Lie Symmetry Analysis of a Nonlinear System Characterizing Endemic Malaria

: In this paper, the integrability of a nonlinear system developing endemic Malaria was demonstrated using Prelle–Singer techniques. In addition, Lie symmetry techniques were employed to identify additional independent variables that led to the modiﬁcation of the nonlinear model and the development of analytical solutions.


Introduction
Malaria is a parasitic disease spread by female Anopheles mosquito bites that is induced by the Plasmodium parasite [1]. It is still one of the most common and deadly human illnesses on the planet. Furthermore, clinical characteristics include the likelihood of infection, severity, and relapse risk. P. falciparum has been identified as the most dangerous of all the species to humans [2]. Malaria-infected areas are home to roughly 40% of the world's population. However, the majority of cases and deaths occur in Sub-Saharan Africa. Every year, 300 to 500 million cases and 1.5 to 2.7 million deaths are estimated to occur over the world. Africa is responsible for 80% of the cases and 90% of the deaths.
In a paper modeling the transmission dynamics of malaria endemic, researchers used rescaling to achieve a cosmetic simplification in order to predict disease propagation [1]. As a result of this scaling, the original five-dimensional system of first-order ordinary differential equations was reduced to the three first-order equations shown below where s h , i h , and i v are rescaled variables that indicate the number of susceptible, infected humans, and infected mosquitoes, respectively, at a given point in time. Nondimensional parameters are described as follows with β h , the rate of human contact with mosquitoes; α h , the rate of human natural death per capita; ρ h , the human disease-induced death rate per capita; γ h , the humans' per capita recovery rate; α v , the natural death rate of mosquitos per capita; β v , the frequency of mosquito contact with humans; µ h , the human population's per capita birth rate; µ v , the mosquitoes' per capita birth rate.
Over the last 40 years, a variety of techniques, including numerical and stability techniques, analytical techniques, approximation techniques, and others, have been utilized to explore and solve nonlinear systems of differential equations. Another major technique, Lie Theory of Symmetry Groups, was utilized to analyze nonlinear differential equations near the end of the twentieth century. Marius Sophus Lie, a talented Norwegian mathematician who lived near the end of the nineteenth century, established the Lie's theory of symmetry groups. Sophus Lie used symmetry groups theory to solve differential equations. He combined all differential equation approaches and deduced that his Lie group theory could account for them all. Lie groups are mathematical objects that represent the properties of groups as defined by group theory. To produce Lie symmetries, the Lie group theory employs appropriate transformations of independent and dependent variables.
The goal of this study, however, is to look at the integrability of a nonlinear system (1). In addition, to employ the modified Prelle-Singer (PS) approach of Chandrasekar et al. [3] to uncover transformations that lead to model linearization. Furthermore, the Lie symmetry technique was used to determine the model's explicit solutions.
This study is organised as follows. Section 2 introduces a heuristic background of the concepts underlying the Prelle-Singer (PS) procedure and Lie symmetry analysis. In Section 3, we used the PS procedure to solve the determining equations of the nonlinear system. In Section 4, we used a Lie symmetry method on the reduced equations to obtain explicit solutions. Section 5 contains the conclusion.

Theorems and Fundamental Concepts
This Section provides a comprehensive review of the Prelle-Singer (PS) procedure and Lie symmetry analysis approaches to solving differential equations. The theory includes the tools that will be used in the following sections of the paper. In [4,5], Matadi provided a fundamental definition and theorems that can be found in the literature (see [6,7]).

The Prelle-Singer (PS) procedure
In [3], Chandrasekar et al. updated the original Prelle-Singer (PS) technique and used it to solve autonomous and non-autonomous nonlinear systems of ordinary differential equations (ODEs) in the following way: Given a three-dimensional system of nonlinear first-order ordinary differential equations [3] Equation (3) can be written as follows: By multiplying the first, second, and third equations in (5) by the functions P, L, and Q we get The following equations are obtained by Comparing Equations (6) and (4) The resulting determining equations for the integrating factors P, L, and Q are derived from the compatibility criteria between Equation (7) with the given condition Integrating Equation (7) produces the given integral of motion with

