Modelling HIV/AIDS and Tuberculosis Coinfection

Size: px
Start display at page:

Download "Modelling HIV/AIDS and Tuberculosis Coinfection"

Transcription

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

GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS

GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model

More information

Thursday. Threshold and Sensitivity Analysis

Thursday. Threshold and Sensitivity Analysis Thursday Threshold and Sensitivity Analysis SIR Model without Demography ds dt di dt dr dt = βsi (2.1) = βsi γi (2.2) = γi (2.3) With initial conditions S(0) > 0, I(0) > 0, and R(0) = 0. This model can

More information

MATHEMATICAL ANALYSIS OF THE TRANSMISSION DYNAMICS OF HIV/TB COINFECTION IN THE PRESENCE OF TREATMENT

MATHEMATICAL ANALYSIS OF THE TRANSMISSION DYNAMICS OF HIV/TB COINFECTION IN THE PRESENCE OF TREATMENT MATHEMATICAL BIOSCIENCES http://www.mbejournal.org/ AND ENGINEERING Volume 5 Number 1 January 28 pp. 145 174 MATHEMATICAL ANALYSIS OF THE TRANSMISSION DYNAMICS OF HIV/TB COINFECTION IN THE PRESENCE OF

More information

Research Article Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant Tuberculosis Dynamics

Research Article Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant Tuberculosis Dynamics International Scholarly Research Network ISRN Biomathematics Volume, Article ID 5989, pages doi:.5//5989 Research Article Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant

More information

Mathematical Model of Tuberculosis Spread within Two Groups of Infected Population

Mathematical Model of Tuberculosis Spread within Two Groups of Infected Population Applied Mathematical Sciences, Vol. 10, 2016, no. 43, 2131-2140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.63130 Mathematical Model of Tuberculosis Spread within Two Groups of Infected

More information

Introduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium

Introduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium Introduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium point and the model reproduction number Prove the stability

More information

GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT

GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 1, Spring 2011 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT HONGBIN GUO AND MICHAEL Y. LI

More information

Mathematical Analysis of Epidemiological Models: Introduction

Mathematical Analysis of Epidemiological Models: Introduction Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010 1. Introduction. The effectiveness of improved sanitation,

More information

Mathematical Analysis of Epidemiological Models III

Mathematical Analysis of Epidemiological Models III Intro Computing R Complex models Mathematical Analysis of Epidemiological Models III Jan Medlock Clemson University Department of Mathematical Sciences 27 July 29 Intro Computing R Complex models What

More information

The death of an epidemic

The death of an epidemic LECTURE 2 Equilibrium Stability Analysis & Next Generation Method The death of an epidemic In SIR equations, let s divide equation for dx/dt by dz/ dt:!! dx/dz = - (β X Y/N)/(γY)!!! = - R 0 X/N Integrate

More information

Mathematical Analysis of HIV/AIDS Prophylaxis Treatment Model

Mathematical Analysis of HIV/AIDS Prophylaxis Treatment Model Applied Mathematical Sciences, Vol. 12, 2018, no. 18, 893-902 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8689 Mathematical Analysis of HIV/AIDS Prophylaxis Treatment Model F. K. Tireito,

More information

Impact of Case Detection and Treatment on the Spread of HIV/AIDS: a Mathematical Study

Impact of Case Detection and Treatment on the Spread of HIV/AIDS: a Mathematical Study Malaysian Journal of Mathematical Sciences (3): 33 347 (8) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Journal homepage: http://einspemupmedumy/journal Impact of Case Detection and Treatment on the Spread

More information

GLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT

GLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 304, pp. 1 8. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY

More information

Available online at J. Math. Comput. Sci. 2 (2012), No. 6, ISSN:

Available online at   J. Math. Comput. Sci. 2 (2012), No. 6, ISSN: Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 6, 1671-1684 ISSN: 1927-5307 A MATHEMATICAL MODEL FOR THE TRANSMISSION DYNAMICS OF HIV/AIDS IN A TWO-SEX POPULATION CONSIDERING COUNSELING

