Cai Liming, Li Xuezhi, Tuncer Necibe, Martcheva Maia, Lashari Abid Ali
College of Mathematics and Statistic Science, Xinyang Normal University, Xinyang, 46400, CHINA.
College of Mathematics and information Science, Xinyang Normal University, Xinyang, 46400, CHINA.
Math Biosci. 2017 Jun;288:94-108. doi: 10.1016/j.mbs.2017.03.003. Epub 2017 Mar 9.
In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R<1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard incidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. Simulations suggest that total prevalence of malaria is the highest if all individuals show symptoms upon infection, but then undergoes an incomplete treatment and the lowest when all the individuals first move to the symptomatic class then treated successfully. Total prevalence is average if more individuals upon infection move to the asymptomatic class. We study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals. Simulations suggest that the optimal control strategies are very dynamic. Although they always lead to decrease in the symptomatic infectious individuals, they may lead to increase in the number of asymptomatic infectious individuals. This last scenario occurs if a large portion of newly infected individuals move to the symptomatic class but many of them do not complete treatment or if they all complete treatment but the superinfection rate of asymptomatic individuals is average.
在本文中，我们引入了一个疟疾模型，该模型在人类群体中有一个无症状类别，在人类和病媒群体中都有暴露类别。该模型假设无症状个体可再次感染并进入有症状类别。在治疗不完全的情况下，有症状个体进入无症状类别。如果成功治疗，有症状个体康复并进入易感类别。基本再生数(R)使用下一代方法计算。该系统有一个无病平衡点（DFE），当(R\lt1)时局部渐近稳定，并且可能有多达四个地方病平衡点。该模型表现出由两种机制产生的向后分支；标准发病率和重复感染。如果该模型不考虑重复感染或因病死亡，那么DFE是全局稳定的，这表明不再可能出现向后分支。模拟表明，如果所有个体在感染时都出现症状，但随后接受不完全治疗，疟疾的总流行率最高；而当所有个体首先进入有症状类别然后成功治疗时，总流行率最低。如果更多感染个体进入无症状类别，总流行率为平均水平。我们研究了将蚊帐使用和治疗作为减少有症状和无症状个体总数的主要工具所应用的最优控制策略。模拟表明，最优控制策略非常动态。虽然它们总是导致有症状感染个体数量减少，但可能导致无症状感染个体数量增加。如果很大一部分新感染个体进入有症状类别但其中许多人未完成治疗，或者如果他们都完成了治疗但无症状个体的重复感染率为平均水平，就会出现最后这种情况。
