Lie Symmetry Analysis of a Nonlinear System Characterizing Endemic Malaria

Abstract

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.

1. 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

$$\begin{array}{ccc}\hfill \frac{d{s}_h}{dt}& =& 1-\beta s_h{i}_h-\alpha s_h\hfill \\ \hfill \frac{d{i}_h}{dt}& =& \beta s_h{i}_h-(\alpha +\gamma )i_h\hfill \\ \hfill ϵ\frac{d{i}_v}{dt}& =& \theta (1-i_v)i_h-\delta i_v\hfill \end{array}$$

(1)

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

$$\beta =\frac{{\beta }_h{N}_v}{{\mu }_h},\alpha =\frac{{\alpha }_h}{{\mu }_h},\gamma =\frac{{\rho }_h+{\gamma }_h}{{\mu }_h},ϵ=\frac{{\alpha }_v}{{\mu }_h},\delta =\frac{{\mu }_v}{{\mu }_h},\theta =\frac{{\beta }_v{N}_h}{{\mu }_v}$$

(2)

with ${\beta }_h$, the rate of human contact with mosquitoes; ${\alpha }_h$, the rate of human natural death per capita; ${\rho }_h$, the human disease-induced death rate per capita; ${\gamma }_h$, the humans’ per capita recovery rate; ${\alpha }_v$, the natural death rate of mosquitos per capita; ${\beta }_v$, the frequency of mosquito contact with humans; ${\mu }_h$, the human population’s per capita birth rate; ${\mu }_v$, 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.

2. 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]).

2.1. 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]

$$\begin{array}{ccc}\hfill \frac{d{x}_1}{dt}& =& \frac{M_1(t,x_1,x_2,x_3)}{N_1(t,x_1,x_2,x_3)}\hfill \\ \hfill \frac{d{x}_2}{dt}& =& \frac{M_2(t,x_1,x_2,x_3)}{N_2(t,x_1,x_2,x_3)},\hfill \\ \hfill \frac{d{x}_3}{dt}& =& \frac{M_3(t,x_1,x_2,x_3)}{N_3(t,x_1,x_2,x_3)},\hfill \end{array}$$

(3)

Given $x_1,x_2,x_3$ with $M_i$’s and $N_i$’s, $i=1,2,3$ analytic functions. Equation (3) admits a first integral $I(t,x_1,x_2,x_3)=K$, on the solutions, with K constant, resulting in a total differential of

$$dI=I_{t}dt+I_{x_1}d{x}_1+I_{x_2}d{x}_2+I_{x_3}d{x}_3=0$$

(4)

Equation (3) can be written as follows:

$$\begin{array}{ccc}\hfill \frac{M_1}{N_1}dt-d{x}_1& =& 0,\hfill \\ \hfill \frac{M_2}{N_2}dt-d{x}_2& =& 0,\hfill \\ \hfill \frac{M_3}{N_3}dt-d{x}_3& =& 0.\hfill \end{array}$$

(5)

By multiplying the first, second, and third equations in (5) by the functions P, L, and Q we get

$$dI=(P{\varphi }_1+K{\varphi }_2+Q{\varphi }_3)dt-Pd{x}_1-Ld{x}_2-Qd{x}_3=0,$$

(6)

where ${\varphi }_i=\frac{M_i}{N_i},i=1,2,3$. The following equations are obtained by Comparing Equations (6) and (4)

$$\begin{array}{ccc}\hfill I_t& =& (P{\varphi }_1+L{\varphi }_2+Q{\varphi }_3),\hfill \\ \hfill I_{x_1}& =& -P,\hfill \\ \hfill I_{x_2}& =& -L\hfill \\ \hfill I_{x_3}& =& -Q.\hfill \end{array}$$

(7)

The resulting determining equations for the integrating factors P, L, and Q are derived from the compatibility criteria between Equation (7)

$$\begin{array}{ccc}\hfill P_t+{\varphi }_1{P}_{x_1}+{\varphi }_2{P}_{x_2}+{\varphi }_3{P}_{x_3}& =& -(P{\varphi }_{1x_1}+L{\varphi }_{2x_1}+Q{\varphi }_{3x_1}),\hfill \\ \hfill L_t+{\varphi }_1{L}_{x_1}+{\varphi }_2{L}_{x_2}+{\varphi }_3{L}_{x_3}& =& -(P{\varphi }_{1x_2}+L{\varphi }_{2x_2}+Q{\varphi }_{3x_2}),\hfill \\ \hfill Q_t+{\varphi }_1{Q}_{x_1}+{\varphi }_2{Q}_{x_2}+{\varphi }_3{Q}_{x_3}& =& -(P{\varphi }_{1x_3}+L{\varphi }_{2x_3}+Q{\varphi }_{3x_3}),\hfill \\ \hfill P_{x_2}& =& L_{x_1},\hfill \\ \hfill P_{x_3}& =& Q_{x_1},\hfill \\ \hfill L_{x_3}& =& Q_{x_2}.\hfill \end{array}$$

