具有垂直传染的HIV/TB两种混合传染病模型的数学分析
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摘要
本文使用关于种群数目的确定性模型从许多重要的流行病学和生物学特性上对艾滋病毒/艾滋病和结核分枝杆菌两种病原体之间的协同作用进行了研究。我们研究了可以母婴传播的艾滋病毒/艾滋病和结核病相结合的模型,我们也研究了三种带有有效控制策略的模型,包括高活性抗逆转录病毒治疗艾滋病的控制策略和预防母婴传播母艾滋病毒的控制策略以及治疗各种结核病的策略(包括潜伏性和活动性的结核病).在这两种类型模型中我们试图分析两种病原体相结合的模型,而不是像他人做的那样单个病原体的模型。其中,无艾滋病毒意味着这个人是只感染结核病,相反也是这样。前四个模型表明,当对应的再生数R0小于1时,模型有全局渐近稳定的无病平衡点。此外临界值RHT是这些再生数的一个凸组合.有意义的是,作为初始条件的控制参数值的任何可行值都能使得疾病清除。最后一个模型表明,该模型没有一个无病的全局渐近稳定平衡点,它出现了倒向的分支,也就是说当疾病预期已经被消灭时,实际上由这一分支可知疾病仍然存在于宿主当中,尽管相应的再生数R0小于1。分析各自的再现比率已经表明,使用的抗逆转录病毒疗法的策略可以有效的控制艾滋病毒因为和未经治疗的艾滋病感染者相比它减少了个体的相对传染性。数值模拟证实了我们得到的分析结果。另外仿真分析显示,当这两种病原体能很好的被治疗时人口将不会突然的衰减。
在我们的模型中,我们考虑了两种病原体交叉感染和母婴传播这两种情形。由于母婴传播是艾滋病毒/艾滋病的一种重要的传播方式,所以我们将母婴传播引入到艾滋病毒/艾滋病模型当中从而使我们的模型不同于其他人的模型。下面是对我们工作的一个总结：
1.联合感染模型(2.0.1).自治微分方程JSS=Λ—λHJSS—λTJSS—μJSS, JlS=λHJSS—φTλJIS一(μ+δH)JIS (0.0.1) JST=λTJSS—φHλHJST—(μ+δT)JST JIT=φTλTJIs+φHλHJST一(μ+δ)JIT对于任意的t≥0系统(2.0.1)所有的初始条件位于正区域的解都是有界的。我们得到如下的无病平衡点
使用[21,77]的方法,我们还可以得到HIV和TB的再生数：and RO=max(ROH,ROT)进而,无病平衡点是局部渐进稳定的当且仅当R0<1,是不稳定的如果R0>1.我们使用Lyapunov函数JST÷JIT得到了无病平衡点的全局渐进稳定性条件
定理0.1联合感染模型(2.0.1)的无病平衡点E0是全局渐进稳定的如果R0<1.
我们还得到了非平凡的平衡点E1=(J*SS,J*IS,J*ST,J*IT),这里
通过使用[54]的方法将平衡点带入到方程得到了一个单调递减函数。生物学上讲,这意味着随着两种病原体致人死亡后意味着易感人群的数目得到了补充。
2.交叉感染模型(3.0.1).JSS=Λ—λHJSS—λJSS—λHTJSS—μJSS, JIS=λHJSS—φTλTIS—(μ+δH)JIs (0.0.6) JST=λJSS—φHλHJST—(μ+δT)JST JIT=φTλTJIS+φHλHJST+λHTJSS—(μ+δ)JIT
我们给出无病平衡点E2=(∧/μ,0,0,0)和非平凡的平衡点E3=(J*SS,J*IS,J*ST,J*IT这里使用[21,77]的方法,我们得出
定理0.2无病平衡点E2是局部渐进稳定的如果R0<1,是不稳定的如果R0>1,这里RO=mox{ROH,ROT,ROHT}.建立如下的Lyapunov函数
且满足L1(J*SS,J*IS,J*ST,J*IT)=0.我们有
定理0.3交叉感染模型(3.0.1)的唯一非平凡平衡点E3是全局渐进稳定的如果R0>1.
