Modelling HIV/AIDS and Tuberculosis Coinfection
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1 Modelling HIV/AIDS and Tuberculosis Coinfection C. P. Bhunu 1, W. Garira 1, Z. Mukandavire 1 Department of Applied Mathematics, National University of Science and Technology, Bulawayo, Zimbabwe 1 Abstract An HIV/AIDS and TB coinfection model which considers anti-retroviral therapy for the AIDS cases and treatment of all forms of TB that is, latent and active forms of TB is presented. We start by presenting an HIV/AIDS-TB coinfection model and analyze the TB and HIV/AIDS submodels separately without any intervention strategy. The TB only model is shown to exhibit backward bifurcation when its corresponding reproduction number is less than unity. On the other hand, the HIV/AIDS only model has a globally asymptotically stable disease-free equilibrium when its corresponding reproduction number is less than unity. We proceed to analyze the full HIV-TB coinfection model and extend the model to incorporate anti-retroviral therapy for the AIDS cases and treatment of active and latent forms of TB. The thresholds and equilibria quantities for the models are determined and stabilities analyzed. From the study we conclude that treatment of AIDS cases result in a significant reductions of numbers of individuals progressing to active TB. Further, treatment of latent and active forms of TB results in delayed onset of the AIDS stage of HIV infection. Key words: Threshold quantity, Stability, Treatment, Anti-retroviral therapy, Coinfection. 1 Introduction Tuberculosis TB) is an air-borne transmitted disease and in human beings is caused by Mycobacterium tuberculosis bacteria Mtb). Mtb droplets are released into the air by caughing and/ or sneezing infectious individual. Tubercle bacillus carried by such droplets live in the air for a short period of time Song et al. [37]), about 2 hours) and therefore it is believed that occasional contacts with an infectious case rarely leads to an infection. TB is described as a slow disease because of its long and variable latency period distribution and its short and relatively narrow infectious period distribution. However in the presence of HIV infection, the TB latency period tends to be short with infected individuals becoming sick with TB quickly as their immune system is compromised. Most secondary infections are a result of prolonged and sustained close contacts with a primary case or exogenous re-infection Feng et al. [17], Styblo [38]). There is strong evidence that TB transmission occurs in groups of close associates of infectious individuals and that such a risk is limited to the life of the epidemiologically active cluster to which they belong. As the world is experiencing the devasting effects of HIV/AIDS epidemic, it is now necessary to ask why we have so far failed to control TB and define the limits of the global TB control programmes Raviglione and Pio [35]). Currently half of the people living with HIV are TB co-infected and three quarters of all dually infected people live in Sub-Saharan Africa. In Sub-Saharan Africa the face of HIV/AIDS is TB. HIV/AIDS and TB fuel one another. Preventive therapy of TB in HIV infected individuals is highly recommended WHO [43]) and could dramatically reduce the impact of HIV on Corresponding author: Modelling Biomedical Systems Research Group, Department of Applied Mathematics, National University of Science and Technology, P. O. Box AC 939 Ascot, Bulawayo, Zimbabwe. address: cpbhunu1762@nust.ac.zw, cpbhunu@gmail.com, cpbhunu@yahoo.com 1
2 TB epidemiology, but its implementation is limited in developing countries because of complex logistical and practical difficulties Frieden [19]). Control programmes have continued to function as if the TB epidemiological situation is stable and indeed all approaches including Directly Observed Treatment Short Course DOTS) strategy have so far failed to control TB in areas of high HIV/AIDS prevalence De Cork and Chaisson [13]). The implementation of a universal strategy is thus challenged on operational, epidemiological, economic and social grounds. The question posed is whether TB control should remain a bio-medical strategy only, focusing on treatment without efforts to understand and fulfill patient needs social and economic needs). The causes behind recent observed increases of active TB cases are the source of many studies Aparicio et al.[4], Porco and Blower [34], Davis [12]). Active TB cases may be pulmonary or extra pulmonary, but it s only pulmonary cases that are infectious and form the bulk of most cases of active TB. The usual symptoms of active TB include tiredness, high fever, and a cough, but confirmation of active TB requires a positive sputum culture. Extra pulmonary TB accounts for between 5% and 30% of the total cases and may affect any part of the body. Pulmonary cases affect the lungs. Recently infected individuals have a high chance of developing active TB within 5 years and these are classified as primary TB cases and those who progress to active TB many years after infection as a result of endogenous reactivation and/ or exogenous re-infection are classified as secondary active TB cases. The vast majority 90%) of people infected with Mtb do not develop TB disease. HIV is the most powerful risk for progression from TB infection to TB disease Naresh and Tripath [40]). An HIV positive person infected with Mtb has a 50% chance of developing active TB against a 10% chance for the HIV negative Naresh and Tripath [40]). Therefore HIV infection increases the development of TB fivefold. TB is the most common serious opportunistc infection occuring among HIV positive individuals and occurs in more than 50% of the AIDS cases in developing countries Naresh and Tripath [40]). Coinfection is the simultenous infection of the same host with two different pathogens or two different strains of the same pathogen and leads to coexistence of strains pathogens) at population level May and Nowak [28]). A lot of ground work has been covered in the mathematical modeling of coinfection of different pathogens strains) [20, 27, 28, 33, 42] though very little was done in the modeling of HIV-TB coinfection. Naresh and Tripath [40] studied an HIV-TB coinfection model which assumes that AIDS cases are non-infectious and did not include all stages of HIV and TB interaction. In their work they did not include anti-hiv treatment. In this paper we incorporate all aspects of TB transmission dynamics as well as aspects of HIV transmission dynamics to come up with a distinct detailed coinfection model for HIV and TB. This paper among other aspects incorporates anti-retroviral therapy for the AIDS cases and analyse its implications on TB. We investigate the implications of treatment of all forms TB on HIV. The paper is organized as follows. Section 2 presents description for the HIV and TB coinfection model. Sections 3 and 4 present TB and HIV/AIDS submodels respectively and their corresponding analysis. In Section 5 there is analysis of the full model and in Section 6 we extend the full coinfection model in Section 2 to incorporate treatment of AIDS cases and treatment of all forms of TB and analyze the model. Section 7 presents numerical simulations and finally we present summary and concluding remarks. 