(8)

with the given condition

$$\begin{array}{c}\hfill {\varphi }_{3x_1}\left({\varphi }_{2t}+{\varphi }_2{\varphi }_{2x_1{x}_2}+{\varphi }_3{\varphi }_{2x_1{x}_3}-{\varphi }_{2x_3}{\varphi }_{3x_1}-{\varphi }_{2x_1}{\varphi }_{2x_2}\right)\\ \hfill -{\varphi }_{2x_1}\left({\varphi }_{3t}+{\varphi }_2{\varphi }_{3x_1{x}_2}+{\varphi }_3{\varphi }_{3x_1{x}_3}-{\varphi }_{2x_1}{\varphi }_{3x_2}-{\varphi }_{3x_1}{\varphi }_{3x_3}\right)=0\end{array}$$

(9)

Integrating Equation (7) produces the given integral of motion

$$I=r_1+r_2+r_3-\int [Q+\frac{d}{d{x}_3}(r_1+r_2+r_3)]d{x}_3,$$

(10)

with

$$\begin{array}{ccc}\hfill r_1& =& \int (P{\varphi }_1+L{\varphi }_2+Q{\varphi }_3)dt\hfill \\ \hfill r_2& =& -\int (P+\frac{d{r}_1}{d{x}_1})d{x}_1\hfill \\ \hfill r_3& =& -\int (L+\frac{d(r_1+r_2)}{d{x}_2})d{x}_2\hfill \end{array}$$

(11)

2.2. Lie Symmetry Procedure

In accordance with the theory of Lie symmetry, the given three dimensional system of first-order differential equation [5]

$$\begin{array}{ccc}\hfill \dot{x_1}& =& f_1(t,x_1,x_2,x_3),\hfill \\ \hfill \dot{x_2}& =& f_2(t,x_1,x_2,x_3),\hfill \\ \hfill \dot{x_3}& =& f_3(t,x_1,x_2,x_3),\hfill \end{array}$$

admits the following Lie group of transformations of one-parameter (a) [5]

$$\begin{array}{ccc}\hfill \tilde{t}& \approx & t+aT(t,x_1,x_2,x_3),\hfill \\ \hfill \widetilde{x_1}& \approx & x_1+a{X}_1(t,x_1,x_2,x_3),\hfill \\ \hfill \widetilde{x_2}& \approx & x_2+a{X}_2(t,x_1,x_2,x_3),\hfill \\ \hfill \widetilde{x_3}& \approx & x_3+a{X}_3(t,x_1,x_2,x_3),\hfill \end{array}$$

with infinitesimal Lie operators [5]

$$G=T\frac{\partial }{\partial t}+X_1\frac{\partial }{\partial x_1}+X_2\frac{\partial }{\partial x_2}+X_3\frac{\partial }{\partial x_3}.$$

(12)

The group transformations $\tilde{t}$, $\widetilde{x_1}$, $\widetilde{x_2}$ and $\widetilde{x_3}$ are obtained by solving the following Lie equations [4,5]

$$\begin{array}{ccc}\hfill \frac{d\tilde{t}}{da}& =& T(t,x_1,x_2,x_3),\hfill \\ \hfill \frac{d\widetilde{x_1}}{da}& =& X_1(t,x_1,x_2,x_3),\hfill \\ \hfill \frac{d\widetilde{x_2}}{da}& =& X_2(t,x_1,x_2,x_3),\hfill \\ \hfill \frac{d\widetilde{x_3}}{da}& =& X_3(t,x_1,x_2,x_3),\hfill \end{array}$$

with the initial conditions:

$$\tilde{t}{\mid }_{a=0}=t,\widetilde{x_1}{\mid }_{a=0}=x_1,\widetilde{x_2}{\mid }_{a=0}=x_2,\widetilde{x_3}{\mid }_{a=0}=x_3.$$

The first extension of Lie operators above is defined as follows [5]

$$G^{\left[1\right]}=G+X_1^{\left[t\right]}\frac{\partial }{\partial {\dot{x}}_1}+X_2^{\left[t\right]}\frac{\partial }{\partial {\dot{x}}_2}+X_3^{\left[t\right]}\frac{\partial }{\partial {\dot{x}}_3},$$