More information

Stability of SEIR Model of Infectious Diseases with Human Immunity

Stability of SEIR Model of Infectious Diseases with Human Immunity Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1811 1819 Research India Publications http://www.ripublication.com/gjpam.htm Stability of SEIR Model of Infectious

More information

Global compact attractors and their tripartition under persistence

Global compact attractors and their tripartition under persistence Global compact attractors and their tripartition under persistence Horst R. Thieme (joint work with Hal L. Smith) School of Mathematical and Statistical Science Arizona State University GCOE, September

More information

Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005

Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005 Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor August 15, 2005 1 Outline 1. Compartmental Thinking 2. Simple Epidemic (a) Epidemic Curve 1:

More information

Epidemics in Two Competing Species

Epidemics in Two Competing Species Epidemics in Two Competing Species Litao Han 1 School of Information, Renmin University of China, Beijing, 100872 P. R. China Andrea Pugliese 2 Department of Mathematics, University of Trento, Trento,

More information

Introduction to SEIR Models

Introduction to SEIR Models Department of Epidemiology and Public Health Health Systems Research and Dynamical Modelling Unit Introduction to SEIR Models Nakul Chitnis Workshop on Mathematical Models of Climate Variability, Environmental

More information

Global Stability of a Computer Virus Model with Cure and Vertical Transmission

Global Stability of a Computer Virus Model with Cure and Vertical Transmission International Journal of Research Studies in Computer Science and Engineering (IJRSCSE) Volume 3, Issue 1, January 016, PP 16-4 ISSN 349-4840 (Print) & ISSN 349-4859 (Online) www.arcjournals.org Global

More information

Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment

Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment Bulletin of Mathematical Biology (2010) 72: 1 33 DOI 10.1007/s11538-009-9435-5 ORIGINAL ARTICLE Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment Zhipeng Qiu a,, Zhilan

More information

Behavior Stability in two SIR-Style. Models for HIV

Behavior Stability in two SIR-Style. Models for HIV Int. Journal of Math. Analysis, Vol. 4, 2010, no. 9, 427-434 Behavior Stability in two SIR-Style Models for HIV S. Seddighi Chaharborj 2,1, M. R. Abu Bakar 2, I. Fudziah 2 I. Noor Akma 2, A. H. Malik 2,

More information

A Model on the Impact of Treating Typhoid with Anti-malarial: Dynamics of Malaria Concurrent and Co-infection with Typhoid

A Model on the Impact of Treating Typhoid with Anti-malarial: Dynamics of Malaria Concurrent and Co-infection with Typhoid International Journal of Mathematical Analysis Vol. 9, 2015, no. 11, 541-551 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.412403 A Model on the Impact of Treating Typhoid with Anti-malarial:

More information

Accepted Manuscript. Backward Bifurcations in Dengue Transmission Dynamics. S.M. Garba, A.B. Gumel, M.R. Abu Bakar

Accepted Manuscript. Backward Bifurcations in Dengue Transmission Dynamics. S.M. Garba, A.B. Gumel, M.R. Abu Bakar Accepted Manuscript Backward Bifurcations in Dengue Transmission Dynamics S.M. Garba, A.B. Gumel, M.R. Abu Bakar PII: S0025-5564(08)00073-4 DOI: 10.1016/j.mbs.2008.05.002 Reference: MBS 6860 To appear

More information

AN ABSTRACT OF THE THESIS OF. Margaret-Rose W. Leung for the degree of Honors Baccalaureate of Science in Mathematics

AN ABSTRACT OF THE THESIS OF. Margaret-Rose W. Leung for the degree of Honors Baccalaureate of Science in Mathematics AN ABSTRACT OF THE THESIS OF Margaret-Rose W. Leung for the degree of Honors Baccalaureate of Science in Mathematics presented on June 5, 2012. Title: A Vector Host Model for Coinfection by Barley/Cereal

More information

MATHEMATICAL ANALYSIS OF A MODEL FOR HIV-MALARIA CO-INFECTION. Zindoga Mukandavire. Abba B. Gumel. Winston Garira. Jean Michel Tchuenche

