



ORIGINAL ARTICLE 

Year : 2021  Volume
: 5
 Issue : 1  Page : 713 

Analysis for malaria transmission dynamic between human and mosquito population, part II: Effective infection rate using new technique
S Saravana Kumar^{1}, L Maragatham^{1}, A Eswari^{2}
^{1} Department of Mathematics, Sri Ramakrishna Institute of Technology, Coimbatore, Tamil Nadu, India ^{2} Department of Physical Sciences and IT, AEC and RI, TNAU, Coimbatore, Tamil Nadu, India
Date of Submission  27Nov2020 
Date of Acceptance  07Dec2020 
Date of Web Publication  15Sep2021 
Correspondence Address: Prof. S Saravana Kumar Department of Mathematics, Sri Ramakrishna Institute of Technology, Coimbatore, Tamil Nadu India
Source of Support: None, Conflict of Interest: None
DOI: 10.4103/MTSP.MTSP_15_20
Objective : This article presents the seven equation SEIRSIR model for the dynamics of malaria parasite transmission in both mosquito and human. It defines the presence of area in which the model is epidemiologically feasible. Material and Methods: This paper is to find the approximate solution of the above models using qhomotopy analysis method. It is a flexible method that is used to solve a variety of differential equations. Results: Numerical simulations are carried out to confirm the analytic results and explore the possible behavior of the formulated model. Conclusions: The results of our study are that, Malaria can be controlled by reducing the rate of contact between humans and mosquitoes, the use of active malaria drugs, insecticides and mosquito nets treated with mosquitoes can also help reduce mosquito populations and malaria transmission respectively.
Keywords: Infection rate, malaria transmission, nonlinear model, qhomotopy analysis method, simulation
How to cite this article: Kumar S S, Maragatham L, Eswari A. Analysis for malaria transmission dynamic between human and mosquito population, part II: Effective infection rate using new technique. Matrix Sci Pharma 2021;5:713 
How to cite this URL: Kumar S S, Maragatham L, Eswari A. Analysis for malaria transmission dynamic between human and mosquito population, part II: Effective infection rate using new technique. Matrix Sci Pharma [serial online] 2021 [cited 2022 May 24];5:713. Available from: https://www.matrixscipharma.org/text.asp?2021/5/1/7/326041 
Introduction   
Malaria, an infectious disease caused by a virus contracted by the bite of a female mosquito called Anopheles mosquito, is one of the most prevalent vector transmitted diseases in the world. Given the decades of global efforts to eliminate the disease and to monitor reappear in areas where preventive measures have been successful and appears in epidemicfree zone.^{[1],[2],[3],[4],[5]} Mathematical models for malaria transmission dynamics are useful to provide a clearer view of the epidemic, to plan for the future, and to suggest effective control steps. Models played a major role in the growth of the disease epidemiology. The study on malaria using mathematical modeling originated from the works of a recent study.^{[6]} According to Ross, if the mosquito population can be reduced to below a certain threshold, then malaria can be eradicated. MacDonald did some modifications to the model and included superinfection.^{[7],[8]} He has shown that the number of mosquitoes has no impact on the malaria epidemiology in areas with severe transmission.
Based on previous studies add two classes of men in their mathematical models, namely those with low recovery rates (more infections, vulnerability is greater) and high recovery rate (less infection, less susceptibility).^{[8],[9]} Compartmental models of malaria and differential equations are designed to shape the disease.^{[8],[10],[11],[12],[13],[14]} Studies on malaria vectors with bifurcation. Forms of malaria transmission that integrate immunity in humans have been studied.^{[11],[12],[13],[14],[15]} Only epidemiological models were developed for the dissemination of antimalarial resistance.^{[15]} Anderson and May proposed a malaria model assuming that the immunity obtained in malaria is independent of the period of exposure.^{[16]} They also examined the different control measures and the role of transmission rates in the prevalence of disease. Ross and Macdonald's works have been expanded further by a research with the common generalized SEIR malaria model, which involves human and mosquito interactions.^{[10]} Recently, AlRahman et al. analyzed the mathematical model for malaria transmission using stability analysis.^{[17]} Currently, previous research has solved a nonlinear model in different mathematical areas using asymptotic method.^{[18],[19],[20],[21]}
Our goal in this work is to investigate and understand the effect of logistic growth model due to the transmission of malaria in a fully homogeneous population and to inspect the influence of the embedded parameters in the dynamic of the malaria disease model. In this paper, we present a model formulation of malaria transmission in part 2, where the general mathematical framework, notation and equations SEIR modelSEI analyzed. In section 3, Application of qHAM. In Part 4, we perform numerical simulation model with a graphic illustration. Part 5 consists of the results of the discussion and gives a closing speech in Part 6.
Mathematical Analysis of SeirSei Model   
The SEIRSEI Model for the transmission of malaria between humans and mosquitoes is relevant to this work,^{[17]} using a sevendimensional (ODE'S) Ordinary Differential Equations method to model the transmission of plasmodium falciparum malaria between humans and mosquitoes with nonlinear infection rate in the form of saturated incidence rates. This model is formulated for both human population and mosquito population at time t. We divide the human population into (4) four classes: Susceptible S_{H}, Exposed E_{H}, Infectious I_{H}, and Recovery Human R_{H}, and that of the population the mosquitoes is divided into three classes they are Susceptible S_{V}, Exposed E_{V}, and Infectious I_{V}, respectively. The compartmental model which shows the mode of transmission of malaria between the two interacting populations is depicted in the [Figure 1]. The above model equations are given by:  Figure 1: Schematic diagram of malaria transmission between humans and mosquitoes