Lie Symmetry Procedure
In accordance with the theory of Lie symmetry, the given three dimensional system of first-order differential equation [5] x admits the following Lie group of transformations of one-parameter (a) [5] with infinitesimal Lie operators [5] The group transformationst,x 1 ,x 2 andx 3 are obtained by solving the following Lie equations [4,5] with the initial conditions: The first extension of Lie operators above is defined as follows [5] G [1] where with D t representing the total differential operator describe as follows The infinitesimals transformation obtained will be used to solve the following equation [5] Tr t + X 1 r x 1 + X 2 r x 2 + +X 3 r x 3 = 0, Equation (14) will provide a set of new independent variable, r, and dependent variables, u, v and w, which can be used to transform the nonlinear system (1) to a linear system. The following section explore the existence of integrals to the nonlinear system (1).

Application of (PS) Procedure to Nonlinear System (1)
Considering the three-dimensional equation (1) Case 1: I s h = 0 and I t , The substitution of Equation (15) into (9) gives From Equation (16) we have, δ = 0 or β = 0 or α = 0. The determining equation for Q in (9) becomes in which we have taken Q s h = 0 (since I s h = 0). A simple solution for (17) is Q = −i v i h with γ = θ. Using the restriction δ = β = α = 0 and γ = θ, the solution of the determining equation for P, L is given by P = i h , L = 0. Hence, from Equation (11), we obtain therefore, the integral of motion is given by Case 2: I i h = 0, I t , I s h , I i v = 0 and I i v = 0, I t , I s h , I i h = 0 According to [3], the determining equations and conditions is obtained by introducing the following transformation P = SQ and L = UQ, with

Lie Symmetry of one Dimensional Second-Order Differential Equation
From the first equation of the nonlinear system (1), we obtain differentiating Equation (22) with respect to t, we obtaiṅ the substitution of Equation (23) into the second equation of the nonlinear system (1) giveṡ Lie group analysis to (24) is performed using SYM packages [4][5][6], yielding the following cases Case 1: β = 0 and α + γ = 0 The determining equations for the classical symmetries of the nonlinear Equation (24) are The coefficients of the infinitesimal generator are obtained by solving the overdetermining Equation (25) as a result, the two-dimensional Lie algebra is given by   (1) gives with G = F + Lδ, H = E + Kδ, E = θ A, F = θB, L = βd, K = βc Hence, the number of infected mosquitoes is given by Case 2: β = 0 and α + γ = 0 The determining equations for the classical symmetries of the nonlinear Equation (24) are given by from the above overdetermining equation yields the coefficients of the infinitesimal generator As a result, we have the two-dimensional Lie algebra shown below Solving the nonlinear Equation (24) for case 1, we obtain the number of susceptible humans The substitution of (43) and (44) gives the first-order nonlinear first-order ordinary differential equation with The solution to Equation (45) is a hypergeometric function, and the numerical solution can be found in [8].

Lie Symmetry of Three Dimensional System of First-Order Differential Equation
Equations (12) and (13) are applied to the Non-dimensional model Equation (1), yielding the following: The extended infinitesimal transformation is obtained by using Equation (13) with In general, solving nonlinear system (48) is challenging. As a result, it is required to use special solutions [5]. In the case of T = T (t), S = S(s h ), I 1 = I 1 (i h ), I 2 = I 2 (i v ), the non-linear Equations (48) are simplified as The following second-order partial differential equation is obtained by considering the partial derivative of Equation (51) with regard to τ [5] T [tt] = 0, solving Equation (52), we obtain T (t) = lt + m, with l and m being the integration constants. Substituting Equation (52) into (49)-(51) yields After twice partially differentiating Equation (54) with respect to i h , we obtain Hence, Substituting Equation (56) into (53), we obtain As Equation (57) is dependent on the values of s h and i h , we obtain Hence, From (55), we obtain Hence, As a result, the infinitesimal transformations are as follows: S(s h ) = c 1 exp [(βq − α)s h ] + l −βq + α , It is worth noting that these infinitesimal transformations are not unique. There is, however, an infinite number of infinitesimal transformations [5]. As a result, Equation (12) becomes