A SEIRS-SEI Mathematical Model for Malaria Disease Transmission.

Abstract

Malaria is an infectious disease caused by a parasite of the genus Plasmodium of Anopheles mosquito. The malaria is transmitted to human through mosquito bites which is very dangerous for our health. In this research, a system of ordinary differential equations for the spread of malaria in human and mosquito populations is presented. In the model formulated by Chitnis, the human population is divided into four classes, namely susceptible, exposed, infected, and recovered. The mosquito population is divided into three classes, namely susceptible, exposed, and infected. Susceptible human can be infected when they are bitten by infectious mosquitos. In this study, the exposed humans considered to have been open to be infected by parasites so they are classified into human exposed class. After a period of incubation elapsed, those in human exposed class might be infected so that they are classified into infected class. Those in infected class might be recovered after a latent period passed on so that they are classified into recovered class, or may return back to the susceptible class when the immunity decreased. Those in recovered class will have temporary immunity so that again be susceptible human in a given period. In this study, this model is a modification of a previous model is proposed and analyzed by adding a recovery rate of infected subclass into susceptible subclass of human. The simulation study showed the existence of two equilibrium points, i.e. the disease-free equilibrium and the endemic equilibrium points. Next, the stability analysis of the equilibrium points were conducted by considering the basic reproduction number ( ). The basic reproduction number is the expected value of infections per unit of time. The number is considered as a benchmark of disease transmission in the population. If then on average each infected individual will be infecting less than one newly individual, so that the disease will disappear. If , then on average each infected individual will generate more than one newly infected individuals, so that the disease will spread. Numerical analysis and simulation results showed that the number of each class of human and mosquito reaches a stable condition approaching the diseasefree equilibrium and obtained , and approaching the stable condition of the endemic equilibrium with the value of In addition, the increase of human recovery rate will decrease the . Therefore, the rate of disease transmission decreases. The human recovery rate indicates the proportion of infected human who get recovered of the disease and converted back into the susceptible subclass. Furthermore, it has been showen that for the human population, if the abovementioned recovery rates increase, then the number of susceptible human become exposed decrease. Similarly for the mosquito population, if the recovery rate increase, then the number of susceptible mosquitoes become exposed also decrease. As a consequence, disease will be vanished from population.