(13)

where

$$\begin{array}{ccc}\hfill X_1^{\left[t\right]}& =& D_t\left(X_1\right)-{\dot{x}}_1{D}_t\left(T\right),\hfill \\ \hfill X_2^{\left[t\right]}& =& D_t\left(X_2\right)-{\dot{x}}_2{D}_t\left(T\right),\hfill \\ \hfill X_3^{\left[t\right]}& =& D_t\left(X_3\right)-{\dot{x}}_3{D}_t\left(T\right),\hfill \end{array}$$

with $D_t$ representing the total differential operator describe as follows

$$D_t=\frac{\partial }{\partial t}+\dot{x_1}\frac{\partial }{\partial x_1}+\dot{x_2}\frac{\partial }{\partial x_2}+\dot{x_3}\frac{\partial }{\partial x_3}+\ddot{x_1}\frac{\partial }{\partial \dot{x_1}}+\ddot{x_2}\frac{\partial }{\partial \dot{x_2}}+\ddot{x_3}\frac{\partial }{\partial \dot{x_3}}+...$$

The infinitesimals transformation obtained will be used to solve the following equation [5]

$$\begin{array}{ccc}\hfill T{r}_t+X_1{r}_{x_1}+X_2{r}_{x_2}++X_3{r}_{x_3}& =& 0,\hfill \\ \hfill T{u}_t+X_1{u}_{x_1}+X_2{u}_{x_2}++X_3{u}_{x_3}& =& 0,\hfill \\ \hfill T{v}_t+X_1{v}_{x_1}+X_2{v}_{x_2}++X_3{v}_{x_3}& =& 0,\hfill \\ \hfill T{w}_t+X_1{w}_{x_1}+X_2{w}_{x_2}++X_3{w}_{x_3}& =& 1.\hfill \end{array}$$

(14)

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).

3. Application of (PS) Procedure to Nonlinear System (1)

Considering the three-dimensional Equation (1)

$$\begin{array}{ccc}\hfill \frac{d{s}_h}{dt}& =& 1-\beta s_h{i}_h-\alpha s_h={\varphi }_1\hfill \\ \hfill \frac{d{i}_h}{dt}& =& \beta s_h{i}_h-(\alpha +\gamma )i_h={\varphi }_2\hfill \\ \hfill ϵ\frac{d{i}_v}{dt}& =& \theta (1-i_v)i_h-\delta i_v={\varphi }_3\hfill \end{array}$$

(15)

Case 1: $I_{s_h}=0$ and $I_t,I_{i_h},I_{i_v}\ne 0$

The substitution of Equation (15) into (9) gives

$$-\alpha \delta {\beta }^2{i}_h^2=0.$$

(16)

From Equation (16) we have, $\delta =0$ or $\beta =0$ or $\alpha =0$. The determining equation for Q in (9) becomes

$$Q_t-\alpha s_h{Q}_{s_h}-(\alpha +\gamma )i_h{Q}_{i_h}+\theta (1-i_v)i_h=\theta i_h,$$

(17)

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 $\gamma =\theta$. Using the restriction $\delta =\beta =\alpha =0$ and $\gamma =\theta$, the solution of the determining equation for $P,L$ is given by $P=i_h,L=0$. Hence, from Equation (11), we obtain

$$\begin{array}{ccc}\hfill r_1& =& i_{h}t-\theta (1-i_v)i_v{i}_h^{2}t\hfill \\ \hfill r_2& =& -i_h{s}_h\hfill \\ \hfill r_3& =& -(i_h+\theta (1-i_v)i_v{i}_h^2)t\hfill \end{array}$$

(18)

therefore, the integral of motion is given by

$$I=-i_h{s}_h-\frac{1}{2}{i}_v^2{i}_h$$

(19)

Case 2: $I_{i_h}=0$, $I_t,I_{s_h},I_{i_v}\ne 0$ and $I_{i_v}=0$, $I_t,I_{s_h},I_{i_h}\ne 0$

According to [3], the determining equations and conditions is obtained by introducing the following transformation

$$P=SQ\mathrm{and}L=UQ,$$

(20)

with

$$U=-\frac{{\varphi }_{3s_h}}{{\varphi }_{2s_h}}\mathrm{and}S=-\frac{({\varphi }_3+{\varphi }_{2}U)}{{\varphi }_1}$$

(21)

4. Lie Symmetry Analysis of the System (1)

4.1. Lie Symmetry of one Dimensional Second-Order Differential Equation

From the first equation of the nonlinear system (1), we obtain

$$i_h=-\frac{{\dot{s}}_h}{\beta s_h}+\frac{1}{\beta s_h}-\frac{\alpha }{\beta },$$