2 Model Description The model subdivides the human population into the following sub-population of susceptible individuals S), those exposed to TB only E T ), individuals with symptoms of TB, I T ), those who have recovered with temporal immunity R T ), those infected with HIV only but show no clinical symptoms of AIDS I H ), HIV infected individuals pre-aids) exposed to TB E TH ), HIV infected displaying AIDS symptoms A H ), AIDS individuals exposed to TB E AT ), HIV infected individuals pre-aids) displaying TB symptoms I TH ) and AIDS individuals dually infected with TB A AT ). It is assumed that susceptible humans are recruited into the population at per capita rate. Susceptible individuals acquire HIV infection following contact with HIV infected individuls at a rate λ H and and acquire TB infection following contact with an infectious individual at a rate λ T. The total population size at time 2
3 time t is Nt) and is given by, Nt) = St) + E T t) + I T t) + R T t) + I H t) + A H t) + E TH t) + E AT t) + I TH t) + A AT t). 1) It is assumed that invidividuals suffering from TB may naturally recover and enter the recovered class R T ) at constant rate p. All individuals in different human subgroups suffer from natural death at a constant rate µ). The force of infection λ H ), associated with HIV infection is λ H = β H N [I H + E TH + I TH ) + η A A H + η TH θ TH A AT + E AT ))]. 2) In 2), β H is the effective contact rate for HIV infection contact sufficient to result in HIV infection) the modification parameter η TH 1 accounts for the relative infectiousness of individuals infected with HIV in the AIDS stage exposed to TB E AT ), in comparison to those solely infected with HIV in the AIDS stage A H ). Further, θ TH 1 models the fact that dually infected people in the AIDS stage displaying symptoms of TB A AT ) are more infectious than the corresponding dually infected individuals in the AIDS stage who are only exposed to TB, E AT ). Finally the parameter η A > 1 captures the fact that individuals who are in the AIDS stage of infection are more infectious than HIV-infected individuals with no AIDS symptoms. This is so because people in the AIDS stage have a higher viral load compared to other HIV infected individuals with no symptoms and there is a positive correlation between viral load and infectiousness. Similarly the rate of TB infection in humans is λ T = β T ci T + I TH + A AT ) N where β T is the probability that one individual being infected with with one infectious individual and c is per capita contact rate. Susceptibles infected with Mtb enter the latency class at rate λ T and then progress to active TB at rate k. Individuals latently infected with Mtb also progress to active TB, as a result of re-infection at rate ψ 1 λ T with ψ 1 0, 1) since primary infection confers some degree of immunity in the absence of HIV infection. Individuals infected with TB can acquire HIV at rates λ H and δλ H for the exposed and infectives respectively. Individuals with TB suffer disease induced death at rate d T and recover at rate p. Individuals infected with HIV only with no symptoms) are generated following infection at rate λ H. This further following progression to AIDS at rate ρ 1 and through being infected with Mtb to enter the E TH. Individuals infected with HIV exposed to TB develop active TB at a constant rate t 1 k and develop AIDS at a constant rate γ 1 ρ 1 with the modification parameters t 1 > 1 and γ 1 > 1 respectively. These individuals in E TH also develop TB as a result of re-infection at rate ψ 2 λ T with ψ 2 > 0 being the modification parameter. Individuals in I TH class die due to TB at rate d T and progress to AIDS at rate γ 2 ρ 1 with the modification parameter γ 2 > 1. Individuals exposed to TB in the AIDS stage of HIV infection develop active TB at a constant rate t 2 k with the modification parameter t 2 > 1. These individuals also develop TB at rate ψ 3 λ T with ψ 3 > 1. The population of individuals with AIDS alone is generated following progression to AIDS with people infected with HIV only at rate ρ 1. The population of individuals with AIDS and symptoms of TB is generated by progression to AIDS by individuals dually infected with HIV and TB at rate γ 2 ρ 1 and by progression to active TB by individuals exposed to TB and in the AIDS stage of HIV infection at a rate t 2 k. Individuals infected with HIV and in the AIDS stage die at a rate d A. The assumptions result in the following differential equations that describe 3) 3
4 the interaction of the two diseases model. ds = λ T S λ H S µs, di H da H = λ H S + λ H R T ρ 1 + µ)i H λ T I H = ρ 1 I H σλ T A H µ + d A )A H de TH = λ H E T + λ T I H γ 1 ρ 1 + t 1 k + µ)e TH ψ 2 λ T E TH de AT di TH = γ 1 ρ 1 E TH + σλ T A H t 2 k + µ + d A )E AT ψ 3 λ T E AT = δλ H I T + t 1 ke TH µ + d T + γ 2 ρ 1 )I TH + ψ 2 λ T E TH 4) da AT = γ 2 ρ 1 I TH + t 2 ke AT µ + d T + ɛd A )A AT + ψ 3 λ T E AT de T di T dr T = λ T S + λ T R T ψ 1 λ T E T λ H E T µ + k)e T = ψ 1 λ T E T + ke T µ + p + d T )I T δλ H I T + qr T = pi T qr T λ T R T λ H R T µr T. The model flow diagram is shown in Figure 1. 4
5 The model has initial conditions given by, Figure 1: Structure of model. S0) = S 0 0, E T 0) = E T0 0, I T 0) = I T0 0, R T 0) = R T0 0, I H 0) = I H0 0, A H 0) = A H0 0, E TH 0) = E HT0 0, E AT 0) = E AT0 0, 5) I TH 0) = I HT0 0, A AT 0) = A AT0 0. Based on biological considerations model sytem 4) will be studied in the following region, { G = S, E T, I T, R T, I H, A H, E TH, E AT, I TH, A AT ) R 10 + : Nt) }, 6) µ which is positively invariant with respect to model system 4). 5