Click here to view 
with
Where ∧_{H} denotes recruitment rate for humans, ∧_{V} is recruitment rate for mosquitoes, α_{1}_{H} is developing rate of exposed (humans) becoming infectious, α_{2}_{H} is recovery rate of humans (removal rate), μ_{H} is natural death rate for humans, δ is inducing death rate for humans, α_{1}_{V} is developing rate of exposed (mosquitoes) becoming infectious, μ_{V} is natural death rate for mosquito, and β_{H} and β_{V} are infection rate q_{H} × η_{V} of humans q_{V} × η_{V }and mosquitoes, respectively, where q_{H} and q_{V} are probability of transmission of infection from an infectious mosquito to a susceptible human and an infectious human to a susceptible mosquito, respectively. For qham solution, we choose the linear operator:^{[22],[23]}
Which satisfies the following conditions:
Where c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6} and c_{7} are integral constants. According to the zerothorder deformation equations and the mthorder deformation equations with the initial conditions,
Where
Now, the solution of system of equations (4)–(10) for m ≥ 1. We now successively obtain
These solutions (5)–(11) are used to construct the analytical solutions given by
Numerical Solution   
Here, we analyze the behavior of the system numerically using the surveillance estimate of susceptible, exposed, infectious, and recovered mosquito population, as shown in [Table 1], and by considering initial conditions, such as.^{[16],[17]}  Table 1: Parameters descriptions for SEIRSEI model (AlRahman et al., 2017)
Click here to view 
The numerical simulations are performed using MATLAB, and the results are shown in [Figure 2] to illustrate the behavior of the system with respect to the different values of the model parameters. It can be easily verified that the condition of nonlinear differential equation (1) could be solved using analytical method and compared with the help of function pdex in MATLAB. Satisfactory results are observed.  Figure 2(a,b,c,d): Comparison between analytical and Simulation result of the model (1), showing malaria transmission population against time for values of embedded parameters indicated in (Al Rahman et al., 2017)
Click here to view 
Discussion   
The analytical results show that in epidemic situation of model (1), both the human and mosquito populations will exist and get infected. These results are helpful in predicting malaria transmission and how to find an effective way of malaria prevention and control in the model. In [Figure 2], it is inferred that the susceptible human and mosquito population increases as time increases and its carrying capacity remains stable, while the exposed and infected human and mosquito population decreases over time. In reality, the plot reveals that the susceptible human populations continue to grow, which means that the disease will remain endemic throughout the population.
Conclusions   
In this paper, we have formulated and analyzed a compartmental malaria transmission model for nonlinear infection rates in human and mosquito populations. The nonlinear model is analyzed using the asymptotic tool qHAM. The human population has been divided into four compartments: susceptible, exposed, infectious, and recovered, while the mosquito population has been divided into three compartments: susceptible, exposed, and infectious, because the mosquitoes remain lifeinfectious and have not recovered.The conditions for the system of analytical solution have been determined using qhomotopy analysis method, and these conditions are further justified numerically by considering a given set of parameter values, see Appendix 1: Basic idea of the qhomotopy analysis of the problem. Infection rates are not linear in the model formulated shows the incidence rate is not saturated like ordinary mass action and the common occurrence. We established a region where the model is epidemiologically feasible and mathematically wellposed.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
Appendix 1: Basic idea of the qhomotopy analysis of the problem
Considering the following system of differential equation of the form
Where N_{i} = (i = 1, 2, 3, ...n) are nonlinear operators, t is independent variable, f_{i} (t) (i = 1, 2, 3, ...n) are
Known functions and x_{i} (t) (i = 1, 2, 3, ...n) are an unknown function. Note that, not necessary in the nonlinear operator N_{i} contains nonlinear term. Let us construct the socalled zerothorder deformation equation
Where denotes the socalled embedding operator, L is an auxiliary linear operator with the property L[f] = 0 when f = 0, h ≠ 0 is an auxiliary parameter, x_{i,0} (t) are initial guesses of x_{i} (t) and H_{i} (t) denotes a nonzero auxiliary function. It is obvious that when q = 0 and equation (A2) becomes
respectively. Thus, as q increases from 0 to the solution varies from the initial guess x_{i}_{,0} (t) to the solution x_{i} (t). Having the freedom to choose x_{i}_{,0} (t), L, h, H (t), we can assume that all of them can be properly chosen so that the solution of equation (A2) exists for
Expanding in Taylor series with respect to q,one has
Where
Assume that h, H(t),u_{i,0}(t),L are properly chosen such that the series (A4) converges at and:
Define the vector
Differentiating equation (A2)m times with respect to the embedding operator q and then setting q = 0 and finally dividing them by m!, we have the socalled mthorder deformation equation
Where
and
It should be emphasized that is x_{i,m} (t) m ≥ 1 governed by the linear equation (A7) with the linear boundary conditions that come from the original problem. Due to the existence of the factor ,more chances for convergence may occur or even much faster convergence can be obtained better than standard HAM. It should be noted that in the case of n = 1 in equation (A2), standard HAM can be reached.
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[Figure 1], [Figure 2]
[Table 1]
