1. Malaria Model with Periodic Mosquito Birth Rate *Bassidy Dembele, ** Avner Friedman, * Abdul-Aziz Yakubu *Department of Mathematics, Howard University, Washington, D.C **Mathematical Biosciences Institute, The Ohio State University, OHIO Abstract In this study, we introduce a model of malaria, a disease that involves a complex life cycle of parasites, requiring both human and mosquito hosts. The novelty of the model is the introduction of a periodic coefficient into the system of ODEs, which accounts for the seasonal variations (wet and dry seasons) in the mosquito birth rate. We define a basic reproduction number R0 which depends on the periodic coefficient and prove that if R0 1 then the disease is endemic and may even be periodic. 1. Introduction Malaria is one the most devastating diseases and one of the leading causes of death in the tropical regions of the world. Worldwide, there are more than 500 million clinical cases of malaria a year and almost 2.7 million deaths, a large percentage of them children, according to the 2006 UNICEF report on malaria. The symptoms of the disease are fever, chills, sweats, headache, nauseas and vomiting, body aches, and general malaise. If not treated by drugs, the symptoms become more severe, as blood cells are continuously being destroyed, and may lead to death. Malaria is spread in three ways. The most common is by the bite of an infected female Anopheles mosquito. However, malaria can also be spread through a transfusion of infected blood or by sharing a needle with an infected person. In this project, we focus on infection via human-mosquito interaction. Malaria is caused by a parasite called plasmodium sporozoite. The parasite has a complex life cycle that requires both a human host and an insect host namely, female anopheline mosquito. Humans can only be infected by bites from infected mosquitoes and Mosquitoes can only be infected by biting infected humans. Animals can also get the malaria disease. However, animal malaria cannot spread to humans, and human malaria cannot spread to animals. The human-mosquito interaction can be described by the following diagram 2. Malaria Model with a Periodic Mosquito Birth Rate S h = the proportion of susceptible individuals (humans); I h = the proportion of infected individuals; R h = the proportion of recovered individuals; S m = the proportion of susceptible mosquitoes; I m = the proportion of infected mosquitoes. By definition S h + I h + R h = 1 and S m + I m = 1. The susceptible individuals are those who are not sick but they can become infected through bites from infected mosquitoes; The infected individuals are those who were bitten by infected mosquitoes and show symptoms of the disease; The recovered individuals are those who have been treated by ACT; The susceptible mosquitoes are those who do not carry any parasite in their salivary glands; The infected mosquitoes are those who carry a Multitude of parasites in their salivary glands and are thus able to infect humans. Our malaria model with periodic mosquito birth rate is given by the following system of equations: dSh/dt = λh (Ih + Rh) + βhRh − ρShIm, (2.1) dIh/dt = ρShIm – (λh +αh)Ih, (2.2) dRh/dt = αhIh – (λh+ βh)Rh, (2.3) dIm/dt = αIhSm – λ(t)Im, (2.4) dSm/dt = λ(t)Im– αIhSm, (2.5) The parameter βh represents the rate of loss of immunity. The parameter λh represents birth rate; children of infected and of recovered individuals are born as susceptible individuals, and thus there is a loss of the infected and recovered populations at rate λh. On the other hand, for susceptibles, it is assumed that birth rate does not reduce or increase their relative proportions. Newborn mosquitoes are assumed to be susceptible and thus new births decreases Im at a rate λ(t) and increases Sm at a rate λ(t). Let ParameterDescription α hm infectivity of the human (the probability that a bite by a susceptible mosquito on an infected human will transfer the infection to the mosquito) α mh infectivity of the mosquito ( the probability that a bite by an infected mosquito of a susceptible human will transfer the infection to the human) λhλh the birth-rate for humans λ(t)mosquito birth rate, λ(t+T)= λ(t) βhβh loss of immunity rate αhαh human recovery rate a=N m /N h the ratio of total population of mosquitoes to humans bmbm the man-biting rate of the mosquitoes (the average number of bites given to humans by each mosquito per unit time) ρ=ab m α mh infectious rate of humans by infected mosquitoes, relative to population of human-mosquito α=b m α hm infectious rate of mosquitoes by a susceptible mosquito biting of infected humans Table 1: Parameters for the Malaria model Note that the average infectious period of a single person is 1/(λ h +α h ). Also the average infectious period of a single mosquito is Hence, R 0 may be viewed as the average value of the expected number of secondary cases produced by a single infected individual entering the population at the DFE. R 0 is called the basic reproduction number. 3. Results Theorem 1. If R 0 < 1 then the disease free equilibrium point, DFE = (1, 0, 0), is globally asymptotically stable. Theorem 2. If R0 > 1 then there exist numbers δ0 > 0 and δ1 > 0 such that for any initial values Ih(0), Im(0) there is a time Ť = Ť(Ih(0), Im(0) ) such that Ih(t)>δ0 and Im(t) > δ1 for all t  Ť(Ih(0), Im(0) ). Theorem 3. If R 0 > 1 then there exists at least one T-periodic solution of the system (2.1)-(2.5); that is, R 0 > 1 implies the persistence of the infective human and mosquito populations on periodic solutions. 4. Examples Figures 2. The parameter values are: λ h = 0.5, β h = 0.4, α h =0.5, μ = 0.6, α= 0.4, δ= 0.3, ω = 2π, and ρ = μ(λ h + α h )/ α − 0.3 = 1.2. The disease goes extinct ( R 0 < 1). Figures 3. The parameter values are: λ h = 0.5, β h = 0.4, α h =0.5, μ = 0.6, α= 0.4, δ= 0.3, ω = 2π, and ρ = μ(λ h + α h )/ α + 0.3 = 1.8. The disease persists ( R 0 > 1). 5. Conclusions In this paper, we studied a mathematical model of malaria consisting of a system of ODEs. The novelty of the model is in the introduction of a time – dependent periodic coefficient which accounts for the seasonal variation (wet and dry seasons) in the birth rate of the mosquito population. We defined the basic reproduction number R 0 which naturally depends on this periodic coefficient, and proved by a rigorous mathematical analysis that if R 0 1 then the disease remains endemic. 6. Future Work It will be interesting to test our results with the real population data. On the other hand, it will also be interesting to take account of spatial variations in the mosquito population, as well as their migration from one location to another. This material is based upon work supported by the National Science Foundation under Agreement No. 0112050 and by The Ohio State University.