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 ${\mathcal{R}}_0$ 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 ${\mathcal{N}}_0$ needed for the global stability of the disease free equilibrium point which is achieved when ${\mathcal{R}}_0⩽1$ and ${\mathcal{N}}_0⩽1$. It is further shown that the phage absorption is a possible cause of bi-stability, since in its absence, the condition ${\mathcal{R}}_0⩽1$ is sufficient for cholera to die out. The existence of endemic equilibrium points depends on the range of both ${\mathcal{R}}_0$ and ${\mathcal{N}}_0$. 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间接控制霍乱疾病的最佳控制的影响。噬菌体与细菌之间的相互作用是由捕食者－捕食系统使用史密斯功能响应描述的，其中考虑到了细菌结合位点的数量。该研究分两个步骤完成，即没有控制的模型和有控制的模型。对于第一种情况，我们显式计算基本复制数${\mathcal{[R}}_0$它用作稳定性阈值和分叉参数。所提出的模型由于存在反向分叉而表现出双稳定性现象，这意味着使复制数量统一的经典要求虽然必要，但不再足以从人口中消除霍乱。我们直观地引入了新的阈值${\mathcal{ñ}}_0$ 无病平衡点的整体稳定性所需的时间 ${\mathcal{[R}}_0⩽\mathrm{1个}$ 和 ${\mathcal{ñ}}_0⩽\mathrm{1个}$。进一步表明，噬菌体吸收是双稳定性的可能原因，因为在没有这种稳定性的情况下${\mathcal{[R}}_0⩽\mathrm{1个}$足以使霍乱消亡。地方平衡点的存在取决于两者的范围${\mathcal{[R}}_0$ 和 ${\mathcal{ñ}}_0$。关于扩展到最优控制问题的模型，该模型涉及使用强毒噬菌体减少或消除细菌种群，我们使用最优控制理论技术。我们建立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。Pontryagin的最大原理用于表征最佳控制。数值模拟表明，溶菌性噬菌体的释放可以显着减少疾病的传播。我们讨论了噬菌体疗法作为治疗某些无副作用的细菌传播疾病的机会。