6 2.1 Positivity and boundedness of solutions Model system 4) describes human population and therefore it is necessary to prove that all the variables St), E T t), I T t),, A AT t) are non-negative for all time. Solutions of the model system 4) with positive initial data remains positive for all time t 0 and are bounded in G. Theorem 1. Let St) 0, E T t) 0,, A AT t) 0. The solutions S, E T,, A AT of model system 4) are positive for t 0. For the model system 4), the region G is positively invariant and all solutions starting in G approach, enter, or stay in G. Proof. Under the given initial conditions, it is easy to prove that the components of solutions of model system 4) are positive if not we assume a contradiction that there exist a first time t 1 : St 1 ) = 0, S T t 1) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 1 or there exists a t 2 : E T t 2 ) = 0, E T t 2) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 2 or there exists a t 3 : I T t 3 ) = 0, I T t 3) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 3 or there exists a t 4 : R T t 4 ) = 0, R T t 4) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 4 or there exists a t 5 : I H t 5 ) = 0, I H t 5) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 5 or there exists a t 6 : A H t 6 ) = 0, A H t 6) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 6 or there exists a t 7 : E TH t 7 ) = 0, E T H t 7 ) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 7 or there exists a t 8 : E AT t 8 ) = 0, E A T t 8 ) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 8 or there exists a t 9 : I TH t 9 ) = 0, I T H t 9 ) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 9 or there exists a t 10 : A AT t 10 ) = 0, A A T t 10 ) < 0 S T t) > 0, E T t) > 0,, A AT t) > 0 for 0 < t < t 10. In the first case we have, S t 1 ) = > 0, 7) which is a contradiction meaning that S T, remains positive. In the second case we have, E T t 2) = St 2 ) + R T t 2 ) ) λt 2 ) > 0, 8) which is a contradiction meaning that E T, remains positive. In the third case we have, I T t 3) = ke T t 3 ) + qr T t 3 ) > 0, 9) which is a contradiction meaning that I T, remains positive. In the fourth case we have, R T t 4) = pi T t 4 ) > 0, 10) which is a contradiction meaning that R T, remains positive. In the fifth case we have, I Ht 5 ) = β H N ETH t 5 ) + I TH t 5 ) + η A AH t 5 ) + η TH θ TH A AT t 5 ) + E AT t 5 )) )) St 5 ) + R T t 5 )) > 0, which is a contradiction meaning that I H, remains positive. In the sixth case we have, 11) A H t 6) = ρ 1 I H t 6 ) > 0, 12) which is a contradiction meaning that A H, remains positive. In the seventh case we have, E T H t 7 ) = β H N IH t 7 ) + I TH t 7 ) + η A AH t 7 ) + η TH θth A AT t 7 ) + E AT t 7 ) ))) + λ T t 7 )I H t 7 ) > 0, 13) 6
7 which is a contradiction meaning that E TH, remains positive. In the eighth case we have, E A T t 8 ) = γ 1 ρ 1 E TH t 8 ) + σλt 8 )A H t 8 ) > 0, 14) which is a contradiction meaning that E AT, remains positive. In the nineth case we have, I T H t 9 ) = δ β H N IH t 9 ) + E TH t 9 ) + η A AH t 9 ) + η TH θth A AT t 9 ) + E AT t 9 ) ))) I T t 9 ) + t 1 ke TH t 9 ) + ψ 2 E TH t 9 ) > 0, 15) which is a contradiction meaning that I TH, remains positive. In the final case we have, A A T t 10 ) = γ 2 ρ 1 I TH t 10 ) + t 2 ke AT t 10 ) + ψ 3 λ T E AT t 10 ) > 0, 16) which is a contradiction meaning that A AT remains positive. Thus in all cases S, E T,, A AT remain positive. Since Nt) I T t) + E AT t) + I TH t) + A AT t) + A H t), then µ + d T + d A + ɛd A )N N t) µn, 17) implies that Nt) is bounded and all solutions starting in G approach, enter or stay in G. 3 TB submodel We have the TB submodel when E TH = E AT = I TH = A AT = I H = A H = 0 and is given by, S t) = λs µs, E T t) = λs + R T ) ψ 1 λe T µ + k)e T, I T t) = ψ 1λE T + ke T µ + d T + p)i T + qr T, 18) R T t) = pi T qr T λr T µr T, with S0) = S 0 0, E T 0) = E T0 0, I T 0) = I T0 0, R T 0) = R T0 0 as the initial conditions and λ = β T ci T N T is the force of infection and the total population is given by N T t) = St) + E T t) + I T t) + R T t). Based on biological considerations submodel system 18) will be studied in the following region, { G = S, E T, I T, R T ) R 4 + : N T t) }. 19) µ It can be easily shown that the solutions S, E T, I T, R T of submodel system 18) are positive for t 0 and that, the region G is positively invariant and solutions starting in G approach, enter or stay in G. 3.1 Disease free equilibrium and stability analysis The disease free equilibrium is given as, Q 0 = S 0, E T0, I T0, R T0 ) = ), 0, 0, 0. 20) µ The basic reproduction number is defined as the number of secondary infections produced by a single infectious individual during his or her entire infectious period. Mathematically the reproduction number 7
8 is defined as a spectral radius. The spectral radius R 0 which is a threshold quantity for disease control defines the number of new infections generated by a single infected individual in a fully susceptible population van den Driessche and Watmough [41]). In our case the reproduction number, R 1 is defined as the number of TB infections produced by active TB case. We use the later s approach to determine the reproductive number of the submodel system 18). Thus we have, F = λs + R T ) and V = ψ 1 λe T + µ + k)e T µ + d T + p)i T qr T ke T ψ 1 λe T µ + q)r T + λr T pi T µs + λs The infected compartments are E T, I T and R T. Thus 0 β T c 0 µ + k 0 0 F = and V = k µ + d T + p) q p µ + q). 21). 22) Thus the TB induced reproduction number, which is the spectral radius of dominant eigenvalue is ρf.v 1 ) = R 1 = Theorem 2 follows from [41] Theorem 2). β T cµ + q) µ + k) µ + q)µ + d T + p) pq). 23) Theorem 2. The disease-free equilibrium, Q 0 is locally asymptotically stable when R 1 < 1 and unstable for R 1 > 1. Theorem 2 can also be proven using the Jacobian matrix as follows. The Jacobian matrix of the submodel system 18) at Q 0 is given as, µ 0 β T c 0 JQ 0 ) = 0 µ + k) β T c 0 0 k µ + d T + p) q, 0 0 p µ + q) and T race[jq 0 )] < 0. 24) Det[JQ 0 )] = µ + k) µ + q)µ + q + d T ) pq) kµ + q)β T c > 0, when µ + q)kβ T c µ + k) µ + q)µ + d T + p) pq) < 1. µ + q)kβ T c Thus R 1 = µ + k) µ + q)µ + d T + p) pq) < 1, which means that eigenvalues of Det[JQ 0) λ] = 0 have negative real parts implying Q 0 is locally asymptotically stable whenever R 1 < 1. We now list two conditions that if met, also guarantee the global asymptotic stability of the disease free state. Rewriting model system 18) as, dx = F X, Z), dz 25) = GX, Z), GX, 0) = 0, where X = S, R T ) and Z = E T, I T ) ), with X R 2 denoting its components) the number of uninfected individuals and Z R 2 denoting its components) the number of infected individuals including the latent and the 8