(22)

differentiating Equation (22) with respect to t, we obtain

$${\dot{i}}_h=-\frac{{\ddot{s}}_h}{\beta s_h}+\frac{{\dot{s}}_h^2}{\beta s_h^2}-\frac{{\dot{s}}_h}{\beta s_h^2},$$

(23)

the substitution of Equation (23) into the second equation of the nonlinear system (1) gives

$${\dot{s}}_h^2-\beta {\dot{s}}_h-{\ddot{s}}_h{s}_h-\beta s_h^2+\beta {\dot{s}}_h{s}_h^2+\alpha \beta s_h^3+(\alpha +\gamma )s_h-(\alpha +\gamma ){\dot{s}}_h{s}_h+\alpha (\alpha +\gamma )s_h^2$$

(24)

Lie group analysis to (24) is performed using SYM packages [4–6], yielding the following cases

• Case 1: $\beta =0$ and $\alpha +\gamma =0$

The determining equations for the classical symmetries of the nonlinear Equation (24) are

$$\begin{array}{ccc}\hfill \frac{\partial \xi }{\partial s_h}+s_h\frac{{\partial }^2\xi }{\partial s_h^2}& =& 0\hfill \\ \hfill \frac{\partial \eta }{\partial t}+s_h\frac{{\partial }^2\eta }{\partial t^2}& =& 0\hfill \\ \hfill \eta +s_h(2\frac{\partial \eta }{\partial t}-\frac{\partial \xi }{\partial t}-2s_h\frac{{\partial }^2\xi }{\partial t\partial s_h})& =& 0\hfill \\ \hfill -\eta +s_h(\frac{\partial \eta }{\partial s_h}-2\frac{\partial \xi }{s_h}-s_h\frac{{\partial }^2\eta }{\partial s_h^2}+2s_h\frac{{\partial }^2\xi }{\partial t\partial s_h})& =& 0\hfill \end{array}$$

(25)

The coefficients of the infinitesimal generator are obtained by solving the overdetermining Equation (25)

$$\begin{array}{ccc}\hfill \xi (s_h,t)& =& c_1+c_{2}t\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill \eta (s_h,t)& =& c_2{s}_h,\hfill \end{array}$$

(27)

as a result, the two-dimensional Lie algebra is given by

$$\begin{array}{ccc}\hfill G_1& =& {\partial }_t\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill G_2& =& t{\partial }_t+s_h{\partial }_{s_h},\hfill \end{array}$$

(29)

This case reduces Equation (24) to

$${\dot{s}}_h^2-{\ddot{s}}_h{s}_h=0$$

(30)

Equation (38) can be linearized using the transformation

$$S=\frac{1}{s_h}$$

(31)

Hence, we obtain

$$\frac{d^{2}S}{d{t}^2}=0$$

(32)

The solution to Equation is

$$S\left(t\right)=ct+d$$

(33)

where $c,d$ are constant of integration. Substituting Equation (33) into transformation (31) results in the number of susceptible humans

$$s_h\left(t\right)=\frac{1}{ct+d}$$

(34)

Equation (34) is substituted into Equation (22) to obtain the number of infected persons, $i_h$.

$$i_h=\frac{At+B}{\beta (ct+d)}$$

(35)

with $A=-\alpha c,B=1-c-\alpha d$. The substitution of Equations (34) and (35) into the last equation in (1) gives

$$\frac{d{i}_v}{dt}+\left(\frac{Et+F}{Kt+L}\right)i_v=\frac{Ht+G}{Kt+L}$$

(36)

with $G=F+L\delta ,H=E+K\delta ,E=\theta A,F=\theta B,L=\beta d,K=\beta c$ Hence, the number of infected mosquitoes is given by

$$\begin{array}{ccc}\hfill i_v& =& c_{1}exp\left[\frac{Et}{K}-\frac{(FK-EL)ln(L+Kt)}{K^2}\right]\hfill \\ & & +\frac{1}{K^2}exp\left[-\frac{F+Et}{K}-\frac{(FK-EL)ln(L+Kt)}{K^2}\right]\hfill \\ & & \times {(L+Kt)}^{\frac{(FK-EL)}{K^2}}{\left(-\frac{L+Kt}{K^2}\right)}^{\frac{(FK-EL)}{K^2}}\hfill \end{array}$$

(37)