3.带有母婴传播的交叉感染模型.JSS=Λ+bω(JIS+QJIT)—λHJSS—λTJSs—λHTJSS—μJSS+γ1JST, JIS=λHJSS—φTλTJIS—(μ+δH)JIS+b(1—ω)(JIS+QJIT)+γ2JIT, JST=λTJSS—φHλHJST—(μ+δT+γ1)JST, JIT=φTλTJIS+φHλHJST+λHTJSS—(μ+δ+γ2)JIT.(0.0.11)令系统(4.0.1)右边等于零,我们可以得到系统的稳定状态解如下
系统(4.0.1)有无病平衡点D.F.E和非平凡平衡点E5=(J*SS,J*IS,J*ST,J*IT).
由[21,77],我们计算再生数如下
定理0.4无病平衡点E0是局部渐进稳定的如果R0<1,是不稳定的如果R0>1,这里R0=mαx{R01,R02,R03].
我们建立如下的Lyapunov函数且满足L2(J*SS,J*IS,J*ST,J*IT)=0。通过计算L2(JSS,JIS,JST,JIT)在无病平衡点附近的导数我们有,
定理0.5系统(4.0.1)的无病平衡点是全局渐进稳定的如果RH<1,R0TT<1,RHT<1,也就是R0<1.
4.带有母婴传播的9维联合感染模型(Long et.al).这里,
GSV=b(1—ω)(JIS+η1JIL+η2JIT)+b(1—ω)φ(JAS+η1JAL+η2JAT),GIV=bω(JIS+η1JIL+η2JIT)+bωφ(JAS+η1JAL+η2JAT).
通过使用[21,77]的方法,我们使用mathematica计算系统的再生数如下系统有无病平衡点
我们还给出系统的非平凡平衡点E7=(J*SS,J*IS,J*ST,J*IT),这里
定理0.6／[7]系统(5.0.28)的不动点E6=(X0,0)是全局渐进稳定平衡点如果R0<1并且满足(5.0.30)的假设条件.
我们研究系统(5.0.1)的全局稳定性时发现了倒向的分支,也就是说当疾病预期已经被消灭时,实际上疾病仍然存在于宿主当中,尽管相应的再生数R0小于1。除了第二章以外,每章的末尾我们都进行数字模拟.
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)JIT
All 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 by
After 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/*), where
By 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)JIT
Disease-free equilibrium is given by and endemic equilibrium point is given by E3=(JSS/*,JIS/*,JST/*,JIT/*),where
Using 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.
在我们的模型中,我们考虑了两种病原体交叉感染和母婴传播这两种情形。由于母婴传播是艾滋病毒/艾滋病的一种重要的传播方式,所以我们将母婴传播引入到艾滋病毒/艾滋病模型当中从而使我们的模型不同于其他人的模型。下面是对我们工作的一个总结：
1.联合感染模型(2.0.1).自治微分方程JSS=Λ—λHJSS—λTJSS—μJSS, JlS=λHJSS—φTλJIS一(μ+δH)JIS (0.0.1) JST=λTJSS—φHλHJST—(μ+δT)JST JIT=φTλTJIs+φHλHJST一(μ+δ)JIT对于任意的t≥0系统(2.0.1)所有的初始条件位于正区域的解都是有界的。我们得到如下的无病平衡点
使用[21,77]的方法,我们还可以得到HIV和TB的再生数：and RO=max(ROH,ROT)进而,无病平衡点是局部渐进稳定的当且仅当R0<1,是不稳定的如果R0>1.我们使用Lyapunov函数JST÷JIT得到了无病平衡点的全局渐进稳定性条件
定理0.1联合感染模型(2.0.1)的无病平衡点E0是全局渐进稳定的如果R0<1.