9 infectious. The disease free equilibrium is now denoted by, Q 0 = X, 0), where X = ) µ, 0. 26) The conditions H1) and H2) in equation 27) must be met to guarantee local asymptotic stability. H1 For dx = F X, 0), X is globally asymptotic stable g.a.s) H2 GX, Z) = BZ ĜX, Z), ĜX, Z) 0 for X, Z) G, 27) where, B = D Z GX, 0) is an M-matrix the off diagonal elements of B are nonnegative) and G is the region where the model makes biological sense. If system 25) satisfies the conditions in 27) then Theorem 3 holds. Theorem 3. The fixed point Q 0 = X, 0) is a globally asymptotically stable equilibrium of system 25) provided that R 1 < 1 and assumptions in 27) are satisfied. Proof. In Theorem 2 we have proved that for R 1 < 1, Q 0 is locally asymptotically stable. Consider [ ] µs F X, 0) =, 28) 0 Then GX, Z) = BZ ĜX, Z), B = [ µ + k) βt c k µ + d T + p) Ĝ 1 X, Z) ĜX, Z) = = Ĝ 2 X, Z) β T ci T 1 S + R ) T β T ci T + ψ 1 E T N T N T ψ 1 λe T qr T ]. 29). 30) Thus Ĝ2X, Z) < 0 and this implies that ĜX, Z) is not greater or equal to zero. Conditions in 27) are not satisfied thus Q 0 may not be a globally asymptotically stable. Backward bifurcation Feng et al., [17]) occurs at R 1 = 1 and a double endemic equilibria exists for R c < R 1 < 1, where R c is a positive constant. But in the absence of exogenous re-infection and disease relapse Q 0 is globally asymptotically stable. 3.2 Endemic equilibria stability analysis The endemic equilibrium point is here denoted by Q where, Q = S 2, E T2, I T2, R T2 ), and the quantities S 2, E T2, I T2, R T2 in terms of the equilibrium value of the force of λ are given by, S 2 = µ + λ, R T2 = pλ ψ 1 λ + k) ψ 1 λ + µ + k) µ + d T + p)µ + q + λ ) qp) pλ ψ 1 λ + k), 31) I T2 = E T2 = λ ψ 1 λ + k)µ + q + λ ) ψ 1 λ + µ + k) µ + d T + p)µ + q + λ ) qp) pλ ψ 1 λ + k) µ + d T + p)µ + q + λ ) qp) λ ψ 1 λ + µ + k) µ + d T + p)µ + q + λ ) qp) pλ ψ 1 λ + k). 9
10 Substituting equation 31) into the equilibrium value for the force of infection λ in we obtain λ gλ ) = λ Aλ 2 + Bλ + C) = 0, 32) where λ = 0 corresponds to the disease free equilibrium and gλ ) = 0 corresponds to the existence of endemic equilibria where A = ψ 1 µ + d T β T cµ), B = ψ 1 µ + d T + p)µ + q) pq) µ + q)β T cµ) + µ + d T + p)µ + k) kp + β T cµ), 33) C = µ + k) µ + d T + p)µ + q) pq) β T cµkµ + q). By examining the quadratic equation we see that there is a unique endemic equilibrium if A > 0, B < 0 and C = 0 or A > 0 and B 2 4AC = 0 or A < 0, B > 0 and C = 0 or A < 0 and B 2 4AC = 0. There are two if A > 0, C > 0, B < 0 and A > 0, B 2 4AC > 0, and there is non-otherwise. The coefficient A is positive or negative if µ + d T is greater than or less than β T cµ and C is positive or negative if µr 1 is less than or greater than one respectively. We therefore rewrite these conditions in Lemma 1. Lemma 1. Model system 18) has precisely one unique endemic equilibrium if A > 0, B < 0 and C = 0 or A > 0 and B 2 4AC = 0 or A < 0, B > 0 and C = 0 or A > 0 and B 2 4AC = 0, precisely two endemic equilibria if A > 0, C > 0, B < 0 and B 2 4AC > 0 or A < 0, C < 0, B > 0 and B 2 4AC > 0, otherwise there are none. To find the backward bifurcation point, we set the discriminant B 2 4AC = 0 and make R 1 the subject of the formulae to obtain R1 c = 1 B 2 ) 1, 34) µ 4Aµ + k) µ + d T + p)µ + q) pq) from which it can be shown that backward bifurcation occurs for values of R 1 in the range R c 1 < R 1 < 1. 4 HIV/AIDS submodel We have the HIV/AIDS submodel when E TH = E AT = I TH = A AT = E T = I T = R T = 0 and is given by, S t) = λ H1 S µs, I H t) = λ H 1 S ρ 1 + µ)i H, A H t) = ρ 1I H µ + d A )A H, 35) with S0) = S 0 0, I H 0) = I H0 0, A H 0) = A H0 0 as the initial conditions and λ H1 is the force of infection and is given by, λ H1 = β H I H + η A A H ), N H 36) and the total population for the submodel is N H t) = St) + I H t) + A H t). Based on biological considerations submodel system 35) will be studied in the following region, { G = S, I H, A H ) R 3 + : S + I H + A H µ which is positively invariant with respect to submodel system 35). }, 37) 10
11 4.1 Persistence In this section we look for the conditions under which the host population and disease will persist. We state Theorem 4 and Theorem 5 whose proofs are given in Thieme [39] and use them for proving Theorem 6. Theorem 4. Let X be a locally compact metric space with metric d. Let X be the disjoint union of two sets X 1 and X 2 such that X 2 is compact. Let Φ be a continous semiflow on X 1. Then X 2 is a uniform strong repeller for X 1, whenever it is a uniform weak repeller for X 1. Theorem 5. Let D be a bounded interval in R and g : t 0, ) D R be bounded and uniformly continuous. Further, let x : t 0, ) D be a solution of x = gt, x), which is defined on the whole interval t 0, ). Then there exist sequences s n, t n such that lim gs n, x ) = 0 = lim gt n, x ). n n We now state Corollary 1 for use in proving Theorem 6. Corollary 1. Let assumptions of Theorem 5 be satisfied. Then a) lim inf t b) lim inf t gt, x ) 0 lim sup gt, x ), t gt, x ) 0 lim sup gt, x ). t Rewriting submodel system 35) as S t) = β HN H )I H + η A A H ) S µs, N H I Ht) = β HN H )I H + η A A H ) N H S ρ 1 + µ)i H, 38) A H t) = ρ 1I H µ + d A )A H, β H N H ) can take various forms and in view of that we make the following assumptions. a) β H N H ) is a continous for N H 0 and continously differentiable in N H > 0. b) β H N H ) is monotone nondecreasing in N H. c) β H N H ) > 0 if N H > 0. It is convenient to reformulate the model in terms of the fractions of the susceptible, infected and sick parts of the population, x = S, y = I H, z = A H, 39) N H N H N H and express 38) in these terms to obtain, N H = µ + d A z)n H x = µ 1 x) β HN H )y + η A xz) + d A xz y = β H N H )yx + η A xz) ρ 1 + N H ) y + d A yz 40) z = ρ 1 y + d A zz 1) N H z 11