• Case 2: $\beta \ne 0$ and $\alpha +\gamma \ne 0$

The determining equations for the classical symmetries of the nonlinear Equation (24) are given by

$$\begin{array}{ccc}\hfill \frac{\partial \xi }{\partial s_h}+s_h\frac{{\partial }^2\xi }{\partial s_h^2}& =& 0\hfill \\ \hfill \eta -s_h\left[\frac{\partial \eta }{\partial s_h}+2(\beta s_h-1)\frac{\partial \xi }{\partial s_h}\right]-s_h\left[\frac{{\partial }^2\eta }{\partial s_h^2}-2\frac{{\partial }^2\xi }{\partial t\partial s_h}\right]& =& 0\hfill \\ \hfill -\eta +3s_h^2[(\beta -(\alpha +\gamma )s_h)]\frac{\partial \xi }{\partial s_h}-2\frac{\partial \eta }{\partial t}+\frac{\partial \xi }{\partial t}-\beta s_h\frac{\partial \xi }{\partial t}& =& 0\hfill \\ \hfill -\eta (\alpha +\gamma )s_h+s_h[-\beta +(\alpha +\gamma )s_h]\frac{\partial \eta }{\partial s_h}+\frac{\partial \eta }{\partial t}-\beta s_h\frac{\partial \eta }{\partial t}& =& 0\hfill \\ \hfill \beta s_h\frac{\partial \eta }{\partial t}+2\beta s_h\frac{\partial \xi }{\partial t}-2(\alpha +\gamma )s_h^2\frac{\partial \xi }{\partial t}+s_h\frac{{\partial }^2\eta }{\partial t^2}& =& 0\hfill \end{array}$$

(38)

from the above overdetermining equation yields the coefficients of the infinitesimal generator

$$\begin{array}{ccc}\hfill \xi (s_h,t)& =& \frac{exp\left(\sqrt{\alpha +\gamma }\right)t}{\alpha +\gamma }{c}_1+c_2\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill \eta (s_h,t)& =& exp\left(\sqrt{\alpha +\gamma }\right)t{c}_1{s}_h,\hfill \end{array}$$

(40)

As a result, we have the two-dimensional Lie algebra shown below

$$\begin{array}{ccc}\hfill G_1& =& \left[exp\left(\sqrt{\alpha +\gamma }\right)t+exp\left(\sqrt{\alpha +\gamma }\right)t{s}_h\right]{\partial }_t\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill G_2& =& {\partial }_{s_h},\hfill \end{array}$$

(42)

Solving the nonlinear Equation (24) for case 1, we obtain the number of susceptible humans

$$s_h\left(t\right)=\frac{-1+Aexp(At+B)}{A},$$

(43)

with A and B constants of integration. The number of infected humans, $i_h$ is calculated by substituting Equation (43) into Equation (22)

$$i_h=\frac{A(A-\alpha \beta )exp(At+B)+A-\alpha \beta }{\beta \left[Aexp(AT+B)-1\right]}$$

(44)

The substitution of (43) and (44) gives the first-order nonlinear first-order ordinary differential equation

$$ϵ\frac{d{i}_v}{dt}+i_v\left(\frac{Fk\left(t\right)+G}{Dk\left(t\right)-\beta }\right)=\frac{Lk\left(t\right)+N}{Dk\left(t\right)-\beta }$$

(45)

with

$$\begin{array}{ccc}\hfill k\left(t\right)& =& exp(At+B)\hfill \\ \hfill F& =& L-M\hfill \\ \hfill G& =& N-P\hfill \\ \hfill L& =& \theta C\hfill \\ \hfill M& =& \delta D\hfill \\ \hfill N& =& \theta B\hfill \\ \hfill P& =& \delta B\hfill \end{array}$$

The solution to Equation (45) is a hypergeometric function, and the numerical solution can be found in [8].

4.2. 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:

$$\begin{array}{ccc}\hfill G(1-\beta s_h{i}_h-\alpha s_h)& =& -\beta S{I}_1-\alpha S,\hfill \\ \hfill G(\beta s_h{i}_h-(\alpha +\gamma )i_h)& =& \beta S{I}_1-(\alpha +\gamma )I_1,\hfill \\ \hfill G(\theta (1-i_v)i_h-\delta i_v)& =& -\theta I_1{I}_2-\delta I_2.\hfill \end{array}$$

(46)

