DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018113 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 9, November 2018 pp. 3837{3853 THE IMPACT OF RELEASING STERILE MOSQUITOES ON MALARIA TRANSMISSION Hongyan Yin School of Mathematics and Statistics Central China Normal University Wuhan 430079, China School of Mathematics and Statistics South-Central University for Nationalities Wuhan 430074, China Cuihong Yang∗ and Xin'an Zhang School of Mathematics and Statistics Central China Normal University Wuhan 430079, China Jia Li Department of Mathematical Science University of Alabama in Huntsville Huntsville AL 35899, USA (Communicated by Yuan Lou) Abstract. The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing the malaria transmission. To study the impact of releasing sterile mosquitoes on the malaria transmission, we first formulate a simple SEIR (susceptible-exposed-infected-recovered) malaria transmission model as our baseline model, derive a formula for the reproductive number of infection, and determine the existence of endemic equilibria. We then include sterile mosquitoes in the baseline model and consider the case of constant releases of sterile mosquitoes. We examine how the releases affect the reproductive numbers and endemic equilibria for the model with interactive mosquitoes and investigate the impact of releasing sterile mosquitoes on the malaria transmis- sion. 1. Introduction. Mosquito-borne diseases, such as malaria, transmitted between humans by mosquitoes, are big concerns for the public health. Malaria is the fifth cause of death from infectious diseases worldwide (after respiratory infections, HIV/AIDS, diarrheal diseases, and tuberculosis), and the second leading cause of death from infectious diseases in Africa after HIV/AIDS. About three billion and four thousand million people (half of the world's population) are at risk of malaria transmission living in 106 countries and territories.There were 207 million cases of malaria and 627,000 deaths in 2012, mostly young children in sub-Saharan Africa. 2010 Mathematics Subject Classification. Primary: 92D25, 92D30; Secondary: 34C60. Key words and phrases. Malaria, mathematical modeling, sterile mosquitoes, reproductive number, endemic equilibrium. ∗ Corresponding author: Cuihong Yang. 3837 3838 HONGYAN YIN, CUIHONG YANG, XIN'AN ZHANG AND JIA LI In 2015, an estimated 214 million cases of malaria occurred worldwide and 438,000 people died due to malaria infection [20]. Between 1957 and 2015, 63 outbreaks of locally transmitted mosquito-borne malaria have occurred, and about 1,700 cases of malaria are diagnosed each year in the United States [7]. Malaria is not spread from a human to a human as influenza or HIV, but trans- mitted between humans by blood-feeding mosquitoes. No vaccines are available and an effective way to prevent malaria is to control mosquitoes. Among the mosquitoes control measures, the sterile insect technique (SIT) has been proved to be useful and effective [3]. The SIT is a method of biological control in which the natural reproductive process of mosquitoes is disrupted. Utilizing radical or other chemical or physical methods, male mosquitoes are genetically modified to be sterile so that they are incapable of producing offspring despite being sexually active. These sterile male mosquitoes are then released into the environment to mate with the present wild female mosquitoes. A wild female that mates with a sterile male will either not reproduce, or produce eggs, but the eggs will not hatch. Repeated releases of genetically modified mosquitoes or releasing a significantly large number of sterile mosquitoes may eventually wipe out a wild mosquito population, although it is often more realistic to consider controlling the population rather than eradicating it [1,4, 21]. While the new development of the SIT has been promising, particularly in lab- oratories, many questions have to be answered before the techniques can be im- plemented and the sterile mosquitoes can be deployed in the field. Such questions include the investigation and assessment of the impact of releasing sterile mosquitoes on the malaria transmission. Mathematical models have proven useful in population dynamics and epidemi- ology. To gain insight into such challenging questions, mathematical models have been formulated for the interactive dynamics of the wild and sterile mosquitoes and the dynamical features of these models have been well studied [3,6,8,9, 14, 15, 19]. However, the impact of releases of sterile mosquitoes on the malaria transmission is to be studied. In this paper, we focus on the impact of releases of sterile mosquitoes on the malaria transmission. We first formulate a simple SEIR malaria transmission model as our baseline model in