Nonlinear Analysis of Malaria Epidemics
Abstract
Malaria is a parasitic vector-borne disease caused by the plasmodium parasite, which is transmitted to people via the bites of infected female mosquitoes of the genus Anopheles. The life cycle of plasmodium involves incubation periods in two hosts, the human and the mosquito. Therefore, mathematical modelling of the spread of malaria usually focuses on the feedback dynamics from mosquito to human and back. In this paper we have to understand the important parameters in the transmission and spread of endemic malaria disease, and try to prevent and control by applying mathematical modelling. We have to derive the equilibrium points of the model and investigate their stability. This paper shows that if the reproduction number, , is less than 1, the disease-free equilibrium point is stable, so that the disease dies out. If is larger than 1, then the disease-free equilibrium is unstable. After deriving the reproduction number, they determine its dependence on human travel rates. Their analysis shows that reproduction number varies consistently with movement of exposed, infectious and recovered humans. Analysis of the threshold helps us design effective control measures to reduce disease transmission.
Keywords
Malaria; Reproduction number; Mathematical modelling; System of nonlinear differential equations; Homotopy analysis method.
1. Introduction
Malaria is one of the deadliest infectious diseases that have claimed million of lives around the world. Globally, 3.3 billion people or half of the world’s population in 104 countries are at the risk of getting infected by malaria disease. It has been estimated that between 300 and 500 million individuals of all ages are infected annually and between 1.5 and 2.7 million people die of malaria every year Malaria is widely spread in tropical and subtropical regions, including Africa ,Asia, Latin America, the middle East and some parts of Europe. The most cases and deaths occur in sub-Saharan Africa. In particular, thirty countries in sub-Saharan Africa account for 90% of global malaria deaths [6]. Shockingly, the disease kills an African child every 30 seconds and over2,000 young lives are lost daily across the globe and For example, malaria accounts for 60% of outpatient visits and 30% of hospitalizations among children under ﬁve years of age in Nigeria .The disease, malaria, which remains one of the most prevalent and lethal human infection worldwide, is caused by infection with single-celled (protozoan)parasites of genus Plasmodium and is characterized by paroxysms of chills, fever, headache, pain and vomiting. The parasites are transmitted to humans through the bites of infected female Anopheles mosquitoes (vectors)[15]. Of the ﬁve parasite species (Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, Plasmodium malaria and Plasmodium that cause malaria in humans, Plasmodium falciparum is the most deadly form and it predominates in Africa The parasite is responsible for the greatest number of deaths and clinical cases in the tropics. Its infection can lead to serious complications aﬀecting brain, lungs, kidneys and other organs. Mathematical models for transmission dynamics of malaria are useful in providing better insights into the behaviour of the disease. The models have played great roles in inﬂuencing the decision making processes regarding intervention strategies for preventing and controlling the insurgence of malaria. The study on malaria using mathematical modelling began in 1911[16] with Ronald Ross. He introduced the ﬁrst deterministic two-dimensional model with one variable representing human and the other representing mosquitoes where it was shown that reduction of mosquito population below a certain threshold was suﬃcient to eradicate malaria. In the Ross’s model was modiﬁed by considering the latency period of the parasites in mosquitoes and their survival during that period. For the purpose of estimating infection and recovery rates, Macdonald [3] used a model in which he assumed the amount of infective material to which a population is exposed remains unchanged. Bailey [14] and Aron [8, 9] models take into account that acquired immunity to malaria depends on exposure (i.e. that immunity is boosted by additional infections). Tumwiine [11] used SIS and SI models in the human hosts and mosquito vectors for the study of malaria epidemics that last for a short period in which birth and immunity to the disease were ignored. They observed that the system was in equilibrium only at the point of extinction that was neither stable nor unstable. Some recent papers have also included environmental effects [10, 4, 5], and the spread of resistance to drugs [12, 7]. Recently, Ngwa and Shu [13] and Ngwa [2] proposed an ODE compartmental model for the spread of malaria. Addo [1], Tuwiine, Mugisha and Luboobi [15] developed a compartment model for the spread of malaria with susceptible-infected-recovered-susceptible (SIRS) pattern for human and susceptible-infected (SI) pattern for mosquitoes. Yang, Wei, and Li [17] proposed SIR for the human and SI for vector compartment model. Addo [1], Tuwiine, Mugisha and Luboobi [15] and Yang, Wei, and Li [17], define the reproduction number, R0 and show the existence and stability of the disease-free equilibrium and an endemic equilibrium. From the model in [17], we can see that the number of births for human and mosquito are independent of the total human and mosquito populations. However, in this case, it was shown that reducing the number of mosquitoes is an in eﬃcient control strategy that would have little elect on the epidemiology of malaria in areas of intense transmission. Further extension was described by Anderson and May, where the latency of infection in humans was introduced by making the additional exposed class in humans. This modiﬁcation further reduces the long term prevalence of both the infected humans and mosquitoes. The model presented in this work consists of four compartments in humans and three compartments in mosquitoes, with inclusions of nonlinear forces of infection in form of saturated incidence rates in both the host and vector populations. The disease-induced death rates for humans and mosquitoes are also incorporated into the model. The main objective of the study is to understand the important parameters in the transmission and spread of malaria disease, try to develop effective solutions and strategies for its prevention and control, and eventually how to eradicate it.
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About the Author
Dr. V. Ananthaswamy
^{1}V.Ananthaswamy^{*}, ^{2}E.Sreejee, ^{3}M. Subha ^{1}Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India ^{2}M. Phil., Mathematics, The Madura College, Madurai, Tamil Nadu, India. ^{3}Department of Mathematics, MSNPMW College, Poovanthi, Tamil Nadu, India. ^{*}Corresponding author e-mail: ananthu9777@rediffmail.com |