12 Equations 39) suggests that x + y + z = 1. 41) The manifold x + y + z = 1, x, y, z 0 is forward invariant under the solution of flow of 40) which implies that, for any initial data satisfying 41), the system 40) has a global solution satisfying 39). We now show conditions under which the host population will persist. Theorem 6. Let β H 0) = 0, N H 0) > 0. Then the population is uniformly persistent that is with ɛ > 0 not depending on initial data. Proof. We have to show that the set is a uniform strong repeller for lim inf t N Ht) ɛ, X 2 = {N H = 0, x 0, y 0, z 0, x + y + z = 1}, X 1 = {N H > 0 x 0, y 0, z 0, x + y + z = 1}. As assumptions of Theorem 4 are satisfied it is enough to show that X 2 is a uniform weak repeller for X 1. Let r = y + z then, r = β H N H )yx + η A xz) β H N H )1 + η A ) N H r + d A zr d A z N H r + d A r 1), using the fact that x, y, z, r 1. This implies that, NH r + 1 r )d A β H NH )1 + η A ) β H N H ) From the N H equation in 40) we have, lim inf t 1 dn H N H Hence N H increases exponentially unless N H N H µ + d A r, that is r N H 1 + η A) + 1 r )d A 1 + η A. µ + d A z ) 1 d A N H Combining 43) and 44) we obtain ) β H NH ) d A NH 1 + η A) 1 N H N H µ + d A r ). 42) 43) ) µ r. 44) ) µ + d A. 45) 1 + η A As β H 0) = 0 and β H N H ) is continuous at 0, NH From 45) we see that we can relax β H 0) = 0 and require ) β H 0) < d A NH 1 + η A) 1 ɛ > 0 with ɛ not depending on the initial data. N H ) µ + d A. 1 + η A We now look for conditions under which the disease is persistent or endemic in the population. The disease is persistent in the population if the fraction of the infected and AIDS cases is bounded away from zero. If the population dies out and the fraction of the infected and AIDS remains bounded away from zero, we would still say that the disease is persistent in the population. 12
13 Proposition 1. Let β H )1 + η A ) far as with ɛ > 0 being independent of the initial data, provided that r0) > 0. The proof of Proposition 1 is outlined in Thieme [39]. 4.2 Disease-free equilibrium and stability analysis The disease-free equilibrium point of model system 35) is N r. Then the disease is uniformly weakly persistent in so r = lim sup rt) ɛ, 46) t A 0 = S 0, I H0, A H0 ) = µ, 0, 0 ). 47) The basic reproduction number, R 2 for model system 35) is defined as the number of secondary HIV/AIDS cases produced by one HIV positive individual during his/ her entire life. We employ Van den Driessche and Watmough [41]) to determine R 2. Thus we have F = λ H1 S 0 0 and V = ρ 1 + µ)i H ρ 1 I H + µ + d A )A H λ H + µ)s. 48) The infected compartments are I H and A H are the only infected components. Thus, [ ] [ ] βh β F = H η A ρ1 + µ) 0 and V =. 49) 0 0 ρ 1 µ + d A ) Thus the HIV/AIDS induced reproduction number, which is the spectral radius of the dominant eigenvalue is, ρf.v 1 ) = R 2 = β Hµ + d A + η A ρ 1 ) ρ 1 + µ)µ + d A ). 50) For model system 35), it can be established that the disease free equilibrium is locally asymptotically stable whenever R 2 < 1 and unstable when R 2 > 1. Theorem 7 follows from [41] Theorem 2) Theorem 7. The disease-free equilibrium, A 0 is locally asymptotically stable when R 2 < 1 and unstable for R 2 > 1. We now list two conditions that if met, also guarantee the global asymptotic stability of the disease free state. Rewriting model system 35) as, dx = F X, Z), dz 51) = GX, Z), GX, 0) = 0, where X = S) and Z = I H, A H ) ), with X R 1 + denoting its components) the number of uninfected individuals and Z R 2 + denoting its components) the number of infected individuals including the latent and the infectious. The disease free equilibrium is now denoted by, A 0 = X, 0), where X = ). µ 52) 13
14 The conditions H1) and H2) in equation 53) must be met to guarantee local asymptotic stability. H1 For dx = F X, 0), X is globally asymptotic stable g.a.s) H2 GX, Z) = AZ ĜX, Z), ĜX, Z) 0 for X, Z) G, 53) where, A = D Z GX, 0) is an M-matrix the off diagonal elements of A are nonnegative) and G is the region where the model makes biological sense. If system 51) satisfies the conditions in 53) then Theorem 8 holds. Theorem 8. The fixed point A 0 = X, 0) is a globally asymptotically stable equilibrium of system 51) provided that R 2 < 1 and assumptions in 53) are satisfied. Proof. In Theorem 7, A 0 is locally asymptotically stable for R 2 < 1. Consider Then F X, 0) = [ µs ], 54) [ ] GX, Z) = AZ ĜX, Z), A = βh ρ 1 + µ) β H η A. 55) ρ 1 µ + d A ) Ĝ 1 X, Z) ĜX, Z) = = Ĝ 2 X, Z) β H 1 1 ) I H + η A A H ) N 0. 56) Thus Ĝ1X, Z) 0 and Ĝ2X, Z) = 0 ĜX, Z) 0. Conditions in 53) are satisfied thus A 0 is globally asymptotically stable for R 2 < Stability analysis of the endemic equilibrium The endemic equilibrium is given by A = S, I H, A H) = NH R 2, R 2 µn H ρ 1 + µ)r 2, Theorem 9. The endemic equilibrium, A exists whenever R 2 > 1. Proof. The disease is endemic when I H t) > 0 and A H t) > 0 that is and I H < β H I H + η A A H S < β H N H ρ 1 + µ ρ 1 + µ I H + η A A H ), I H < β H ρ 1 + µ I H + η A A H ), using the fact that A H > ) ρ 1 R 2 µn H ). 57) R 2 ρ 1 + µ)µ + d A ) S N H 1, 58) ρ 1I H µ + d A. 59) Substituting 59) into 58) we obtain, I H < β H ρ 1 + µ I H + η A A H ) < β H I H + ρ ) 1η A I H ρ 1 + µ µ + d A I H < β Hµ + d A + ρ 1 η A ) ρ 1 + µ)µ + d A ) I H = R 2 I H 60) 1 < R 2. Thus the endemic equilibrium exists whenever R 2 > 1. 14
15 a) b) c) Figure 2: Graphs showing the behavior of the susceptibles S, the infected I H and those suffering from AIDS A H as R 2. Figures a) represents the susceptible population, b) represents the infected population, c) represents the sick. In this analysis we used µn = 0.01, = 0.029, ρ 1 = 0.1, d A = 1. Figure 2 is a graphical representation of the components of the equilibrium point A in equation 57) and shows changes in the susceptibles S ), HIV positive IH ) and AIDS cases A H ) as the reproduction number, R 2 ) is varied. In the absence of any intervention strategy the susceptible population, will N decrease to zero as R 2 that is = 0 that is in the endemic state of the disease the lim R 2 R 2 susceptible population is reduced to zero as the reproduction number becomes large. The HIV infected individuals, IH and those suffering from AIDS related symptoms increase and then stabilise when they ρ 1 reach and µ + ρ 1 ρ 1 + µ)µ + d A ) respectively as R 2. 15
