Abstract
Malaria is an infectious disease caused by Plasmodium parasite and it is transmitted among humans through bites of female Anopheles mosquitoes. In this paper, a new deterministic mathematical model for the endemic malaria disease transmission that incorporates imperfect quarantine and optimal control is proposed. Impact of various intervention strategies in the community with varying population at time t are analyzed using mathematical techniques. Further, the model is analyzed using stability theory of differential equations and the basic reproduction number is obtained from the largest eigenvalue of the next-generation matrix. Conditions for local and global stability of disease free, local stability of endemic equilibria and bifurcations are determined in terms of the basic reproduction number. The Center manifold theory is used to analyze the bifurcation of the model. It is shown that the model exhibit both a backward and a forward bifurcation. Reducing the biting rate of the quarantined people is advice able to minimize the spread of endemic malaria disease. The optimal control is designed by applying Pontryagins’s Maximum Principle (PMP) with four control strategies namely, insecticide treated nets, screening, treatment and indoor residual spray. The best strategy to control endemic malaria disease is the combination that incorporated all four control strategies.
Highlights
Malaria is the dangerous one among infectious disease
IN recent, reduction in the number of malaria related cases are due to the global efforts of the current malaria interventions, such as decreasing mosquito breeding sites, sleeping under insecticide-treated nets (ITN), indoor residual spraying (IRS) with insecticides, are used for reducing malaria vectors and their bites, timely treatment with artemisinin-based combination therapies (ACTs) and chemoprevention for most vulnerable such as intermittent preventive treatment for pregnant women (IPTp) recommended by WHO
We formulated and analyzed a deterministic model that incorporates both imperfect quarantine and optimal control strategy to investigate their roles in case of endemic malaria disease control and elimination
Summary
Malaria is the dangerous one among infectious disease. It is caused by plasmodium parasites that are transmitted among humans through the bites of female Anopheles mosquitoes. Mathematical models of the dynamics of malaria transmission are useful in providing a better insight into the behavior of the disease These models have played a great role in influencing the decision making processes regarding intervention strategies for controlling and eliminating the spread of malaria. Okosun et al (2013) derived and analyzed a malaria disease transmission mathematical model that includes insecticide treated net, treatment and indoor residual spray and applied optimal control strategy to study a possible treatment of infective humans that blocks transmission to mosquitoes in controlling the spread of malaria [18]. The purpose of the study of endemic malaria disease model system (1) with imperfect quarantine strategy is to reduce the number of susceptible mosquitoes bites from or contacts with malaria infectious humans and explore the effect of the strategy in the malaria control and elimination
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