Research Unit for Mathematics and Applications, Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67 Dschang, Cameroon; IRD UMI 209 UMMISCO, University of Yaounde I, P.O. Box 337 Yaounde, Cameroon and LIRIMA-EPITAG Team Project, University of Yaounde I, P.O. Box 812Yaounde, Cameroon.
Research Unit for Mathematics and Applications, Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67 Dschang, Cameroon; Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa; IRD UMI 209 UMMISCO, University of Yaounde I, P.O. Box 337 Yaounde, Cameroon and LIRIMA-EPITAG Team Project, University of Yaounde I, P.O. Box 812Yaounde, Cameroon.
J Theor Biol. 2021 Mar 7;512:110537. doi: 10.1016/j.jtbi.2020.110537. Epub 2020 Nov 13.
In this work, we assess the impact of the phage-bacteria infection and optimal control on the indirectly transmitted cholera disease. The phage-bacteria interactions are described by predator-prey system using the Smith functional response, which takes into account the number of bacteria binding sites. The study is done in two steps, namely the model without control and the model with control. For the first scenario, we explicitly compute the basic reproduction number R which serves as stability threshold and bifurcation parameter. The proposed model exhibits a bi-stability phenomenon via the existence of backward bifurcation, which implies that the classical requirement of bringing the reproduction number under unity, while necessary, is no longer sufficient for cholera elimination from the population. We intuitively introduce a new threshold number N needed for the global stability of the disease free equilibrium point which is achieved when R⩽1 and N⩽1. It is further shown that the phage absorption is a possible cause of bi-stability, since in its absence, the condition R⩽1 is sufficient for cholera to die out. The existence of endemic equilibrium points depends on the range of both R and N. Regarding the model extended to an optimal control problem, which involves the use of virulent vibriophages to reduce or eliminate the bacteria population, we use optimal control theory techniques. We establish the conditions under which the spread of cholera can be stopped, and examine the impact of control measures on the transmission dynamic of cholera. The Pontryagin's maximum principle is used to characterize the optimal control. Numerical simulations suggest that, the release of lytic vibriophages can significantly reduce the spread of the disease. We discuss opportunities for phage therapy as treatment of some bacterial-borne diseases without side effects.
在这项工作中，我们评估了噬菌体-细菌感染和最优控制对间接传播的霍乱疾病的影响。噬菌体-细菌相互作用通过使用 Smith 功能反应来描述捕食者-猎物系统，该反应考虑了细菌结合位点的数量。研究分两步进行，即无控制模型和有控制模型。对于第一种情况，我们明确计算了基本繁殖数 R，作为稳定性阈值和分岔参数。所提出的模型通过存在反向分岔表现出双稳定性现象，这意味着将繁殖数降低到单位以下的经典要求，虽然是必要的，但对于从人群中消除霍乱已不再足够。我们直观地引入了一个新的阈值数 N，当 R ⩽1 和 N ⩽1 时，该数用于疾病无平衡点的全局稳定性。进一步表明，噬菌体的吸收可能是双稳定性的原因，因为在不存在噬菌体的情况下，R ⩽1 的条件足以使霍乱消失。地方病平衡点的存在取决于 R 和 N 的范围。关于扩展到最优控制问题的模型，其中涉及使用毒性噬菌体来减少或消除细菌种群，我们使用最优控制理论技术。我们确定了可以阻止霍乱传播的条件，并研究了控制措施对霍乱传播动态的影响。庞特里亚金最大值原理用于表征最优控制。数值模拟表明，释放裂解噬菌体可以显著减少疾病的传播。我们讨论了噬菌体治疗作为治疗某些无副作用的细菌传播疾病的机会。