The extended infinitesimal transformation is obtained by using Equation (13)

$$\begin{array}{cc}\hfill -\beta S{I}_1-\alpha S& =S^{\left[t\right]}+s_h^{{}^{\prime }}{S}^{\left[s_h\right]}+i_h^{{}^{\prime }}{S}^{\left[i_h\right]}+i_v^{{}^{\prime }}{S}^{\left[i_v\right]}\hfill \\ \hfill & -s_h^{{}^{\prime }}({\mathcal{T}}^{\left[t\right]}+s_h^{{}^{\prime }}{\mathcal{T}}^{\left[s_h\right]}+i_h^{{}^{\prime }}{\mathcal{T}}^{\left[i_h\right]}+i_v^{{}^{\prime }}{\mathcal{T}}^{\left[i_v\right]}),\hfill \\ \hfill \beta S{I}_1-(\alpha +\gamma )I_1& =I_1^{\left[t\right]}+s_h^{{}^{\prime }}{I}_1^{\left[s_h\right]}+i_h^{{}^{\prime }}{I}_1^{\left[i_h\right]}+i_v^{{}^{\prime }}{I}_1^{\left[i_v\right]}\hfill \\ \hfill & -i_h^{{}^{\prime }}({\mathcal{T}}^{\left[t\right]}+s_h^{{}^{\prime }}{\mathcal{T}}^{\left[s_h\right]}+i_h^{{}^{\prime }}{\mathcal{T}}^{\left[i_h\right]}+i_v^{{}^{\prime }}{\mathcal{T}}^{\left[i_v\right]}),\hfill \\ \hfill -\theta I_1{I}_2-\delta I_2& =I_2^{\left[t\right]}+u_1^{{}^{\prime }}{I}_2^{\left[u_1\right]}+u_2^{{}^{\prime }}{I}_2^{\left[i_h\right]}+i_v^{{}^{\prime }}{I}_2^{\left[i_v\right]}\hfill \\ \hfill & -i_v^{{}^{\prime }}({\mathcal{T}}^{\left[t\right]}+s_h^{{}^{\prime }}{\mathcal{T}}^{\left[s_h\right]}+i_h^{{}^{\prime }}{\mathcal{T}}^{\left[i_h\right]}+i_v^{{}^{\prime }}{\mathcal{T}}^{\left[i_v\right]}).\hfill \end{array}$$

(47)

with

$$s_h^{{}^{\prime }}=\frac{d{s}_h}{dt};i_h^{{}^{\prime }}=\frac{d{u}_2}{dt};i_v^{{}^{\prime }}=\frac{d{u}_3}{dt}.$$

Substituting Equation (1) into (47) yields

$$\begin{array}{cc}\hfill -\beta S{I}_1-\alpha S& =S^{\left[\tau \right]}+\left(1-\beta s_h{i}_h-\alpha s_h\right)\left(S^{\left[s_h\right]}-{\mathcal{T}}^{\left[t\right]}\right)\hfill \\ & +\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)S^{\left[i_h\right]}\hfill \\ & +\left(\theta (1-i_v)i_h-\delta i_v\right)S^{\left[i_v\right]}\hfill \\ & -{\left(1-\beta s_h{i}_h-\alpha s_h\right)}^2{\mathcal{T}}^{\left[s_h\right]}\hfill \\ & -\left(1-\beta s_h{i}_h-\alpha s_h\right)\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right){\mathcal{T}}^{\left[i_h\right]}\hfill \\ & -\left(1-\beta s_h{i}_h-\alpha s_h\right)\left(\theta (1-i_v)i_h-\delta i_v\right){\mathcal{T}}^{\left[i_v\right]},\hfill & \hfill \\ \hfill \beta S{I}_1-(\alpha +\gamma )I_1& =T_1^{\left[t\right]}+\left(1-\beta s_h{i}_h-\alpha s_h\right)I_1^{\left[s_h\right]}\hfill \\ & +\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)\left(I_1^{\left[i_h\right]}-{\mathcal{T}}^{\left[t\right]}\right)\hfill \\ & +\left(\theta (1-i_v)i_h-\delta i_v\right)I_1^{\left[i_v\right]}\hfill \\ & -\left(1-\beta s_h{i}_h-\alpha s_h\right)\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right){\mathcal{T}}^{\left[s_h\right]}\hfill \\ & -{\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)}^2{\mathcal{T}}^{\left[i_v\right]}\hfill \\ & -\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)\left(\theta (1-i_v)i_h-\delta i_v\right){\mathcal{T}}^{\left[i_v\right]},\hfill & \hfill \\ \hfill -\theta I_1{I}_2-\delta I_2& =U_3^{\left[t\right]}+\left(1-\beta s_h{i}_h-\alpha s_h\right)I_2^{\left[s_h\right]}\hfill \\ & +\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)I_2^{\left[i_h\right]}\hfill \\ & +\left(\theta (1-i_v)i_h-\delta i_v\right)\left(I_2^{\left[i_v\right]}-{\mathcal{T}}^{\left[t\right]}\right)\hfill \\ & -\left(1-\beta s_h{i}_h-\alpha s_h\right)\left(\theta (1-i_v)i_h-\delta i_v\right){\mathcal{T}}^{\left[s_h\right]}\hfill \\ & -\left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)\left(\theta (1-i_v)i_h-\delta i_v\right){\mathcal{T}}^{\left[i_h\right]}\hfill \\ & +{\left(\theta (1-i_v)i_h-\delta i_v\right)}^2{\mathcal{T}}^{\left[i_v\right]}.\hfill \end{array}$$

