Optimal control problem and backward bifurcation on malaria transmission with vector bias

Abstract

This article aims to apply a mathematical model to investigate the spread of malaria by considering vector bias, saturated treatment, and an optimal control approach. A mathematical analysis of the equilibrium points and an investigation of the basic reproduction number show that if the basic reproduction number ( is less than one, the disease-free equilibrium is locally asymptotically stable. Furthermore, the center-manifold theory is applied to analyze the stability of the endemic equilibrium when 1 . We find that our model performs a backward bifurcation phenomenon when the saturated treatment or vector bias parameter is larger than the threshold. Interestingly, we found that uncontrolled fumigation could increase the chance of the appearance of backward bifurcation. From the sensitivity analysis of , we find that the fumigation and vector bias are the most influential parameters for determining the magnitude of . Using the Pontryagin maximum principle, the optimal control problem is constructed by treating fumigation and medical treatment parameters as the time-dependent variable. Our numerical results on the optimal control simulation suggest that time-dependent fumigation and medical treatment could suppress the spread of malaria more efficiently at minimum cost.

Dynamical Analysis on a Malaria Model with Relapse Preventive Treatment and Saturated Fumigation

“…As we mentioned before, a smaller value a indicates a more prepared community to the increasing number of infected individuals. From the expression of R 0 in (15), it can be seen that a does not 13 Computational and Mathematical Methods in Medicine appear in R 0 . Hence, we conclude that a does not impact the size of R 0 .…”

“…The proposed dynamic model for malaria transmission in this article is inspired by our previous work in [ 15 ], by taking into account two important factors. The first factor involved in our new model is the involvement of a malaria relapse prevention drug intervention (for example, tafenoquine [ 28 ]).…”

“…They found that increasing the transmission blocking drugs is a more pronounced effect compared to treatment intervention. Another important factors have been discussed through mathematical models such as vector-bias effect [15,17], relapse [18,19], reinfection [20,21], fumigation [15,22], temperature and seasonality [23][24][25], impact of Wolbachia [26], and coinfection [27]. However, the best that we know, there is no mathematical model considering the impact of potential new treatment (tafenoquine) into their model.…”

“…They used Floquet theory to analyze the stability of their model. Recently, author in [ 15 ] proposed a malaria model with optimal control on saturated treatment rate. Another new strategy of transmission blocking drugs for malaria is modelled by Wu and Hu in [ 16 ].…”

An Optimal Control Model to Understand the Potential Impact of the New Vaccine and Transmission-Blocking Drugs for Malaria: A Case Study in Papua and West Papua, Indonesia

“…They found that as infection rate has the greatest impact on the basic reproduction number compared to other model parameters, it is important to reduce this infection rate, such as using insecticide-treated bed nets, spraying insecticides, clearing stagnant water, etc. In a more recent mathematical model, authors include more recent facts on malaria transmission such as the effect of vector-bias [ 18 ], asymptotic carriers [ 19 ], age-structured [ 20 ], competitive strains [ 21 ], seasonal factor [ 22 , 23 ], and coinfection of malaria with COVID-19 [ 24 ]. Furthermore, intervention models also have been widely introduced by authors, such as the use of fumigation [ 25 ], insecticide-treated bed nets [ 26 ], and vaccines with waning immunity [ 27 ], or transmission-blocking vaccines [ 28 ].…”