MATHEMATICAL ANALYSIS OF A MODEL FOR HIV-MALARIA CO-INFECTION. Zindoga Mukandavire. Abba B. Gumel. Winston Garira. Jean Michel Tchuenche MATHEMATICAL BIOSCIENCES doi:1.3934/mbe.29.6.333 AND ENGINEERING Volume 6 Number 2 April 29 pp. 333 362 MATHEMATICAL ANALYSIS OF A MODEL FOR HIV-MALARIA CO-INFECTION Zindoga Mukandavire Department of Applied

More information

Global Analysis of an SEIRS Model with Saturating Contact Rate 1

Global Analysis of an SEIRS Model with Saturating Contact Rate 1 Applied Mathematical Sciences, Vol. 6, 2012, no. 80, 3991-4003 Global Analysis of an SEIRS Model with Saturating Contact Rate 1 Shulin Sun a, Cuihua Guo b, and Chengmin Li a a School of Mathematics and

More information

Non-Linear Models Cont d: Infectious Diseases. Non-Linear Models Cont d: Infectious Diseases

Non-Linear Models Cont d: Infectious Diseases. Non-Linear Models Cont d: Infectious Diseases Cont d: Infectious Diseases Infectious Diseases Can be classified into 2 broad categories: 1 those caused by viruses & bacteria (microparasitic diseases e.g. smallpox, measles), 2 those due to vectors

More information

DYNAMICAL MODELS OF TUBERCULOSIS AND THEIR APPLICATIONS. Carlos Castillo-Chavez. Baojun Song. (Communicated by Yang Kuang)

DYNAMICAL MODELS OF TUBERCULOSIS AND THEIR APPLICATIONS. Carlos Castillo-Chavez. Baojun Song. (Communicated by Yang Kuang) MATHEMATICAL BIOSCIENCES http://math.asu.edu/ mbe/ AND ENGINEERING Volume 1, Number 2, September 2004 pp. 361 404 DYNAMICAL MODELS OF TUBERCULOSIS AND THEIR APPLICATIONS Carlos Castillo-Chavez Department

More information

Hepatitis C Mathematical Model

Hepatitis C Mathematical Model Hepatitis C Mathematical Model Syed Ali Raza May 18, 2012 1 Introduction Hepatitis C is an infectious disease that really harms the liver. It is caused by the hepatitis C virus. The infection leads to

More information

Australian Journal of Basic and Applied Sciences. Effect of Education Campaign on Transmission Model of Conjunctivitis

Australian Journal of Basic and Applied Sciences. Effect of Education Campaign on Transmission Model of Conjunctivitis ISSN:99-878 Australian Journal of Basic and Applied Sciences Journal home page: www.ajbasweb.com ffect of ducation Campaign on Transmission Model of Conjunctivitis Suratchata Sangthongjeen, Anake Sudchumnong

More information

Mathematical Epidemiology Lecture 1. Matylda Jabłońska-Sabuka

Mathematical Epidemiology Lecture 1. Matylda Jabłońska-Sabuka Lecture 1 Lappeenranta University of Technology Wrocław, Fall 2013 What is? Basic terminology Epidemiology is the subject that studies the spread of diseases in populations, and primarily the human populations.

More information

Modeling Co-Dynamics of Cervical Cancer and HIV Diseases

Modeling Co-Dynamics of Cervical Cancer and HIV Diseases Global ournal of Pure Applied Mathematics. SSN 093-8 Volume 3 Number (0) pp. 05-08 Research ndia Publications http://www.riblication.com Modeling o-dynamics of ervical ancer V Diseases Geomira G. Sanga

More information

SUBTHRESHOLD AND SUPERTHRESHOLD COEXISTENCE OF PATHOGEN VARIANTS: THE IMPACT OF HOST AGE-STRUCTURE

SUBTHRESHOLD AND SUPERTHRESHOLD COEXISTENCE OF PATHOGEN VARIANTS: THE IMPACT OF HOST AGE-STRUCTURE SUBTHRESHOLD AND SUPERTHRESHOLD COEXISTENCE OF PATHOGEN VARIANTS: THE IMPACT OF HOST AGE-STRUCTURE MAIA MARTCHEVA, SERGEI S. PILYUGIN, AND ROBERT D. HOLT Abstract. It is well known that in the most general