16 5 Analysis of the full model In this section we now analyse the full model system 4). 5.1 The Disease Free Equilibrium Point and its stability The disease free equilibrium point is given by X 0 = S 0, E T 0, I T 0, R T 0, I H0, A H0, E T H0, E AT 0, I HT 0, A AT 0 ) = ), 0, 0, 0, 0, 0, 0, 0, 0, 0. 61) µ The basic reproduction number, R 0 is defined as the number of secondary infections produced by one infectious individual during his or her entire infectious period. Mathematically R 0 is defined as a spectral radius. The spectral radius R 0 which is a threshold quantity for disease control defines the number of new infections generated by a single infected individual in a fully susceptible population Van den Driessche [41]). In our case R 0 defines the number of secondary TB or HIV) infections due to a single TB infective or single HIV positive individual). We use the later s approach to determine the reproduction number of the model system 4). F = Let, λ T S 0 0 λ H E T + λ T I H σλ T A H δλ H I T 0 λ H S + R T ) 0 0 and V = h 1 = µ + k, h 2 = µ + p + d T, h 3 = µ + q ψ 1 λ T E T + λ H E T + µ + k)e T λ T R T µ + p + d T )I T + δλ H I T qr T ke T ψ 1 λ T E T pi T + µ + q)r T + λ H + λ T )R T γ 1 ρ 1 + t 1 k + µ)e TH + ψ 2 λ T E TH t 2 k + µ + d A )E AT γ 1 ρ 1 E TH + ψ 3 λ T E AT γ 2 ρ 1 + µ + d T )I TH t 1 ke TH ψ 2 λ T E TH µ + d T + ɛd A )A AT t 2 ke AT γ 2 ρ 1 I TH ψ 3 λ T E AT ρ 1 + µ)i H + λ T I H µ + d A )A H + σλ T A H ρ 1 I H λ H + µ)s + λ T S 62). F = V = h 4 = γ 1 ρ 1 + t 1 k + µ, h 5 = t 2 k + µ + d A, h 6 = γ 2 ρ 1 + µ + d T h 7 = µ + d T + ɛd A, h 8 = ρ 1 + µ, h 9 = µ + d A. 0 β T c β T c β T c β H β H η TH η A β H η A η TH θ TH β H β H β H η A h k h 2 q p h h γ 1 ρ 1 h t 1 k 0 h t 2 k γ 2 ρ 1 h h ρ 1 h 9. 63) 64) 16
17 The dominant eigenvalues of F.V 1 are R 1 = β T ckµ + q) µ + k) µ + q)µ + p + d T ) pq ), R 2 = β Hµ + d A + ρ 1 η A ) ρ 1 + µ)µ + d A ), 65) and these correspond to the reproduction numbers for TB transmission model and HIV/AIDS transmission model respectively. Thus the basic reproduction number, R 0 for the full model is given by: Theorem 10 follows from [41] Theorem 2). R 0 = max {R 1, R 2 }. 66) Theorem 10. The disease-free equilibrium point, X 0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1. Theorem 10 can be proven using the Jacobian matrix. The Jacobian matrix at X 0 of model system 4) is given by JX 0 ) = µ 0 β T c 0 β H η TH η A β H β H β T c η TH η A θ TH β H β T c β H β H η A 0 h 1 β T c β T c β T c k h2 q p h h v1 h v2 0 h v4 v3 h β H η TH η H β H β H η TH θ TH η A β H β H h8 β H η A ρ 1 h9. T race[jx 0 )] = h 1 + h 2 + h 3 + h 4 + h 6 + h 7 + h 9 + µ) + β H ρ 1 + µ), < 0 for β H < ρ 1 + µ. Det[JX 0 )] = h 4 h 5 h 6 h 7 β H ρ 1 η A + h 9 β H h 8 h 9 ) h 1 h 2 h 3 + h 3 β T ck) + h 1 pq) µ > 0, for R 2 = β Hρ 1 η A + h 9 ) h 3 β T ck < 1, R 1 = h 9 h 8 h 1 h 2 h 3 pq) < 1, that is R 1 = β T ckµ + q) µ + k) µ + q)µ + p + d T ) pq) < 1, R 2 = β Hµ + d A + ρ 1 η A ) < 1. µ + d A )ρ 1 + µ) 67) It is noted that X 0 is locally asymptotically stable if R 1 < 1 and R 2 < 1, thus the disease dies out. The basic reproduction number is R 0 = max {R 1, R 2 } < 1 from equation 67). Thus disease free equilibrium point X 0 is locally asymptotically stable whenever R 0 < 1 and unstable if either of R i > 1, i = 1, Global stability conditions for the disease free equilibrium when R 0 < 1. For model system 4), it can be established that the disease free equilibrium is locally asymptotically stable whenever R 0 < 1 and unstable when R 0 > 1. In this section we list two conditions that if met, 17
18 also guarantee the global asymptotic stability of the disease free state for the full HIV-TB coinfection model. Rewriting model system 4) as, dx = F X, Z), dz 68) = GX, Z), GX, 0) = 0, where X = S, R T ) and Z = ET, I T, E TH, E AT, I H, I TH, A H, A TH ), with X R 2 + denoting its components) the number of uninfected individuals and Z R 8 + denoting its components) the number of infected individuals including the latent and the infectious. The disease free equilibrium is denoted by, U 0 = X, 0), where X = ) µ, 0. 69) The conditions H1) and H2) in equation 70) must be met to guarantee global asymptotic stability of the disease free equilibrium of model system 4). H1 For dx = F X, 0), X is globally asymptotic stable g.a.s) H2 GX, Z) = AZ ĜX, Z), ĜX, Z) 0 for X, Z) G, 70) where, A = D Z GX, 0) is an M-matrix the off diagonal elements of A are nonnegative) and G is the region where the model makes biological sense. If system 68) satisfies the conditions in 70) then Theorem 11 holds. Theorem 11. The fixed point U 0 = X, 0) is a globally asymptotically stable equilibrium of system 51) provided that R 0 < 1 and that assumptions in 70) are satisfied. Proof. From model system 4) and equation 68) we have: [ µs F X, 0) = q + µ)r T A = g k g g γ 1 ρ 1 g β H m 1 g 5 β H β H η A m t 1 k 0 0 g ρ 1 0 g t 2 k 0 γ 2 ρ 1 0 g 8 ], 71) 72) 18
19 and ĜX, Z) = = Ĝ 1 X, Z) Ĝ 2 X, Z) Ĝ 3 X, Z) Ĝ 4 X, Z) Ĝ 5 X, Z) Ĝ 6 X, Z) Ĝ 7 X, Z) Ĝ 8 X, Z) β T ci T 1 S + R T N ) + E T ψ 1 λ T + λ H ) δλ H I T qr T ψ 1 λ T E T λ H E T + λ T I H ) σλ T A H λ T I H + 1 S + R ) T β H I H + β H E TH + m 1 E AT + β H I TH + β H η A A H + m 2 A AT ) N δλ H I T σλ T A H 0 73) where g 1 = µ + k), g 2 = µ + p + d T ), g 3 = γ 1 ρ 1 + t 1 k + µ), g 4 = t 2 k + µ + d A ), g 5 = β H ρ 1 + µ), g 6 = γ 2 ρ 1 + µ + d T ), g 7 = µ + d A ), g 8 = µ + d T + ɛd A ), m 1 = β H η H η TH, m 2 = β H η H η TH θ TH. 74) H2) in 70) is not satisfied since Ĝ3X, Z) < 0, Ĝ 4 X, Z) < 0 and Ĝ6X, Z) < 0. Consequently U 0 may not be globally asymptotically stable. Thus in this case backward bifurcation as proved in Feng et al. [17] occurs at R 0 = 1 and that double endemic equilibria can be supported for R c < R 0 < 1, where R c is a positive constant. 5.3 Endemic Equilibria and Stability Analysis The model system 4) endemic equilibria corresponds to, X 1 = S 1, 0, 0, 0, I H1, A H1, 0, 0, 0, 0) S 1, 0, 0, 0, I H1, A H1, 0, 0, 0, 0) = N R 2, 0, 0, 0, R 2 µn ρ 1 + µ)r 2, ) ρ 1 R 2 µn), 0, 0, 0, 0, R 2 ρ 1 + µ)µ + d A ) the Mtb free equilibrium, which exists when R 2 > µn. 75) 19
