Lie Symmetry and Painlevé test for the SIRD model Maba

Abstract: In this paper, Lie symmetry and Painlevé Techniques are applied to the SIRD (Susceptible, 1 Infected, Recovered and Dead) model. A demonstration of the integrability of the model is provided 2 to present an exact solution. The study revealed that nonlinear system passes Painlevé test and 3 does not possesses complex chaotic behaviour. However, the system fails to pass the Painlevé test 4 while constraints reach values equivalent to the corresponding complex chaotic behaviour. The 5 two-dimensional Lie symmetry algebra and the commutator table of the infinitesimal generators are 6 obtained. Lie symmetry analysis serves to linearize the nonlinear system and find the corresponding 7 invariant solution. 8


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In this paper, we modified the classical SIR model of Kermack and McKendrick [5] by assuming that an individual can be born infected. We assumed that after the recovery process, the disease person becomes resistant and the number of individuals died from the disease are counted. The model divides the total population into three different classes: The susceptible class, S, are those who can get infected with the disease; the infective class, I, are those who can spread the disease after getting infected; the removed class, R, are those individuals who recovered from the disease, resistant or sequestered while waiting for recovery; and the death class, D, are those who die from the disease. Most of the infantile viruses, essentially measles, have a removed and death class [14]. The model flow diagram which represented the disease is given by The model is governed by the following nonlinear system of first order ordinary differential equations 11 [8] 12Ṡ = −βSI + γI − µ 1 S + ν 1 S (1) The discussion in the present paper is organised as follows. In Section 2, we reduced the 23 four-dimensional system (1)-(4) into a single ordinary differential equation of second-order. In Section 24 3, the painlevé-analysis was performed for the solutions of nonlinear second order differential equation; 25 parameters values in which nonlinear system (1)-(4) passes painlevé test is found. In Section 4, we 26 performed a Lie symmetry method of the reduced equations to obtain invariant solutions. The explicit 27 solutions and discussion were established in Section 5 and 6 respectively. In this Section we reduce the four-dimensional system (1)-(4) to a one-dimension second order ordinary differential equation. Since equations (1) and (2) does not dependent on R and D, therefore, we can find the number of individual who are recovered, R once we know the infected individual I, hereafter we can excluded R in any consequent analysis of the system. From (2) we have The derivative of (5) givesṠ The substitution of (5) and (6) into (1) gives We have after some arrangement We may attain the following simplification by means of the given change of variable: The substitution of (9) to (8) gives as well as a system of nonlinear ordinary differential equations from the view point of singularity 39 analysis is the Willpower of the existence of isolated movable singularities whereby one may obtained aspect as far as the location of the singularity is concern. Nevertheless, a more complex equation (or 43 system of equations) involving multifaceted arrangement retained more than one polelike singularity.

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Conversely, when it comes to the system of differential equations with many singularities, one need 45 to assure the existence of a Laurent expansion containing an essential amount of arbitrary constant. Ablowitz [1], developed a standard algorithm in order to analyse a differential equation from the view point of Painlevé. Even if there are some illustrations of specific significance of the Painlevé method in analysing a system of nonlinear first order differential equations which are common in mathematical modelling of epidemics, such that the tactical method approach promoted in [2] is considered. In this section, we will primarily, summarise the typical algorithm due to Ablowitz [1]. Furthermore, different approach will be provided. We will start by considering the following autonomous system of first-order ordinary differential equations with n dependent variables represented by x, the independent variable is denoted by t while σ is 52 the conservative of constraints which constantly appears to increase a system commonly used in  The execution of Painlevé test suggests that solution of the following differential equation with independent variable x = (x 0 , x 1 , x 2 , ..., x n−1 ) has the form below where the functions φ, u j (u 0 = 0) are analytic of x around the region of φ(x) = 0. In the following subsection, Ablowitz's algorithm will be used to find the leading-order behaviour 77 of the nonlinear system (1)-(4) and the reduced equation (10). We will start by substituting x i = 78 σ i τ p i , i = 1, n, with τ = t − t 0 and t 0 the assumed location of the movable singularity, into the system 79 (13) and compare the resulting power. As mentioned in Section 2, the Painlevé analysis of nonlinear equations (1)-(4) start by analyzing the first two equations (1)-(2). In this regard, we commence in the customary manner by substituting into (1)-(2) and obtain the following At τ −2 , we obtain At τ −1 , we obtain    The determination of the resonances is find by substituting [9] into (1)-(2) such that arbitrary constants of integration are obtained. We therefore find that nontrivial solution of the system    exists if the resonance is r = ±1. There is no single arbitrary constant which is introduced at the 83 resonance r = ±1. Therefore, the solution of system (1)-(2) is presented as a general solution of the   Proof of Theorem 1. In order to obtain the formal Laurent series expansion, we substitute equation into equation (10) which gives the following equation: for i = j = k = 0, 1, 2, ... At τ −4 we require We move to the next power, τ −3 , and find Since σ 0 = 1, this gives an arbitrary σ 1 only if From (11) and (23) we have: Hence, the reduced equation (10) passes the Painlevé test under parameter values a = γ + b and ν 1 − µ 1 = ν 2 − µ 2 + α + 2γ and does not possess chaotic behaviour.

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A second order ordinary differential equation admits a one-parameter Lie group of transformations with infinitesimal generator The infinitesimal form ofū¯t,ū (1) are found by the given formulas [3,11]: The functions ζ 0 and ζ i are found by using the prolongation formulas below [12] In [9], Matadi claimed that the equation (25) possesses symmetry (group generator) with X [N] the n-th extension of G. where c 1 and c 2 are arbitrary constants. Thus the Lie algebra of equation (35) is spanned by the 108 following two infinitesimal generator: Computing the Lie bracket we obtain the given commutator table: From the commutator table, we conclude that the reduction of (35) can be made by X 2 only. The Lagrange's system associated to X 2 is given by Solving equation (41) we obtain the new dependent variable, X, and independent variable, Y, namely: Therefore equation (35) becomes dY dX the integration of equation (43) gives The substitution of (42) into (44) gives The integration of (45) gives Substituting (11) and (46) into (36) we have The number of infected population is obtain by subtituting (47) into (9) The substitution of (48) into (5) gives 8 of 11 The substitution of (48) and (49) into (3)-(4) gives and 111

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In this Section we obtain the analytical solution of the system (1)-(4) by combining the first two 113 equations.
Equation (51) becomes The solution of (53) is From (52) and (54) we have Equation (6) becomes With the use of the transformation 120 u = 1 I (57) equation (56) becomes: Since and which has the integrating factor The solution of (59) is From (57) we have and from (55), we have Equation (62) has the integrating factor exp [(ν 3 − µ 3 )t]. Therefore and from (4) we obtain the death component of the population to be 124

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In this section we give a numerical result based on the Susceptibles and Infected component of

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In order to understand physical model, the analysis of a nonlinear differential play an essential 131 role. Ove [15] stated that the by finding a closed form solution of a nonlinear differential, one can arrive 132 at a complete understanding of the phenomena which are modeled. In this paper, four dimensional 133 system of the SIRD epidemial model is reduced into a one dimensional second order differential