Section2. We derive formulas of the reproductive num- ber for the baseline model and prove that there exists an endemic equilibrium as the reproductive number exceeds one. We then include the sterile mosquitoes in the baseline malaria model such that the susceptible mosquitoes consist of the wild and sterile mosquitoes in Section3. We consider the case where the releasing rate of sterile mosquitoes is constant and derive a formula for the reproductive number, and explore the existence of an endemic equilibrium for the extended disease model. We investigate the impact of the releases of sterile mosquitoes on the malaria trans- mission based on the reproductive number and the endemic equilibrium in Section 4. Numerical examples are also provided to illustrate our findings. We finally give brief discussions in Section5. 2. Baseline model for malaria transmission. Before studying the impact of the interactive wild and sterile mosquitoes on the malaria transmission, we consider the following simple SEIR malaria model as our baseline model [12, 13]. We divide the human population into groups of susceptible, latent or incubating, infective, and recovered individuals. Here the latent or incubating period is defined IMPACT OF STERILE MOSQUITOES ON MALARIA TRANSMISSION 3839 as the time from initial infection to the appearance of gametocytes in the blood [2]. Using index h for the humans, we let Sh be the number of susceptible humans, Eh the number of latent or incubating humans who are infected but not infectious yet, Ih the number of infective humans who are infected and also infectious, and Rh the number of the humans who are recovered from infection but partly lose their immunity [11, 16{18]. The model equations for humans are given by dS h = Λ − (µ + λ ) S + θ R ; dt h h h h h h dEh = λhSh − (µh + γh) Eh; dt (1) dI h = γ E − (µ + δ + η ) I ; dt h h h h h h dR h = η I − (µ + θ ) R ; dt h h h h h where Λh is the input flow of the susceptible humans, µh and δh are the natural and disease-induced death rates for humans, respectively, θh is the rate of immunity loss, γh is the developing rate of incubating humans to become infectious, such that 1/γh is the incubation period, ηh is the recovery rate for infectious humans, and λh is the infection rate or the incidence of infection from an infective mosquito to a susceptible human. Notice that if there is no infection, the human population has an asymptotically stable steady state lim Sh = Λh/µh. t!1 For the mosquito population, we divide it into groups of susceptible, latent, and infective individuals. Since the mosquito lifespan is usually shorter than their infective period, no recovered mosquitoes are included. We denote the numbers of susceptible, latent, and infective mosquitoes by Sv, Ev, and Iv, respectively. We further follow the line in [15] and assume, in the absence of infection, that the density dependence occurs in the larvae stages progression, and density independent death rate for wild mosquitoes. Then model equations for the mosquitoes are given by dS v = C(N )α (1 − ξ N )N − (µ + λ )S ; dt v v v v v v v v dE v = λ S − (µ + γ )E ; (2) dt v v v v v dI v = γ E − µ I ; dt v v v v where C(Nv) is the mating rate, αv is the maximum number of offspring reproduced per individual, the term 1 − ξvNv characterizes the density-dependent larvae stages progression with ξv being an early-stage carrying capacity parameter, Nv = Sv + Ev + Iv is the total mosquito population size, µv is the natural death rate of the mosquitoes, γv is the rate of incubating mosquitoes becoming infectious or 1/γv is the extrinsic incubation period of the parasite within the mosquito or the period of sporogony, and λv is the infection rate from an infective human to a susceptible mosquito. As in [15], we assume that the intrinsic growth rate of the wild mosquito population r0 := αv − µv > 0 such that, in the absence of sterile mosquitoes, the wild mosquito population approaches a positive steady state. 3840 HONGYAN YIN, CUIHONG YANG, XIN'AN ZHANG AND JIA LI Let Nh = Sh + Eh + Ih + Rh be the total human population size. The infection rates are given by Ih λv = rβh ; (3) Nh and Nv Iv Iv λh = βvr = rβv ; (4) Nh Nv Nh where r is the number of bites on humans by a single mosquito per unit of time, βh is the transmission probability per bite to a susceptible mosquito from an infective human, and βv is the transmission probability per bite from an infective mosquito to a susceptible human. Suppose that mosquitoes have no difficulty to find their mates such that the mating rate is constant denoted by C(Nv) := c. Then system (2) becomes dS v = α (1 − ξ N )N − (µ + λ )S ; dt v v v v v v v dE v = λ S − (µ + γ )E ; (5) dt v v v v v dI v = γ E − µ I ; dt v v v v where we merge c and αv but still write it as αv.