20 X 2 = S 2, E T2, I T2, R T2, 0, 0, 0, 0, 0, 0), the HIV free equilibrium, where in terms of the equilibrium value of the force of infection λ T we have S 2 = µ + λ, T R T2 = I T2 = pλ T ψ 1λ T + k) ψ 1 λ T + µ + k) µ + d T + p)µ + q + λ T ) qp) pλ T ψ 1λ T + k), λ T ψ 1λ T + k)µ + q + λ T ) ψ 1 λ T + µ + k) µ + d T + p)µ + q + λ T ) qp) pλ ψ 1 λ T + k) 76) E T2 = µ + d T + p)µ + q + λ T ) qp) λ T ψ 1 λ T + µ + k) µ + d T + p)µ + q + λ T ) qp) pλ T ψ 1λ T + k). The analysis of the equilibria X 2 is similar to that of the endemic equilibria Q in Section 3.3 equation 31). X 3 = S 3, E T 3, I T 3, R T 3, I H3, A H3, E HT 3, E AT 3, I HT 3, A AT 3 ), the HIV-Mtb coinfection equilibrium and exists when each component of X 3 is positive. Now we determine conditions under which TB and HIV/AIDS coinfection will not elliminate the whole population. Following a similar approach to Allen et al. [3] and Ackleh et al. [1] we make use of the following assumptions which we use to show that coinfection by TB and HIV/AIDS may not elliminate the whole population. 1. IT 3 > µ0) + d T + I HT 3 + A ) AT 3 AH3 + d A + E AT 3 + A ) AT 3 N 3 N 3 N 3 N 3 N 3 N 3 Assumption 1) requires apriori knowledge of the equilibrium coordinates of X 3. Another stronger condition but much simpler condition was assumed by Ackleh et al. [1]. This condition does not require knowledge of X 3. 77) 2. > µ0) + maxd T, d A ). 78) Conditions 1) and 2) in equations 77) and 78) respectively prevent complete extinction because the recruitment rate exceeds the death rate natural and disease related) when population sizes are small. In all these cases we assume that there is no vertical transmission. Coexistence of TB and HIV/AIDS will occur when R 1 > 1 and R 2 > 1 since neither of the two infections confer cross immunity to the other. Figure 3 is a graphical representation showing the regions in the R 1 R 2 parameter space in which the different disease equilibrium states are stable. 20
21 Figure 3: Equilibrium results showing regions of different outcomes of competition between TB and HIV/AIDS as functions of the basic reproduction numbers R 1 and R 2 ). Neither TB nor HIV exists when both reproduction numbers are less than unity. 6 Effects of TB treatment and antiretroviral therapy Antiretroviral treatment for HIV infection consists of drugs which work against HIV/AIDS infection itself by slowing down the replication of HIV in the body. These fall into the following categories. Non-Nucleoside Reverse Transcriptase Inhibitors NNRTIs) such as Efavirenz, bind to and block the action of reverse transcriptase, a protein that HIV needs to reproduce. Nucleoside Reverse Transcriptase Inhibitors NRTIs) such as Zidovudine, Tenofovir DF and Stavudine are faulty versions of the building blocks that HIV needs to make more copies of itself. When HIV uses NRTI instead of a normal building block, reproduction of the virus is stalled. Protease Inhibitors PIs) such as Lopinavir disable protease, a protein that HIV needs to reproduce itself. Fusion Inhibitors FIs) such as Enfuvirtide are newer treatments that work by blocking HIV entry into cells. As far as HIV/AIDS treatment is concerned there is no one best regimen. It is recommended that taking one or two drugs is not recommended because any decrease in viral load is temporal without three or more drugs. Treatment of HIV/AIDS infection is highly dynamic. Active forms of tuberculosis are treated using first line drugs rifampcin, isoniazid, pyrazinamide, ethambtol) taken daily for two months and followed by a daily intake of rifampcin and isoniazid for a period of four months. Exposed individuals are treated with isoniazid. In this section we extend the coinfection model to incorporate the effects of treatment of AIDS patients with antiretrovirals and TB with antibiotics. In the extended model, α is the rate of treatment of AIDS cases. AIDS patients getting antiretrovirals and become healthy looking and τ 1 and τ 2 are the rates at which the individuals latently infected with Mtb and 21
22 those suffering from TB are treated respectively. In individuals dually infected with TB and HIV/AIDS are concurrently treated using rifabutin in combination with antiretroviral regimen containing Protease Inhibitors as outlined in Narita et. al [32]. Thus with treatment model system 4) becomes, ds = λ T S λ H S µs, di H da H = λ H S + λ H R T ρ 1 + µ)i H λ T I H + αa H + τ 1 E TH + τ 2 I TH, = ρ 1 I H σλ T A H µ + α + d A )A H + τ 2 A AT + τ 1 E AT, de TH = λ H E T + λ T I H γ 1 ρ 1 + t 1 k + τ 1 + µ)e TH + αe AT ψ 2 λ T E TH, de AT di TH = γ 1 ρ 1 E TH + σλ T A H t 2 k + α + τ 1 + µ + d A )E AT ψ 3 λ T E AT, = δλ H I T + t 1 ke TH µ + d T + τ 2 + γ 2 ρ 1 )I TH + αa AT + ψ 2 λ T E TH, 79) da AT = γ 2 ρ 1 I TH + t 2 ke AT µ + d T + ɛd A + α + τ 2 )A AT + ψ 3 λ T E AT, de T di T dr T = λ T S + λ T R T ψ 1 λ T E T λ H E T µ + k + τ 1 )E T, = ψ 1 λ T E T + ke T µ + p + τ 2 + d T )I T δλ H I T + qr T, = τ 1 E T + p + τ 2 )I T qr T λ T R T λ H R T µr T. The disease free equilibrium point is given by, X 0t = S 0t, E T 0t,, A AT 0t ) = ), 0, 0, 0, 0, 0, 0, 0, 0, 0. 80) µ Using the method for finding reproduction numbers outlined in subsection 5.2, we have the treatment induced reproduction number, R 0t as, R 0t = max[r 1t, R 2t ] ) β T ckq + kµ + τ 1 q) = max µ + k + τ 1 ) µ + p + τ 2 + d T )q + µ) qp + τ 2 ) ), β H µ + α + ρ 1 η A + d A ), µ + α + d A )ρ 1 + µ) ρ 1 να 81) where R 1t and R 2t are the treatment induced reproduction numbers for the TB submodel and HIV submodel respectively. We now state Theorem 12 whose proof is similar to the one in Theorem 10. Theorem 12. X 0t is locally asymptotically stable when R 0t < 1 and unstable when R 0t > Analysis of the reproduction number, R 0t Case 1: In the absence of any treatment option lim R 0t = lim max {R 1t, R 2t } = max {R 1, R 2 } = R 0, i = 1, 2. τ i,α) 0,0) τ i,α) 0,0) 82) In this case we revert to pre-treatment reproduction number found in subsection
