ∫ 0 Φ − λ H + λ C ( τ ) d τ ∫ 0 Φ Λ e ⌈ μx + ∫ 0 x λ H + λ C ( τ ) d τ ⌉ dx > 0

Analogously, it's easy to show that H U t , H A t , H T t , A A t , I c t , C c t , H uI t , H AI t , H TI t , H UC t , H AC t , H TC t , A AI t , and A AC t

for all t > 0 are all positive. Theorem 2:

The closed set Φ = { S t + H u t + H A t + H T t + A A t + I c t + C c t + H uI t + H AI t + H TI t + H UC t + H AC t + H TC t + A AI t + A AC t ∈ ℝ + 15 : N ≤ Λ μ } is positively invariant.

It follows from the equation that lim t → ∞ supN t ≤ Λ μ . As a result, the system's dynamics ( 4) will be looked at in light of the region’s biological factors. This is simple to demonstrate as being positively model-invariant.

Therefore as t ⟶ ∞ , Λ μ is the upper limit of N given that N 0 ≤ Λ μ , N t will decline to this level if N 0 > Λ μ . As a result, the region Φ contains all possible system solutions that can enter or remain. Under the flow caused by the system ( 4), the region of biological interest Φ is therefore positively invariant Therefore, since region Φ is positively invariant and the results for the system's existence and uniqueness hold there, it is sufficient to analyze the dynamics of the flow caused by the model ( 4) in region Φ .

From model ( 4), we set to zero variables pertaining to HIV dynamics H U = H A = H T = A A = H UI = H UC = H AI = H AC = H TI = H TC = A AI = A AC = 0 , and is given by

E C ∗ = S ∗ I C ∗ C C ∗ 0 0 0 0 0 0 0 0 0 0 0 0 with S ∗ = Λ η σ c + μ + d c + r g 1 R eC I C ∗ = r + μ + d c c b c 1 − ψξ g 1 S ∗ C C ∗ = η σ c c b c 1 − ψξ g 1 S ∗ (15)

Where g 1 = ημ σ c + η d c σ c + μ 2 + μr + μ d c + η d c σ c η σ c + ω σ c + μ r + μ + d c . Lemma 4:

The unique endemic equilibrium E c * is said to be globally asymptotically stable for model system ( 4) if R e C > 1 and R HC < 1 .

As a result of trace J being negative and the determinat J being positive, the steady state is locally asymptotically stable.

The above statement for trace and determinant is true whenever R ₀ < 1 for DFE and for endemic equilibrium, whenever R ₀ > 1.

In order to demonstrate E C ∗ ′ s global stability, firstly we observe the domain S I C C C ≥ 0 S + I C + C C < Λ μ is positively invariant and attractive for the 3D system. Adopting Bendixson and Dulac's negative criterion to eliminate the presence of the periodic orbits using the expression 1 I C and 1 C C as the Dulac multiplier, we obtain S ′ I C = Λ I C − λ c S S + I C + C C − μS I C C C + ω + r , I C ′ I C = λ c S S + I C + C C − η + ω + μ , C C ′ C C = λ c S S + I C + C C + η − r + μ + d c (18)

When the right side of the first equation is differentiated with regards to S , the second equation with regards to I c and the right side of the third equation is differentiated with regards to C c , − λ c S S + I C + C C + λ c S S + I C + C C 2 − μS I C C C < 0 , − λ c S S + I C + C C 2 < 0 and − λ c S S + I C + C C 2 < 0 (19)

As the sum of these three expressions are negative, there is no existence of periodic orbits. Consequently, E c * is globally asymptotic for R c > 1 and R HC < 1 .

This occur by setting to zero the variables pertaining to HCV dynamics ( I C = C C = H UI = H UC = H AI = H AC = H TI = H TC = A AI = A AC = 0 and is given by S h ∗ , H U ∗ , H A ∗ , H T ∗ , A A ∗ , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 which is present when R H > 1 exists, the endemic steady states can be computed. So that, S ∗ = Λ Λ φ − k 1 Λ μφ − λ k 1 − μ k 1 H U ∗ = Λ λ Λ μφ − λ k 1 − μ k 1 H A ∗ = Λ αλ Λ μφ − λ k 1 − μ k 1 k 2 H T ∗ = α θ 1 k 4 + ρ θ 2 k 2 Λ λ k 2 Λ μυφ θ 2 − Λ μφ k 3 k 4 − λυ k 1 θ 2 + λ k 1 k 3 k 4 − μυ k 1 θ 2 + μ k 1 k 3 k 4 A A ∗ = Λ λ αυ θ 1 + ρ k 2 k 3 k 2 Λ μυφ θ 2 − Λ μφ k 3 k 4 − λυ k 1 θ 2 + λ k 1 k 3 k 4 − μυ k 1 θ 2 + μ k 1 k 3 k 4 (20)