More information

Mathematical Modeling Applied to Understand the Dynamical Behavior of HIV Infection

Mathematical Modeling Applied to Understand the Dynamical Behavior of HIV Infection Open Journal of Modelling and Simulation, 217, 5, 145-157 http://wwwscirporg/journal/ojmsi ISSN Online: 2327-426 ISSN Print: 2327-418 Mathematical Modeling Applied to Understand the Dynamical Behavior

More information

Australian Journal of Basic and Applied Sciences

Australian Journal of Basic and Applied Sciences AENSI Journals Australian Journal of Basic and Applied Sciences ISSN:1991-8178 Journal home page: www.ajbasweb.com A SIR Transmission Model of Political Figure Fever 1 Benny Yong and 2 Nor Azah Samat 1

More information

Supplement to TB in Canadian First Nations at the turn-of-the twentieth century

Supplement to TB in Canadian First Nations at the turn-of-the twentieth century Supplement to TB in Canadian First Nations at the turn-of-the twentieth century S. F. Ackley, Fengchen Liu, Travis C. Porco, Caitlin S. Pepperell Equations Definitions S, L, T I, T N, and R give the numbers

More information

Research Article Modeling Computer Virus and Its Dynamics

Research Article Modeling Computer Virus and Its Dynamics Mathematical Problems in Engineering Volume 213, Article ID 842614, 5 pages http://dx.doi.org/1.1155/213/842614 Research Article Modeling Computer Virus and Its Dynamics Mei Peng, 1 Xing He, 2 Junjian

More information

Delay SIR Model with Nonlinear Incident Rate and Varying Total Population

Delay SIR Model with Nonlinear Incident Rate and Varying Total Population Delay SIR Model with Nonlinear Incident Rate Varying Total Population Rujira Ouncharoen, Salinthip Daengkongkho, Thongchai Dumrongpokaphan, Yongwimon Lenbury Abstract Recently, models describing the behavior

More information

A Modeling Approach for Assessing the Spread of Tuberculosis and Human Immunodeficiency Virus Co-Infections in Thailand

A Modeling Approach for Assessing the Spread of Tuberculosis and Human Immunodeficiency Virus Co-Infections in Thailand Kasetsart J. (at. Sci.) 49 : 99 - (5) A Modeling Approach for Assessing the Spread of Tuberculosis and uman Immunodeficiency Virus Co-Infections in Thailand Kornkanok Bunwong,3, Wichuta Sae-jie,3,* and

More information

Modeling and Global Stability Analysis of Ebola Models

Modeling and Global Stability Analysis of Ebola Models Modeling and Global Stability Analysis of Ebola Models T. Stoller Department of Mathematics, Statistics, and Physics Wichita State University REU July 27, 2016 T. Stoller REU 1 / 95 Outline Background

More information

Australian Journal of Basic and Applied Sciences. Effect of Personal Hygiene Campaign on the Transmission Model of Hepatitis A

Australian Journal of Basic and Applied Sciences. Effect of Personal Hygiene Campaign on the Transmission Model of Hepatitis A Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: 67-73 ISSN:1991-8178 Australian Journal of Basic and Applied Sciences Journal home page: wwwajbaswebcom Effect of Personal Hygiene

More information

TRANSMISSION DYNAMICS OF CHOLERA EPIDEMIC MODEL WITH LATENT AND HYGIENE COMPLIANT CLASS

TRANSMISSION DYNAMICS OF CHOLERA EPIDEMIC MODEL WITH LATENT AND HYGIENE COMPLIANT CLASS Electronic Journal of Mathematical Analysis and Applications Vol. 7(2) July 2019, pp. 138-150. ISSN: 2090-729X(online) http://math-frac.org/journals/ejmaa/ TRANSMISSION DYNAMICS OF CHOLERA EPIDEMIC MODEL

More information

STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL

STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL VFAST Transactions on Mathematics http://vfast.org/index.php/vtm@ 2013 ISSN: 2309-0022 Volume 1, Number 1, May-June, 2013 pp. 16 20 STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL Roman Ullah 1, Gul