(48)

In general, solving nonlinear system (48) is challenging. As a result, it is required to use special solutions [5]. In the case of $\mathcal{T}=\mathcal{T}\left(t\right)$, $S=S\left(s_h\right)$, $I_1=I_1\left(i_h\right)$, $I_2=I_2\left(i_v\right)$, the non-linear Equations (48) are simplified as

$$\begin{array}{cc}\hfill \left(1-\beta s_h{i}_h-\alpha s_h\right)\left(S^{\left[s_h\right]}-{\mathcal{T}}^{\left[t\right]}\right)& =-\beta S{I}_1-\alpha S,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)\left(I_1^{\left[i_h\right]}-{\mathcal{T}}^{\left[t\right]}\right)& =\beta S{I}_1-(\alpha +\gamma )I_1,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill \left(\theta (1-i_v)i_h-\delta i_v\right)\left(T_2^{\left[i_v\right]}-{\mathcal{T}}^{\left[t\right]}\right)& =-\theta I_1{I}_2-\delta I_2.\hfill \end{array}$$

(51)

The following second-order partial differential equation is obtained by considering the partial derivative of Equation (51) with regard to $\tau$ [5]

$${\mathcal{T}}^{\left[tt\right]}=0,$$

solving Equation (52), we obtain

$$\mathcal{T}\left(t\right)=lt+m,$$

(52)

with l and m being the integration constants. Substituting Equation (52) into (49)–(51) yields

$$\begin{array}{ccc}\hfill \left(1-\beta s_h{i}_h-\alpha s_h\right)\left(S^{\left[s_h\right]}-l\right)& =& -\beta S{I}_1-\alpha S,\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill \left(\beta s_h{i}_h-(\alpha +\gamma )i_h\right)\left(I_1^{\left[i_h\right]}-l\right)& =& \beta S{I}_1-(\alpha +\gamma )I_1,\hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill \left(\theta (1-i_v)i_h-\delta i_v\right)\left(I_2^{\left[i_v\right]}-l\right)& =& -\theta I_1{I}_2-\delta I_2.\hfill \end{array}$$

(55)

After twice partially differentiating Equation (54) with respect to $i_h$, we obtain

$$I_1^{\left[i_h{i}_h\right]}=0.$$

Hence,

$$I_1\left(i_h\right)=p{i}_h+q.$$

(56)

Substituting Equation (56) into (53), we obtain

$$\left(1-\beta s_h{i}_h-\alpha s_h\right)\left(S^{\left[s_h\right]}-l\right)=-\beta S(p{i}_h+q)-\alpha S.$$

(57)

As Equation (57) is dependent on the values of $s_h$ and $i_h$, we obtain

$$\begin{array}{ccc}\hfill s_h& :& -(\beta i_h+\alpha )\frac{\partial S}{\partial s_h}=-\beta l{i}_h-\alpha l,\hfill \\ \hfill i_h& :& s_h\left(\frac{\partial S}{\partial s_h}-l\right)=Sp,\hfill \\ \hfill -& :& \frac{\partial S}{\partial s_h}-l=\beta Sq-\alpha S.\hfill \end{array}$$

Hence,

$$S=c_{1}exp\left[\left(\beta q-\alpha \right)s_h\right]+\frac{l}{-\beta q+\alpha }.$$

(58)

From (55), we obtain

$$I_h:\frac{\partial I_2}{\partial i_v}=l.$$

Hence,

$$I_2=l{i}_v+c_2.$$

(59)

As a result, the infinitesimal transformations are as follows:

$$\begin{array}{ccc}\hfill S\left(s_h\right)& =& c_{1}exp\left[\left(\beta q-\alpha \right)s_h\right]+\frac{l}{-\beta q+\alpha },\hfill \\ \hfill I_1\left(i_h\right)& =& p{i}_h+q,\hfill \\ \hfill I_2\left(i_v\right)& =& l{i}_v+c_2.\hfill \end{array}$$

(60)