Here, the system of equations in ( 4) is set to zero and the endemic stable states are determined. S = S ∗ , H A = H A ∗ , H T = H T ∗ , A a = A A ∗ , I c = I c ∗ , C c = C c ∗ , H uI = H uI ∗ , H AI = H AI ∗ , H TI = H TI , H UC = H UC ∗ , H AC = H AC ∗ , H TC = H TC ∗ A AI = A AI ∗ and A AC = A AC ∗ so that; S ∗ = ω 0 + η 0 σ c + γ λ h + μ λ c I C ∗ H U ∗ = Λ λ c τ λ h + k 5 + A φ Λ τ λ c τ λ h + k 5 I C ∗ H A ∗ = A 7 + A 8 I C ∗ H T ∗ = A 24 + A 25 I C ∗ A A ∗ = A 26 + A 27 I C ∗ I C ∗ = Λ λ c τ λ h + k 5 + A λ c τ λ h + k 5 + λ h ω 0 + η 0 σ c + γ λ h + μ λ c + ω ε 1 σ c A 7 + λ 8 + r 2 A 3 + A 4 − δ 1 λ c Λ τ λ h + k 5 + A φ Λ τ λ h + k 5 − k 1 λ c Λ τ λ h + k 5 + A φ Λ λ c τ λ h + k 5 C C ∗ = η 0 σ c τ λ h + k 5 I C ∗ H uI ∗ = δ 1 λ c k 6 φ + δ 1 A + γ λ h Λ φ τ λ h + k 5 k 6 φ Λ τ λ h + k 5 I C ∗ H AI ∗ = A 5 + A 6 I C ∗ H TI ∗ = θ 3 A 5 + A 6 A 32 k 8 + δ 3 λ c A 24 + A 25 A 32 k 8 + θ 4 η ϵ 2 σ c δ 1 λ c k 6 k 9 φ + A 1 A 32 k 8 H uC ∗ = η ε 2 σ c δ 1 λ c k 6 k 9 φ + A 1 I C ∗ H AC ∗ = A 3 + A 4 I C ∗ H TC ∗ = A 9 + A 10 I C ∗ A AI ∗ = A 30 + A 31 I C ∗ A AC ∗ = A 28 + A 29 I C ∗ (21)

Where the values of k 1 − k 13 and A 1 − A 31 are given in appendix A, available from the source code link provided.

We want to consider how the reproduction number of HCV, R C and reproduction number of HIV R H impact one another as follows: ∂ R H ∂ C ∗ ∂ C ∂ R C = ∂ R H ∂ R C = b h M k 5 η 0 σ c + ω 0 σ c + μ k 2 Λφ − k 1 θ 2 μ + μ + ν 1 μ + d a b c η 0 σ c − k 5 (22)

which is the total sum of new HIV infections that one person with HIV will cause in a population where HCV is already common. Even if R C > 1 > R H , HIV will be allowed to spread into a population where HCV is common if R HC is greater than 1. In other words, R HC > 1 which shows the presence of HCV makes it easier for HIV to spread in a community. But for R HC < 1 , HCV is still the biggest health issue, even though HIV has been spread to a population where HCV was already common and vice versa.

Taking the partial derivative of R HC in ( 22) with regards to b h , we have ∂ R HC ∂ b h = c h M k 5 η 0 σ c + ω 0 σ c + μ k 2 Λφ − k 1 θ 2 μ + μ + ν 1 μ + d a c c b c η 0 σ c − k 5 > 0 . (23)

Where M = α 1 θ 1 υ 1 + k 2 k 3 ρ 1 + k 2 μ + ν 1 μ + d a + k 2 θ 2 μ

Any time R c > 1 , equation (14)'s positive result shows that the existences of HCV accelerates the spread of HIV infections in a community and vice versa.

From ( 22) since the partial derivatives with respect to R C is positve, this signifies that as the reproduction number of HCV, R C increases, it impacts the reproduction number of HIV R H . Then, we should simply allow HCV infection to reduce to avoid increased viral load in HIV-infected individuals because any slight increase in HCV will make HIV increase.

Furthermore, from ( 23) since the partial derivatives with respect to b _h is also positive, this simply implies that parameter b _h is sensitive to reproduction number of HIV-HCV. The implication is that any increase in transmission coefficient of HIV; b _h , will result in increase in the reproduction number of HIV-HCV. In the same manner, any increase in transmission coefficient of HCV; b _c , will result in increase in the reproduction number of HIV-HCV. This also suggest that these parameters are parameters to watch out for when controlling the transmission of HIV and HCV coinfection.