More information

Mathematical models on Malaria with multiple strains of pathogens

Mathematical models on Malaria with multiple strains of pathogens Mathematical models on Malaria with multiple strains of pathogens Yanyu Xiao Department of Mathematics University of Miami CTW: From Within Host Dynamics to the Epidemiology of Infectious Disease MBI,

More information

MODELING MAJOR FACTORS THAT CONTROL TUBERCULOSIS (TB) SPREAD IN CHINA

MODELING MAJOR FACTORS THAT CONTROL TUBERCULOSIS (TB) SPREAD IN CHINA MODELING MAJOR FACTORS THAT CONTROL TUBERCULOSIS (TB) SPREAD IN CHINA XUE-ZHI LI +, SOUVIK BHATTACHARYA, JUN-YUAN YANG, AND MAIA MARTCHEVA Abstract. This article introduces a novel model that studies the

More information

Stability Analysis of an SVIR Epidemic Model with Non-linear Saturated Incidence Rate

Stability Analysis of an SVIR Epidemic Model with Non-linear Saturated Incidence Rate Applied Mathematical Sciences, Vol. 9, 215, no. 23, 1145-1158 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.41164 Stability Analysis of an SVIR Epidemic Model with Non-linear Saturated

More information

Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species

Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species Xingfu Zou Department of Applied Mathematics University of Western Ontario London, Ontario, Canada (Joint work

More information

Analysis of SIR Mathematical Model for Malaria disease with the inclusion of Infected Immigrants

Analysis of SIR Mathematical Model for Malaria disease with the inclusion of Infected Immigrants IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 14, Issue 5 Ver. I (Sep - Oct 218), PP 1-21 www.iosrjournals.org Analysis of SIR Mathematical Model for Malaria disease

More information

The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV. Jan P. Medlock

The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV. Jan P. Medlock The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV A Thesis Presented to The Faculty of the Division of Graduate Studies by Jan P. Medlock In Partial Fulfillment of the Requirements

More information

Dynamics of a Population Model Controlling the Spread of Plague in Prairie Dogs

Dynamics of a Population Model Controlling the Spread of Plague in Prairie Dogs Dynamics of a opulation Model Controlling the Spread of lague in rairie Dogs Catalin Georgescu The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD USA

More information

Impact of Heterosexuality and Homosexuality on the transmission and dynamics of HIV/AIDS

Impact of Heterosexuality and Homosexuality on the transmission and dynamics of HIV/AIDS IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 12, Issue 6 Ver. V (Nov. - Dec.216), PP 38-49 www.iosrjournals.org Impact of Heterosexuality and Homosexuality on the

More information

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission P. van den Driessche a,1 and James Watmough b,2, a Department of Mathematics and Statistics, University

More information

STABILITY AND PERSISTENCE IN A MODEL FOR BLUETONGUE DYNAMICS

STABILITY AND PERSISTENCE IN A MODEL FOR BLUETONGUE DYNAMICS STABILITY AND PERSISTENCE IN A MODEL FOR BLUETONGUE DYNAMICS STEPHEN A. GOURLEY, HORST R. THIEME, AND P. VAN DEN DRIESSCHE Abstract. A model for the time evolution of bluetongue, a viral disease in sheep

More information

Apparent paradoxes in disease models with horizontal and vertical transmission

Apparent paradoxes in disease models with horizontal and vertical transmission Apparent paradoxes in disease models with horizontal and vertical transmission Thanate Dhirasakdanon, Stanley H.Faeth, Karl P. Hadeler*, Horst R. Thieme School of Life Sciences School of Mathematical and

More information

Demographic impact and controllability of malaria in an SIS model with proportional fatality

Demographic impact and controllability of malaria in an SIS model with proportional fatality Demographic impact and controllability of malaria in an SIS model with proportional fatality Muntaser Safan 1 Ahmed Ghazi Mathematics Department, Faculty of Science, Mansoura University, 35516 Mansoura,

More information

A dynamical analysis of tuberculosis in the Philippines

A dynamical analysis of tuberculosis in the Philippines ARTICLE A dynamical analysis of tuberculosis in the Philippines King James B. Villasin 1, Angelyn R. Lao 2, and Eva M. Rodriguez*,1 1 Department of Mathematics, School of Sciences and Engineering, University