23 Case 2: Only the TB cases are treated. Then, lim R 0t = lim max {R 1t, R 2t } = max {R 1t, R 2 }. α 0 α 0 83) In this case AIDS epidemic is allowed to grow. Rewriting R 1t as R 1t = H 1 R 1 with H 1 0, 1) where, H 1 = µ + k) kq + µ) + τ 1 q) µ + p + d T )q + µ) pq) kµ + k + τ 1 )q + µ) µ + p + d T + τ 2 )q + µ) qp + τ 2 )), is the factor by which chemoprophylaxis and treatment of infectives reduce the number of secondary TB cases if adopted in a community. If R 1 < 1, TB can not develop into an epidemic and treatment and chemoprophylaxis may not be necessary and for R 1 > 1, we want to determine conditions necessary for slowing down the TB epidemic. Following Hsu Schmitz [22] we have, T = R 1 1 H 1 ) for which T > 0 is expected for slowing down the epidemic and is satisfied for 0 < τ i < 1, i = 1, 2. Differentiating R lt with respect to τ 1 and τ 2 we have, R 1t τ 1 µ + k) µ + p + d T )q + µ) pq) k q)µr 1 = kq + µ) µ + p + d T + τ 2 )q + µ) qp + τ 2 )) µ + k + τ 1 ) 2 R 1t τ 2 = µ + k) µ + p + d T )q + µ) pq) kq + µ) + τ 1 q) µr 1 kq + µ) µ + p + d T + τ 2 )q + µ) qp + τ 2 )) 2 µ + k + τ 1 ) From equation 85) the conditions necessary for slowing TB in people coinfected with HIV are R 1t R 1t T > 0, < 0, < 0 and these are satisfied for τ i 0, 1). Setting R 1t = 1 and solving τ 1 τ 2 for critical chemoprophylaxis and treatment rates we have, τ c 1 = kµ + k)µ + q) γ1 R 1) + τ 2 µ) kqr 1 1) + qµγr 1 kµγ + τ 2 µ) 84) 85) τ c 2 = γµ + k)µ + q)r 1 1) + γτ 1 µ + k)qr 1 µ + q)) µµ + q)µ + k + τ 1 ) 86) where γ = µ + p + d T )q + µ) + pµ. Thus chemoprophylaxis and treatment of infectives would succeed in controlling the TB epidemic in individuals coinfected with HIV if τ i > τ c i. We can write R 1t as R 1t τ 1, τ 2 ), noting that, R 1t τ 1, τ 2 ) < R 1t τ 1, 0) < R 1t 0, τ 2 ) < R 1, 87) suggesting that chemoprophylaxis is more effective than treatment in controlling the TB epidemic in individuals coinfected with HIV though the holistic approach is the most effective. Thus in the presence of chemoprophylaxis and treatment of TB, AIDS only remains as an epidemic so, lim α 0 R 0t = R 2. 88) Case 3: Only the AIDS cases are treated Then, lim τ i 0 R 0t = τi 0 max {R 1t, R 2t } = max {R 1, R 2t }. 89) Rewriting R 2t as, R 2t = H 2 R 2, with H 2 0, 1) where H 2 = µ + α + d A + ρ 1 η A )ρ 1 + µ)µ + d A ) µ + α + d A )ρ 1 + µ) ρ 1 α) µ + d A + η A ρ 1 ) 90) 23
24 is the factor by which treatment of AIDS cases slows the development of the epidemic. Here it is important to note that decreasing R 2 to values less than unity does not result in the reduction of people being infected with HIV. Actually treatment of AIDS cases alone as an intervention strategy can be counter productive as increasing the time an HIV/AIDS infected lives, tends to increase the infective period. Other intervention strategies beyond the scope of this work have to be investigated to analyse their effects in reducing the growth of the AIDS epidemic in individuals coinfected with TB. Now we consider, A = R 2 R 2t = R 2 1 H 2 ) for which A > 0 is expected to reduce the number of secondary AIDS cases and is satisfied for α 0, 1). Differentiating R 2t partially with respect to α we get, R 2t α = ρ 1 + µ)µ + d A ) ρ 1 µ + d A + η A ρ 1 ) ρ 1 + µ)η A ρ 1 ) R 2 µ + d A + η A ρ 1 ) µ + α + d A )ρ 1 + µ) ρ 1 α) 2. 91) From equation 91) treatment of AIDS cases will reduce the burden of the epidemic when, R 2t α < 0. Setting R 2t = 1 and solving for α c we have, α c = µ + d A + η A ρ 1 )µ + d A )ρ 1 + µ)1 R 2 ) µ + ρ 1 )µ + d A )R 2 1) µ + ρ 1 )η A ρ 1 + ρ 1 µ + d A + η A ρ 1 ). 92) Whenever α > α c treatment of AIDS cases results in all AIDS cases leading a near normal life with the AIDS disease. 6.2 The Endemic Equilibria The endemic equilibria of 79) corresponds to, E 1 = S 1, I H1, A H1, 0, 0, 0, 0, 0, 0, 0), the TB free endemic equilibrium point where, S 1 = µnµ + α + d A), R 2t I H1 = 1 β H µ + α + ρ 1 η A + d A ) R 2t µ + α + d A )µn) 93) A H1 = ρ 1 β H µ + α + d A )µ + α + ρ 1 η A + d A ) R 2t µ + α + d A )µn) 24
25 a) b) c) Figure 4: Graphs showing the behavior of the susceptibles S, the infected HIV positive) I H1 and those suffering from AIDS A H1 as the treatment induced reproduction number, R 2t. Figures a) represents the susceptible population, b) represents the infected population, c) represents the AIDS cases. In this analysis we used µn = 0.01, = 0.029, ρ 1 = 0.1, d A = Figure 4 is a graphical representation of the components of the endemic equilibrium point, E 1 and shows the changes in the susceptibles S 1 ), the HIV positive I H1 )and AIDS cases A H1 ) as the treatment induced reproduction number, R 2t is varied. In Figure 4a) the susceptibles are being depleted rapidly as R 2t becomes large. Figures 4b) and c) shows that the HIV positive individuals and the AIDS cases both have a linear relationship with the reproduction number, R 2t and all become large as R 2t as become large. Comparing the susceptibles in Figure 2a) and Figure 4a) it is shown that the susceptibles in later Figure are being depleted more rapidly as their corresponding reproduction becomes large. 25
26 This suggests that treatment of AIDS cases while prolonging life for the sick fuels the pandemic as improving the life of AIDS individuals in turn increases the spread of HIV as a result of lengthening the infectious period. Comparing Figures 2b) and c) with Figures 4b) and c) we find that graphs for the HIV positive and AIDS cases reach a limit in absence of treatment but in the presence of treatment the HIV positive and AIDS cases become large as R 2t gets bigger. E 2 = S 2, 0, 0, 0, 0, 0, 0, E T 2, I T 2, R T 2 ), the HIV free endemic equilibrium point where, S 2 = N 2 µn 2 + β T ci 2, E T 2 = q + µ)µ + p + d T + τ 2 )N 2 I T 2 p + τ 2 )qn 2 I T 2, τ 1 qn 2 + q + µ)β T ci T 2 + N 2 ) R T 2 = τ 1 N 2 2 I T 2 q + µ)µ + p + τ 2 + d T ) p + τ 2 )q) N 2 q + µ) + β T ci T 2 ) τ 1 dn 2 + q + µ)β T ci T 2 + q + µ)n 2 ) 94) + N 2 I T 2 p + τ 2 ) µ + q)n 2 + β T ci T 2 E 3 = S 3, I H3, A H3, E HT 3, E AT 3, I HT 3, A AT 3, E T 3, I T 3, R T 3 ), such that S 3 > 0, I H3 > 0, A H3 > 0, E HT 3 > 0, E AT 3 > 0, I HT 3 > 0, A AT 3 > 0, E T 3 > 0, I T 3, R T 3 > 0 is the equilibrium point for which the two infections coexist. Theorem 13. The endemic equilibria points E 1, E 2, E 3 are locally asymptotically stable when R 0t > 1 and unstable otherwise. Proof. Lets consider showing that E 1 is locally asymptotically when R 0t > 1 and unstable otherwise. The disease is endemic when I H > 0 and A H > 0 that is, β H N I H + η A A H )S ρ 1 + µ)i H + αa H > 0, ρ 1 I H µ + α + d A )A H > 0. 95) From the second inequality of 95) we have, A H < and from the first inequality of 95) we have, ρ 1 + µ)i H < β H N I H + η A A H )S + αa H β H I H + η A A H ) + αa H, using the fact that S N 1 ρ 1 I H µ + α + d A, 96) β H I H + η Aρ 1 I H µ + α + d A ) + αρ 1I H µ + α + d A 97) ρ 1 + µ)µ + α + d A ) αρ 1 < β H µ + α + d A + η A ρ 1 ) 1 < β Hµ + α + d A + η A ρ 1 ) ρ 1 + µ)µ + α + d A ) αρ 1 = R 2t = R 0t, when there is no TB. Thus the endemic equilibrium E 1 is locally asymptotically stable when R 0t > 1 and unstable when R 0t < 1. In the same way we can show that E 2 and E 3 are stable for R 0t > 1 and unstable for R 0t < 1. 26
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