Time-Scale Analysis of Malaria Dynamics in Human-Mosquito Population

More realistic human-mosquito population mathematical model in which re-infected asymptomatic humans are considered is presented. Six possible time-scale of events for model transition from non-endemic to endemic state are analyzed. Results show that the buildup of the latent asymptomatic humans at steady state is the main dynamics of malaria in the endemic region. This become evident in the time scale of about 1-2 weeks and thus influences the mode of infection in the malaria transmission analysis.


Introduction
Malaria is a vector born disease that can be transmitted to humans through infected anopheles mosquito bites. The parasite's life cycle is well described in literatures [1,2]. Symptoms of the disease include vague, anemia, blood stools, convulsion, myalgia, diarrhea, nausea, vomiting and others [3]. The infection can also lead to damage of human organs such as the brain, lungs, kidneys and blood vessels [3,4]. Extensive research efforts have been focused to the study of the transmission of the malaria parasite between humans and mosquitoes and how to control the disease in the endemic regions [5][6][7]. Despite substantial progress, it is still a major global problem in terms of morbidity and mortality especially in the endemic regions. In 2015, the World health organization reported 214 million high risk malaria cases and over 438,000 deaths worldwide [8].
The complexity of the disease's zoonotic nature makes it clinically impossible to understand fully the mechanism in which the disease spreads. Clinically, two methods are in use: one which reduces the risk and intensity of malaria and death and another which uses protective antimalarial immunity to reduce the number of parasites in an infected individual. In both methods, the tendency for malaria patients to become resistant to these methods makes many malaria patients temporarily asymptomatic parasite carriage [9,10]. Although there is no standard definition for "asymptomatic" malaria infections, it is generally accepted to be malarial parasitemia of any density, in the absence of fever or other acute symptoms in individuals [7]. These infected individuals sustain malaria overtime and so are important source of malaria transmission and therefore a major obstacle to eradication programs. Ogutu et al. [11], reported in their that a large proportion of malaria infections are with microscopy-detection level as high as 39 percent on children under 10 years old in endemic regions. Based on findings they hypothesized without testing that a significant reduction of the malaria parasite pool could be obtained through the treatment of the asymptomatic class in endemic population.
Some recent papers have used mathematical modeling approach to provide more insight of the effects of asymptomatic parasite carriage on malaria transmission in human and mosquito populations [2,[12][13][14][15]. These models have played important roles in influencing intervention strategies for preventing and controlling the transmission of malaria. However, of these papers, only one [2] made an attempt to study the evolution of malaria based on the reproduction number ( 0 R ) and a reasonable timescale to demonstrate the existence and effects of the asymptomatic group in malaria dynamics. In their paper, Annan et al. analyzed the transition model by adducing sufficient conditions to show that malaria free state in which asymptomatic group is present is locally and asymptotically stable if 0 1 R < and unstable for 0 1 R > . In addition, they performed a preliminary time scale analysis to show that the asymptomatic individuals at steady state may be one of the main force in transmitting malaria in endemic regions. This paper explores this notion by providing detailed timescale analysis of the model formulated in [2]. The six main timescales predicted by the model are used with appropriate rescaling to explore the dynamics of malaria from early incidence (i.e. when a small infected mosquitoes are introduced into human-mosquito population) to endemic state. Finding the time to develop this asymptomatic status by malaria patients in an endemic state may be essential in planning control and eradication strategies.
We organize the rest of the paper as follows. Dimensionless model is described in section 2, Basic Reproduction Number in section 3 and Timescale Analysis in section 4. Finally, discussions and conclusion of the analysis are given in sections 5 and 6, respectively

Dimensionless Model
We adopt the dimensionless form of the model formulated in [2] as,  ε ≪ it means that asymptomatic humans remain infectious for a longer time compared to the latency period of humans. In the adapted model, the human population is divided into , and infectious ( m I ) compartments. Detailed model description including state variables, schematic flow diagram, and assumption are described in [2]. The rate at which infected mosquitoes bites is m eI , e is a constant value of the biting rate per human per unit time, h β is the number of human infections per bite, a β being the probability that a bite by a susceptible mosquito on an asymptomatic infectious human transfers the infection to the mosquito, and s β depicting the probability that a susceptible mosquito gets infected after biting a symptomatic infectious human.
We also state here that the susceptible mosquitoes are recruited into the mosquito population through a constant birth rate m from the In addition, those who survive receive treatment and are recovered with complete clearance to join the susceptible at a rate s s r I (treatment period 14-days), or only recover from symptoms (after a 3-day mono-therapy) without clearance to join the A class at a rate a s r I . The post symptomatic class A still carry merozoites and produce gametocytes. So, they can infect biting mosquitoes. Since patients can be in this state for several weeks or months they can play an important role in sustaining an epidemic. Thus, consider a putative treatment which removes individual from A and A L classes down to h S and h L , respectively, with the effect of the treatment parameter being