More information

MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL. Hor Ming An, PM. Dr. Yudariah Mohammad Yusof

MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL. Hor Ming An, PM. Dr. Yudariah Mohammad Yusof MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL Hor Ming An, PM. Dr. Yudariah Mohammad Yusof Abstract The establishment and spread of dengue fever is a complex phenomenon with many factors that

More information

6. Age structure. for a, t IR +, subject to the boundary condition. (6.3) p(0; t) = and to the initial condition

6. Age structure. for a, t IR +, subject to the boundary condition. (6.3) p(0; t) = and to the initial condition 6. Age structure In this section we introduce a dependence of the force of infection upon the chronological age of individuals participating in the epidemic. Age has been recognized as an important factor

More information

Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008

Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008 Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008 James Holland Jones Department of Anthropology Stanford University May 3, 2008 1 Outline 1. Compartmental

More information

On CTL Response against Mycobacterium tuberculosis

On CTL Response against Mycobacterium tuberculosis Applied Mathematical Sciences, Vol. 8, 2014, no. 48, 2383-2389 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43150 On CTL Response against Mycobacterium tuberculosis Eduardo Ibargüen-Mondragón

More information

Dynamical Analysis of Plant Disease Model with Roguing, Replanting and Preventive Treatment

Dynamical Analysis of Plant Disease Model with Roguing, Replanting and Preventive Treatment 4 th ICRIEMS Proceedings Published by The Faculty Of Mathematics And Natural Sciences Yogyakarta State University, ISBN 978-62-74529-2-3 Dynamical Analysis of Plant Disease Model with Roguing, Replanting

More information

HETEROGENEOUS MIXING IN EPIDEMIC MODELS

HETEROGENEOUS MIXING IN EPIDEMIC MODELS CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 HETEROGENEOUS MIXING IN EPIDEMIC MODELS FRED BRAUER ABSTRACT. We extend the relation between the basic reproduction number and the

More information

Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response

Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA you@mail.usf.edu IMA Workshop in Memory of George R. Sell June 016 Outline

More information

Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population

Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Ruzhang Zhao, Lijun Yang Department of Mathematical Science, Tsinghua University, China. Corresponding author. Email: lyang@math.tsinghua.edu.cn,

More information

A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage

A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage Applied Mathematical Sciences, Vol. 1, 216, no. 43, 2121-213 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.63128 A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and

More information

Optimal control of vaccination and treatment for an SIR epidemiological model

Optimal control of vaccination and treatment for an SIR epidemiological model ISSN 746-7233, England, UK World Journal of Modelling and Simulation Vol. 8 (22) No. 3, pp. 94-24 Optimal control of vaccination and treatment for an SIR epidemiological model Tunde Tajudeen Yusuf, Francis

More information

Mathematical Modeling and Analysis of Infectious Disease Dynamics

Mathematical Modeling and Analysis of Infectious Disease Dynamics Mathematical Modeling and Analysis of Infectious Disease Dynamics V. A. Bokil Department of Mathematics Oregon State University Corvallis, OR MTH 323: Mathematical Modeling May 22, 2017 V. A. Bokil (OSU-Math)

More information

Project 1 Modeling of Epidemics

Project 1 Modeling of Epidemics 532 Chapter 7 Nonlinear Differential Equations and tability ection 7.5 Nonlinear systems, unlike linear systems, sometimes have periodic solutions, or limit cycles, that attract other nearby solutions.

More information

Research Article A Delayed Epidemic Model with Pulse Vaccination

Research Article A Delayed Epidemic Model with Pulse Vaccination Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2008, Article ID 746951, 12 pages doi:10.1155/2008/746951 Research Article A Delayed Epidemic Model with Pulse Vaccination

More information

Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model

Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model Bulletin of Mathematical Biology (2008) 70: 1272 1296 DOI 10.1007/s11538-008-9299-0 ORIGINAL ARTICLE Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical

More information

Applications in Biology

Applications in Biology 11 Applications in Biology In this chapter we make use of the techniques developed in the previous few chapters to examine some nonlinear systems that have been used as mathematical models for a variety

More information

PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS

PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 4, Winter 211 PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS Dedicated to Herb Freedman on the occasion of his seventieth birthday

More information

Qualitative Analysis of a Discrete SIR Epidemic Model

Qualitative Analysis of a Discrete SIR Epidemic Model ISSN (e): 2250 3005 Volume, 05 Issue, 03 March 2015 International Journal of Computational Engineering Research (IJCER) Qualitative Analysis of a Discrete SIR Epidemic Model A. George Maria Selvam 1, D.