我们还得到了非平凡的平衡点E1=(J*SS,J*IS,J*ST,J*IT),这里
通过使用[54]的方法将平衡点带入到方程得到了一个单调递减函数。生物学上讲,这意味着随着两种病原体致人死亡后意味着易感人群的数目得到了补充。
2.交叉感染模型(3.0.1).JSS=Λ—λHJSS—λJSS—λHTJSS—μJSS, JIS=λHJSS—φTλTIS—(μ+δH)JIs (0.0.6) JST=λJSS—φHλHJST—(μ+δT)JST JIT=φTλTJIS+φHλHJST+λHTJSS—(μ+δ)JIT
我们给出无病平衡点E2=(∧/μ,0,0,0)和非平凡的平衡点E3=(J*SS,J*IS,J*ST,J*IT这里使用[21,77]的方法,我们得出
定理0.2无病平衡点E2是局部渐进稳定的如果R0<1,是不稳定的如果R0>1,这里RO=mox{ROH,ROT,ROHT}.建立如下的Lyapunov函数
且满足L1(J*SS,J*IS,J*ST,J*IT)=0.我们有
定理0.3交叉感染模型(3.0.1)的唯一非平凡平衡点E3是全局渐进稳定的如果R0>1.
3.带有母婴传播的交叉感染模型.JSS=Λ+bω(JIS+QJIT)—λHJSS—λTJSs—λHTJSS—μJSS+γ1JST, JIS=λHJSS—φTλTJIS—(μ+δH)JIS+b(1—ω)(JIS+QJIT)+γ2JIT, JST=λTJSS—φHλHJST—(μ+δT+γ1)JST, JIT=φTλTJIS+φHλHJST+λHTJSS—(μ+δ+γ2)JIT.(0.0.11)令系统(4.0.1)右边等于零,我们可以得到系统的稳定状态解如下
系统(4.0.1)有无病平衡点D.F.E和非平凡平衡点E5=(J*SS,J*IS,J*ST,J*IT).
由[21,77],我们计算再生数如下
定理0.4无病平衡点E0是局部渐进稳定的如果R0<1,是不稳定的如果R0>1,这里R0=mαx{R01,R02,R03].
我们建立如下的Lyapunov函数且满足L2(J*SS,J*IS,J*ST,J*IT)=0。通过计算L2(JSS,JIS,JST,JIT)在无病平衡点附近的导数我们有,
定理0.5系统(4.0.1)的无病平衡点是全局渐进稳定的如果RH<1,R0TT<1,RHT<1,也就是R0<1.
4.带有母婴传播的9维联合感染模型(Long et.al).这里,
GSV=b(1—ω)(JIS+η1JIL+η2JIT)+b(1—ω)φ(JAS+η1JAL+η2JAT),GIV=bω(JIS+η1JIL+η2JIT)+bωφ(JAS+η1JAL+η2JAT).
通过使用[21,77]的方法,我们使用mathematica计算系统的再生数如下系统有无病平衡点
我们还给出系统的非平凡平衡点E7=(J*SS,J*IS,J*ST,J*IT),这里
定理0.6／[7]系统(5.0.28)的不动点E6=(X0,0)是全局渐进稳定平衡点如果R0<1并且满足(5.0.30)的假设条件.
我们研究系统(5.0.1)的全局稳定性时发现了倒向的分支,也就是说当疾病预期已经被消灭时,实际上疾病仍然存在于宿主当中,尽管相应的再生数R0小于1。除了第二章以外,每章的末尾我们都进行数字模拟.
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)JIT
All 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 by
After 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/*), where
By 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)JIT
Disease-free equilibrium is given by and endemic equilibrium point is given by E3=(JSS/*,JIS/*,JST/*,JIT/*),where
Using 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.
引文
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