Mathematical Analysis Of Two Pathogens (HIV/TB) Co-infection Model Coupled With Vertical Transmission

This thesis addresses the synergistic interaction between HIV/AIDS and mycobac-terium tuberculosis (TB) using a deterministic model at population level, which in-cludes many of the very important epidemiological and biological features of the two pathogens. We considered an HIV/AIDS and TB co-infection model with the incor-poration of vertical transmission on HIV infecteds, and also we looked at the effec-tive treatment which considers highly active antiretroviral therapy and Prevention of mother to child transmission for HIV/AIDS cases and treatment of all kinds of TB, that is, latent and active TB. In both models we attempted to analyzing the full model instead of separating the models like other authors did in their models. Results shows that HIV-free implies that individual is infected with TB only and conversely is true. The first four models have shown that there are globally asymptotically stable at disease-free equilibrium when corresponding to the reproduction number less than unity. Further they shows that there is a convex combination associated to a threshold RHT-Implication is, for all choices of initial condition of control parameter values the disease clears out. The last model reveals that the model is not globally asymptotically stable at the disease-free equilibrium, thus it exhibits the phenomena of backward bi-furcation, that is, when the disease expected to clear out but in actual fact according to this phenomena the disease persist in the host population when is associated to the cor-responding reproduction number less than unity. Analysis of respective reproduction ratios has shown that the use of antiretroviral therapy strategy can leads to an effec-tive treatment of HIV if and only if it reduces the relative infectiousness of individuals treated as compared to the untreated HIV-infected individuals. Numerical simula-tion were performed and they confirmed the analytical result we obtained. Further, the simulation analysis shows that when the two pathogens are well treated the popu-lation will not abruptly go to an extinction as compared to when there is no treatment.We considered the idea of incorporating dual-infection and vertical transmission in our co-infection model. We included vertical transmission into co-infection model since it is one of the important transmission mode in HIV/AIDS and this, makes our model to be totally different from other models. The following proceedings are a summary of what we have done:1. Formulation of co-infection model system (2.0.1).The autonomous differential equations of the form Jss=Λ—λHJSS—λJSS—μJSS, JIS=λHJSS—φTλTJIS—(μ+δH)JIS (0.0.18) JST=λTJSS—φHλH JST—(μ+δT)JST JIT=φTλTJIS+φHλHJST—(μ+δ)JITAll solutions of model system (2.0.1) with initial conditions in the positive region is bounded for t>0.We have the disease-free equilibrium point given byAfter computing using the next generation operator by [21,77], we get the repro-duction numbers of HIV and TB: and Ro=max(R0H,R0T).Thus, the disease-free equilibrium point is locally asymp-totically stable whenever R0<1and unstable if R0>1.We use Lyapunov function get the global asymptotically stability condition of the disease-free equilibrium point, and we have定理0.7The disease-free equilibrium point, E0, of the co-infection model system (2.0.1) is global asymptotically stable in Ω0whenever R0<1. The endemic equilibrium point is given by E1=(JSS/*,JIS/*,JST/*,JIT/*), whereBy following[54]methods and substituting into the equations of forces of infection it reveals that the function is a decreasing function.Biologically speaking,this implies that the susceptible population is replenished by the death induced caused by the two pathogens HIV and TB.2.An extension of co-infection model to incorporate dual-infection in model sys-tem(3.0.1). JSS=Λ一λHJSS—λJSS—λHTJSS—μJSS, JIS=λHJSS—φTλTIS—(μ+δH)JIS (0.0.23) JST=λTJSS—φHλHJST—(μ+δT)JST JIT=φTλTJIS+φHλHJST+λHTJSS—(μ+δ)JITDisease-free equilibrium is given by and endemic equilibrium point is given by E3=(JSS/*,JIS/*,JST/*,JIT/*),whereUsing the next generation operator by[21,77],we have,定理0.8The disease equilibrium,E2is locally asymptotically stable when R0<1and unstable whenever R0>1,where R0=max{R0H,R0T,R0HT}.We construct a Lyapunov function of the form satisfying the necessary condition at the new origin L1(JSS/*,JIS/*,JST/*,JIT/*)=0.Thus, we have定理0.9The unique endemic equilibrium E3,of the dual infection model (3.0.1),is globally asymptotically stable in Ω1,whenever R0>1.3.Extension of co-infection model to incorporate vertical transmission coupled with dual-infection.JSS=Λs+bω(JIS+(?)JIT—λHJSS—λTJSS—λHTJSS—μJSS+λ1JST,JIS=λHJSS—φTλTJIS—(μ+δH)JIS+b(1—ω)(JIS+(?)JIT)+λ2JIT,JST=λTJSS—φHλHJST—(μ+δT+γ1)JST,JIT=φTλTJIS+φHλHJST+λHTJSs—(μ+δ+γ2)JIT.(0.0.28) We obtain the steady state solutions of model system (4.0.1) by setting its right hand side to be zero and obtain the following, The disease-free equilibrium point D.F.E which is given and the endemic equilibrium point E5=(JSs/*, JIs/*, JST/*, JIT/*).Using the next generation operator technique [21,77], we compute the model reproduction numbers as定理0.10The disease-free equilibrium, EO is locally asymptotically stable whenever Ro<1and unstable whenever RO>1, where Ro=max{Roi, R02, R03}.We construct the Lyapunov function of the form satisfying L2(JSS/*, JIS/*, JST/*JIT/*)=0at the origin which is the necessary condition and also negative definite. By direct calculation of the time derivative of L(JSS, JIS, JST, JIT) along the disease-free equilibrium solution we have,定理0.11The disease-free equilibrium point of model system (4.0.1) is globally asymp-totically stable if RH<1,ROT<1, RHT<1implying RO<1. 4.Construction of a nine dimensional co-infection model considered by Long et.al [49]coupled with vertical transmission. dt/dJSS=Λ+b(1+ω)GSV-JSS-JSS+γJST, dt/dJSS=λTJSS—λHJSL—(σsL+μ)JS dt/dJSS=σSLJSL-λHJST-(μ+μT+γ)JST, dt/dJSS=λHJSS+bωGIV-λTJIS—(VIS+μ)JIS,(0.0.34) dt/dJSS=λHJSl+λTJIS-(VIL+σIL+μ)JILS, dt/dJSS=λHJST+σILJIL-(VIL+μ+μT)JIT dt/dJAT=VISJIT-λTJAS-(μ+μA)JIT, dt/dJAT=VISJIT-λTJAL-(σAL+μ+μA)JIT, dt/dJAT=VISJIT-σATJAL-(μ+μT+μA)JIT, where,Gsy=b(1—ω)(JIS+η1JIL+η2JIT)+b(1—ω)φ(JAS+η1JAL+η2JAT), GTv=bω(JIS+η1JIL+η2JIT)+bωφ(JAS+η1JAL+η2JAT). Using the next generation operator technique[21,77],we compute the model repro-duction number with help of mathematica as (?)O=max((?)O/H,(?)O/T)In the absence of the disease we have our disease-free equilibrium point given by The endemic equilibrium point is given by E7=(JSS/*, JIS/*, JST/*, JIT/*), where定理0.12[7] The fixed point E6=(X0,0) is globally asymptotically stable equilibrium point of the system (5.0.28) whenever (?)0<1and that assumptions in (5.0.30) are satisfied.We carried out global stability of the model system (5.0.1) and it exhibited a phenomena of backward bifurcation, that is the disease persisted even when (?)0<1, up to a certain point.Numerical simulations are given at the end of every chapters except chapter2.