More information

SIR Epidemic Model with total Population size

SIR Epidemic Model with total Population size Advances in Applied Mathematical Biosciences. ISSN 2248-9983 Volume 7, Number 1 (2016), pp. 33-39 International Research Publication House http://www.irphouse.com SIR Epidemic Model with total Population

More information

Research Article Modeling the Spread of Tuberculosis in Semiclosed Communities

Research Article Modeling the Spread of Tuberculosis in Semiclosed Communities Computational and Mathematical Methods in Medicine Volume, Article ID 689, 9 pages http://dx.doi.org/.//689 Research Article Modeling the Spread of Tuberculosis in Semiclosed Communities Mauricio Herrera,

More information

An Improved Computer Multi-Virus Propagation Model with User Awareness

An Improved Computer Multi-Virus Propagation Model with User Awareness Journal of Information & Computational Science 8: 16 (2011) 4301 4308 Available at http://www.joics.com An Improved Computer Multi-Virus Propagation Model with User Awareness Xiaoqin ZHANG a,, Shuyu CHEN

More information

STUDY OF THE BRUCELLOSIS TRANSMISSION WITH MULTI-STAGE KE MENG, XAMXINUR ABDURAHMAN

STUDY OF THE BRUCELLOSIS TRANSMISSION WITH MULTI-STAGE KE MENG, XAMXINUR ABDURAHMAN Available online at http://scik.org Commun. Math. Biol. Neurosci. 208, 208:20 https://doi.org/0.2899/cmbn/3796 ISSN: 2052-254 STUDY OF THE BRUCELLOSIS TRANSMISSION WITH MULTI-STAGE KE MENG, XAMXINUR ABDURAHMAN

More information

Dynamics of Disease Spread. in a Predator-Prey System

Dynamics of Disease Spread. in a Predator-Prey System Advanced Studies in Biology, vol. 6, 2014, no. 4, 169-179 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/asb.2014.4845 Dynamics of Disease Spread in a Predator-Prey System Asrul Sani 1, Edi Cahyono

More information

LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC

LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic

More information

Simple Mathematical Model for Malaria Transmission

Simple Mathematical Model for Malaria Transmission Journal of Advances in Mathematics and Computer Science 25(6): 1-24, 217; Article no.jamcs.37843 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-851) Simple

More information

GLOBAL STABILITY OF A VACCINATION MODEL WITH IMMIGRATION

GLOBAL STABILITY OF A VACCINATION MODEL WITH IMMIGRATION Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 92, pp. 1 10. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY

More information

Optimal Treatment Strategies for Tuberculosis with Exogenous Reinfection

Optimal Treatment Strategies for Tuberculosis with Exogenous Reinfection Optimal Treatment Strategies for Tuberculosis with Exogenous Reinfection Sunhwa Choi, Eunok Jung, Carlos Castillo-Chavez 2 Department of Mathematics, Konkuk University, Seoul, Korea 43-7 2 Department of

More information

Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay

Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay Ekkachai Kunnawuttipreechachan Abstract This paper deals with stability properties of the discrete numerical scheme

More information

Sensitivity and Stability Analysis of Hepatitis B Virus Model with Non-Cytolytic Cure Process and Logistic Hepatocyte Growth

Sensitivity and Stability Analysis of Hepatitis B Virus Model with Non-Cytolytic Cure Process and Logistic Hepatocyte Growth Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 3 2016), pp. 2297 2312 Research India Publications http://www.ripublication.com/gjpam.htm Sensitivity and Stability Analysis

More information

Dynamical models of HIV-AIDS e ect on population growth

Dynamical models of HIV-AIDS e ect on population growth Dynamical models of HV-ADS e ect on population growth David Gurarie May 11, 2005 Abstract We review some known dynamical models of epidemics, given by coupled systems of di erential equations, and propose

More information

Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis

Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis RESEARCH Open Access Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis Hyun M Yang, Silvia M Raimundo Correspondence: hyunyang@ime. unicamp.br UNICAMP-IMECC.

