A NOTE ON STABILITY ANALYSIS OF COMPARTMENTAL MATHEMATICAL MODEL FOR THE SPREAD OF MALARIA

A. Ali 1 , R. Hussain 1 and M. Anayat 2 . 1. Mirpur University of Science and Technology (MUST), Mirpur-10250(AJK ), Pakistan. 2. International Islamic University Islamabad Pakistan. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

In this study, we present a compartmental model for the spread of malaria in a population where group of individuals were vaccinated. The purpose of this paper is to analyze the transmission dynamics of Malaria by using the compartmental model, including ordinary differential equations for human host and mosquito vector populations. A parallel system is obtained, which has two equilibriums: a diseasefree equilibrium and an endemic equilibrium. The stability of the equilibrium points is verified by the basic reproduction number 0 . Asymptotically stable solution is obtained only for disease-free equilibrium and results are presented graphically.

…………………………………………………………………………………………………….... Introduction:-
Malaria is most common in Africa. The disease prevails in warmer climates and in areas where there is abundance of humidity and rain. According to latest WHO estimates released in December 2015, there were 214 million cases of malaria in 2015 and 438,000 deaths. At present five known species of plasmodium exists which are; P falciparum, P vivax, P ovale, P malariae and P knowlesi among them P. falciparum is responsible for most fatal infection and is widespread in the tropics. Clinical symptoms of malaria include: high fever, muscle and joint aches, headache, vomiting, chills, sweating and anemia etc. Young children and travellers (who do not have developed immunity) are at high risk of malaria. In highly endemic areas people develop a degree of acquired immunity which is boosted-up by new infections. Immunity lasts for a certain period in the absence of new infection. The duration of infection depends on many factors such as degree of infection, method and time of treatment, resistivity of parasite to drugs and biology of host.
Mathematical modeling has been frequently used in epidemiology and several other fields. Epidemiological modeling is very helpful in identifying important model parameters and suggests improvements in models for future predictions.
We modified the model of NidhiNirwani, R. Khandelwal and V.H. Badshah by performing the stability analysis. The purpose of this study is better understanding the dynamics of malaria.
2410 Description of Model:-Let N H and N V denote human and mosquito population respectively with total population at time t. We assume that human and mosquito population has constant size with birth and death rates equal to µ H and µ V .
The human population of size N H is composed of Susceptible S H , Infective I H and Recovered R H whereas vector population consists of Susceptible S V and Infective I V .
We say group of people ρ ,0 ≤ ≥ 1 of newborns host population be vaccinated. Assume that vaccine is not perfect and let the effectiveness of vaccine is s, then µ (1 − ) newborns remain susceptible, and µ directly being removed to . The governing equations are: Human population:- The human population of size N H is formed of susceptible S H , infective I H and recovered R H where vector population is composed of S H and I H . The average infection rate on man by single mosquito. b The proportion of bites on man that produce an infection. c The probability that a mosquito becomes infectious. ϒ The per capita rate of loss of immunity in human hosts. r The rate at which human hosts acquire immunity. ν The rate of recovery of human hosts from the disease.

Conclusion
In this paper we present compartmental model for spread of malaria. An identical system is obtained which has two equilibrium points; a disease-free equilibrium and an endemic equilibrium. We discussed the stability of the diseasefree equilibrium point numerically by computing Eigen-values and by using basic reproduction number. Further we perform stability analysis and graphically verify our results. All the graph indicate that the system is stable.

Future Work
The model presented here is multi-dimensional. In future we can perform stability analysis for endemic-equilibrium. Furthermore we can establish these results by using any of the numerical methods. Also we can work on its sensitivity analysis and parameter estimation.