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

$$G=\left(lt+m\right)\frac{\partial }{\partial t}+(c_{1}exp\left[\left(\beta q-\alpha \right)s_h\right]+\frac{l}{-\beta q+\alpha })\frac{\partial }{\partial s_h}+\left(p{i}_h+q\right)\frac{\partial }{\partial i_h}+\left(l{i}_v+c_2\right)\frac{\partial }{\partial i_v}.$$

Hence, the following Lie generators are found

$$\begin{array}{ccc}\hfill G_1& =& t\frac{\partial }{\partial t}+\frac{1}{\alpha }\frac{\partial }{\partial s_h}+i_v\frac{\partial }{\partial i_v},\hfill \\ \hfill G_2& =& \frac{\partial }{\partial t},\hfill \\ \hfill G_3& =& i_h\frac{\partial }{\partial i_h},\hfill \\ \hfill G_4& =& \frac{1}{(\alpha -\beta )}\frac{\partial }{\partial s_h},\hfill \\ \hfill G_5& =& exp[-\alpha s_h]\frac{\partial }{\partial s_h},\hfill \\ \hfill G_6& =& \frac{\partial }{\partial i_v}.\hfill \end{array}$$

By setting the constant of integration to $l=1$, $m=1$, $p=1$, $q=1$, $c_1=1$, $c_2=1$. Equation (60) becomes

$$\begin{array}{ccc}\hfill S& =& \frac{1}{\alpha -\beta }+exp[\beta -\alpha ],\hfill \\ \hfill I_1& =& 1+i_h,\hfill \\ \hfill I_2& =& 1+i_v,\hfill \\ \hfill \mathcal{T}& =& 1+t.\hfill \end{array}$$

Hence, Equation (14) becomes

$$\begin{array}{ccc}\hfill (1+t)r_t+\left(\frac{1}{\alpha -\beta }+exp[\beta -\alpha ]\right)r_{s_h}+(1+i_h)r_{i_h}+(1+i_v)r_{i_v}& =& 0,\hfill \\ \hfill (1+t)s_h^{\left[t\right]}+\left(\frac{1}{\alpha -\beta }+exp[\beta -\alpha ]\right)S^{\left[s_h\right]}+(1+i_h)S^{\left[i_h\right]}+(1+i_v)S^{\left[i_v\right]}& =& 0,\hfill \\ \hfill (1+t)i_h^{\left[t\right]}+\left(\frac{1}{\alpha -\beta }+exp[\beta -\alpha ]\right)I_1^{\left[s_h\right]}+(1+i_h)I_1^{\left[i_h\right]}+(1+i_v)I_1^{\left[i_v\right]}& =& 0,\hfill \\ \hfill (1+t)i_v^{\left[t\right]}+\left(\frac{1}{\alpha -\beta }+exp[\beta -\alpha ]\right)I_2^{\left[s_h\right]}+(1+i_h)I_2^{\left[i_v\right]}+(1+i_v)I_2^{\left[i_v\right]}& =& 1.\hfill \end{array}$$

(61)

The solution of Equation (61) is given by

$$\begin{array}{ccc}\hfill r& =& \frac{t}{g\left(s_h{i}_h{i}_v\right)},\hfill \\ \hfill s_h& =& lnt+\frac{t}{g\left(s_h{i}_h{i}_v\right)},\hfill \\ \hfill i_h& =& \frac{t}{g\left(s_h{i}_h{i}_v\right)},\hfill \\ \hfill i_v& =& \frac{t}{g\left(s_h{i}_h{i}_v\right)}.\hfill \end{array}$$

The special case is given by

$$\begin{array}{ccc}\hfill r& =& \frac{t}{\left(s_h{i}_h{i}_v\right)},\hfill \\ \hfill s_h& =& lnt+\frac{t}{\left(s_h{i}_h{i}_v\right)},\hfill \\ \hfill i_h& =& \frac{t}{\left(s_h{i}_h{i}_v\right)},\hfill \\ \hfill i_v& =& \frac{t}{\left(s_h{i}_h{i}_v\right)}.\hfill \end{array}$$

5. Conclusions

Understanding physical models requires the analysis of nonlinear differential equations. According to Ove [9], finding a closed form solution of a nonlinear differential requires a thorough comprehension of the phenomena being described. The Prelle–Singer (PS) and Lie symmetry techniques are utilized in this research to show the linear integrability of a mathematical model of endemic malaria and to identify explicit solutions. The results showed that for parameter values $\beta \ne \alpha +\gamma$ and $\alpha +\gamma \ne 0$, the reduced second-order differential equation allows for system linearization and provides the explicit solutions.