More information

Understanding the incremental value of novel diagnostic tests for tuberculosis

Understanding the incremental value of novel diagnostic tests for tuberculosis Understanding the incremental value of novel diagnostic tests for tuberculosis Nimalan Arinaminpathy & David Dowdy Supplementary Information Supplementary Methods Details of the transmission model We use

More information

A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host

A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host A mathematical model for malaria involving differential susceptibility exposedness and infectivity of human host A. DUCROT 1 B. SOME 2 S. B. SIRIMA 3 and P. ZONGO 12 May 23 2008 1 INRIA-Anubis Sud-Ouest

More information

The E ect of Occasional Smokers on the Dynamics of a Smoking Model

The E ect of Occasional Smokers on the Dynamics of a Smoking Model International Mathematical Forum, Vol. 9, 2014, no. 25, 1207-1222 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.46120 The E ect of Occasional Smokers on the Dynamics of a Smoking Model

More information

Smoking as Epidemic: Modeling and Simulation Study

Smoking as Epidemic: Modeling and Simulation Study American Journal of Applied Mathematics 2017; 5(1): 31-38 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20170501.14 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) Smoking as Epidemic:

More information

Global Stability of SEIRS Models in Epidemiology

Global Stability of SEIRS Models in Epidemiology Global Stability of SRS Models in pidemiology M. Y. Li, J. S. Muldowney, and P. van den Driessche Department of Mathematics and Statistics Mississippi State University, Mississippi State, MS 39762 Department

More information

Three Disguises of 1 x = e λx

Three Disguises of 1 x = e λx Three Disguises of 1 x = e λx Chathuri Karunarathna Mudiyanselage Rabi K.C. Winfried Just Department of Mathematics, Ohio University Mathematical Biology and Dynamical Systems Seminar Ohio University November

More information

Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment

Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment American Journal of Applied Mathematics 6; 4(5): -4 http://www.sciencepublishinggroup.com/j/ajam doi:.648/j.ajam.645.4 ISSN: -4 (Print); ISSN: -6X (Online) Epidemic Model of HIV/AIDS Transmission Dynamics

More information

New results on the existences of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator and their applications

New results on the existences of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator and their applications Chen Zhang Journal of Inequalities Applications 2017 2017:143 DOI 10.1186/s13660-017-1417-9 R E S E A R C H Open Access New results on the existences of solutions of the Dirichlet problem with respect

More information

Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity

Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity Daniah Tahir Uppsala University Department of Mathematics 7516 Uppsala Sweden daniahtahir@gmailcom

More information

Stochastic modelling of epidemic spread

Stochastic modelling of epidemic spread Stochastic modelling of epidemic spread Julien Arino Centre for Research on Inner City Health St Michael s Hospital Toronto On leave from Department of Mathematics University of Manitoba Julien Arino@umanitoba.ca

More information

A Mathematical Model for Tuberculosis: Who should be vaccinating?

A Mathematical Model for Tuberculosis: Who should be vaccinating? A Mathematical Model for Tuberculosis: Who should be vaccinating? David Gerberry Center for Biomedical Modeling David Geffen School of Medicine at UCLA Tuberculosis Basics 1 or 2 years lifetime New Infection

More information

Mathematical modelling and analysis of HIV/AIDS and trichomonas vaginalis co-infection

Mathematical modelling and analysis of HIV/AIDS and trichomonas vaginalis co-infection Mathematical modelling and analysis of HIV/AIDS and trichomonas vaginalis co-infection by Chibale K. Mumba Submitted in partial fulfilment of the requirements for the degree Magister Scientiae in the Department

More information

Dynamic pair formation models

Dynamic pair formation models Application to sexual networks and STI 14 September 2011 Partnership duration Models for sexually transmitted infections Which frameworks? HIV/AIDS: SI framework chlamydia and gonorrhoea : SIS framework

More information