The Basic Reproduction Number
Since there are no trivial equilibrium points as long as and The NGM operator approach approximates the number of secondary infections produced by one infected individual and expresses 0 R as the product of the expected duration of the infectious period and the secondary rate infectious. When 0 1 R < , each infected individual produces on average less than one new infected individual so we would expect the disease to die out. On the other hand if 0 1 R > , each individual produces more than one new infected individual so we would expect the disease to spread in the population. This implies that the threshold quantity for eradicating the disease is to reduce 0 R to less than one. Using equation (9), the stability analysis of the equilibrium in the domain Ω is attained from the eigenvalues of the Jacobian matrix evaluated at equilibrium point detailed in [2]. The analysis revealed that the disease-free equilibrium is locally and globally stable if 0 1 R < and an endemic equilibrium is not feasible. However, since normally 0 1 R ≫ and the asymptomatic infectious humans to mosquitoes is significantly large, a possible treatment is to reduce the infectivity of the asymptomatic humans, , d and the symptomatic humans, b by increasing the parameters and θ λ .

⌢⌢
This introduces a reaction of infection from asymptomatic class in the susceptible mosquitoes into the susceptible human population. Thus, creating a stability between the amount of mosquitoes converting to the infectious class and the amount becoming infected by biting humans in the asymptomatic infectious class. However, there is a notable difference between and m m S L with an accelerated rate of mosquito infection from asymptomatic infectious humans when the initial conditions where we have set df δ βη = ⌢⌢ ⌢ ⌢ . If we allow Here, the asymptomatic humans become infected with new asexual parasites after contact with infectious mosquitoes and eventually reduce the number of the asymptomatic class ( ) A to the latent asymptomatic class, ( ) , thus an unstable steady state is Thus, when t t ε = ⌢ and using the following rescaling variables:

Discussions
We have analyzed six main time scales with appropriate rescaling to elucidate the dynamics of malaria disease as they evolve from small infection to endemic state. At 2 ( ) 1 3 days, t ε = Ο ≈ − introducing a small amount of infected mosquitoes into the system and biting susceptible humans causes human to get infected. The early infection indices itself into the population and grows linearly but their effect remains unnoticeable in the latent asymptomatic group. When 4 3 ( ) 7 8 days, t ε = Ο ≈ − the susceptible mosquitoes get infected by biting asymptomatic infectious humans. We noticed that the amount of mosquitoes converting to the infectious group is balanced by the amount of mosquitoes being infected by biting humans in the asymptomatic infectious group. This observation is expected because individuals with clinical malaria have low level of gametocytes. Thus, the early infection of susceptible mosquitoes is likely to come from contact with asymptomatic infectious humans since they have high gametocyte density. The contribution of asymptomatic infectious humans has a significant effect on the dynamics of the disease. Now, as more mosquitoes get infected through contact with asymptomatic infectious population, (i.e. when 5 4 ( ) 9 10 days t ε = Ο ≈ − ), the amount of susceptible mosquitoes become optimized and starts to decline. However the feedback from infectious humans offsets the linear growth of the initial amount of infected mosquitoes introduced. Thus, causing the amount of latent mosquitoes to grow exponentially. the asymptomatic humans become infected with new asexual parasite as a result of contact with infectious mosquitoes. The amount of infected humans to the asymptomatic status is being balanced by the amount of asymptomatic human to the latent asymptomatic group due to a boost in their partial immunity level. Thus, more mosquitoes become infected and the overall distribution of infection results in a fast shift of susceptible humans into the latent group. After this time the disease is now visible with infected humans equal to (1) Ο . Also, the latent asymptomatic humans may still be infectious to the mosquitoes as , , and h s S I A adjust to their equilibrium values and h L decaying exponentially. At approximately two months after the initial introduction of infected mosquitoes, all human classes equilibrate and latent individuals become symptomatic. The mosquito classes also adjust to assume their equilibrium state.

Conclusions
We completed a detailed six main timescale analysis on the model adapted from [2] with appropriate rescaling to elucidate the dynamics of malaria from small infection to endemic state. In our analysis, h S remain proportional to h L , thus showing that the level of the disease depends on the non-immune humans becoming infected. Thus, the buildup of the latent asymptomatic humans at steady state is the main dynamics of malaria in the endemic region. Also, we find that intervention programs may yield better result if implemented before the fourth timescale. During that time, the feedback from infection humans offsets the linear growth effect of the initial amount of infected mosquitoes and this equates to about 2-3 weeks from the initial infection. We also notice that the buildup of the latent asymptomatic humans at steady state is the main dynamics of malaria in the endemic region. This become evident in the time scale of about 1-2 weeks and thus influences the mode of infection in our analysis.