Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models

Transcription

1 Journal of Mathematical Modelling and Application 2011, Vol. 1, No. 4, ISSN: Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models S O Maliki Department of Industrial Mathematics and Applied Statistics Ebonyi State University, Abakaliki, Nigeria somaliki@yahoo.com Abstract We study here a particular case of SIR and SIS epidemic models for a given constant population. These mathematical models are described by coupled nonlinear first order differential equations. Exact solutions are computed by linearizing the term in the resulting differential equation for the infective population. We then compare the exact solution with the numerical, using a Runge-Kutta algorithm implemented by MathCAD software. The profiles of the solutions are provided from which we infer that the numerical and exact solutions agreed very well. However for the SIR model the approximation in the derivation of the exact solution made this to deviate slightly from the numerical, for small initial population of susceptibles. For larger values it is observed that this deviation becomes negligible. KeyWords: SIR and SIS exact solutions, Runge-Kutta algorithm, MathCAD. 1. Introduction Recently Shabbir,Khan and Sadiq [5] gave exact solutions to the SIS and SIR models by Kermack and Mckendrick [1] given as follows; 1.1 The SIS Model We consider a constant population of size N divided into three components: susceptible population S, the infective population I, who are infected and can retransmit the disease to the susceptible population (fig. 1). This is possible especially if the disease confers no immunity against reinfection. This model is suitable for modelling diseases caused by bacteria and also for most sexually transmitted diseases [1,2]. S I S S I R Figure 1. The SIS Model Figure 2. The SIR Model The exact solution of the SIS model is given as; (1.1)

2 52 Sadik Olaniyi Maliki. (1.2) Here is the infective parameter associated with the typical Lotka-Volterra interaction term and the recovery coefficient. 1.2 The SIR Model This model has three compartments similar to the SIS model except that there is the removed population R, corresponding to those that either had the disease and recovered, died or have developed immunity, or have been removed from contact with other populations (fig. 2). The exact solution [5] is given by; (1.3). While it is true that closed-form solutions to mathematical models are desirable and of great importance in epidemiology, the fact is that they are not always easy to come by especially for models that are described by highly non-linear systems of ordinary differential equations. The primary objective of this paper is to solve the above models numerically using a Runge- Kutta algorithm implemented by a powerful software called MathCAD. Secondly the numerical solutions are then compared with the exact solutions in terms of their profiles. It is observed that the behaviour of the solutions were not given in [5]. 2. MathCAD Solution The numerical solution of the SIS and SIR models involves the Runge-Kutta algorithm implemented by MathCAD software [4]. In what follows we employ dummy variables Y 0 and Y 1 in the algorithm to represent the susceptible and infective variables S and I respectively. We superimpose the exact solution for comparison. (1.4) D( t Y) (SIS) D( t Y) Derivative vector (SIR) t0 0 t1 2 Y0 50 Initial value of independent variable Terminal value of independent variable Initial condition vector Number of solution values on [t0, t1]

3 Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models 5 N 0 M Rkadapt ( Y0 t0 t1 N Solution D) matrix t M 0 S M 1 I M 2 Independent variable values Susceptible variable values Infective variable values 2.1 SIS model simulations Y( t) Exact Approx I 0 exp ( t) I Figure 3(a). Figure 3(b). Figure 3(c). Figure 3(d) Figure 3(e). Figure 3(f). From the above solution profiles (fig 3), and the given parameters, it is clear that there is excellent agreement between the exact and approximate solutions generated by the MathCAD software. This becomes more apparent when the initial population of susceptible becomes large. Variation of the infective parameter and the recovery coefficient do not alter the profiles drastically, hence they were kept constant for the simulations.

4 54 Sadik Olaniyi Maliki 2.2 SIR model simulations... Exact Approx 300 Figure 4(a) X( t). Figure 4(b) I 0 I 0 exp ( 1) exp t ( 1) Figure 4(c) X( t) Figure 4(d) I 0 I 0 exp ( 1) exp t ( 1) Fig. 4(e) X( t) Fig. 4(f) I 0 I 0 exp ( 1) exp t ( 1) We remark that in obtaining the exact solution of the SIR model the authors in [2] were forced to make a slight approximation in the solution. To understand this we detail the steps taken to that point. Given Adding equations (2.1) and (2.2) we get (2.1) (2.2) Which gives upon integration

5 Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models 5 (2.3) Where C is an arbitrary constant. Substituting (2.3) into (2.2) we get (2.4) With the change of variable, the above equation is integrated to give, (2.5) D being a constant of integration. For the integral in (2.5) an analytic solution is almost impossible unless we make the following approximation as in [5]. Since, we consider the series expansion of, neglecting higher order terms. Equation (2.5) now becomes; From the last expression the final solution (1.3) and hence (1.4) are obtained. Observing the solution profiles for the SIR model, we note that when the initial populations of are small, the exact and approximate solutions do not agree very well. As a matter of fact the exact solution for the susceptible population in fig 4(a) crosses the time axis. This is the resulting effect of the approximation explained above. However the approximate solution becomes better behaved as becomes larger compared with, as shown in figs 4(e) and 4(f). 3. Conclusion In this paper numerical solutions to a particular case of SIR and SIS models was investigated and compared with the exact solutions provided by Shabbir, Khan and Sadiq [5]. The Runge-Kutta algorithm for the numerical solutions proved very efficient and was implemented by MathCAD software. Both numerical and exact solutions agreed very well, however for the SIR model the approximation in the derivation of the exact solution made this to deviate slightly from the numerical, for small initial population of susceptibles. For larger values it is observed that this deviation becomes negligible. What this paper has demonstrated is that it is possible to obtain solutions to highly nonlinear systems of ordinary differential equations in mathematical modelling with the help of powerful softwares such as MathCAD. We need not worry too much if we are unable to derive analytical solutions because the numerical solutions implemented by MathCAD are accurate enough to give us the profiles of the solutions in the simulations. References Diekmann, O., Heesterbeek, J.A.P. & Metz, J. A. J.1990 On the Definition and Computation of the Basic Reproductive Ratio in Models for Infectious Diseases in Heterogeneous Population, J. Math Biol. 28, Hethcote, H. W. 0 The Mathematics of Infectious Diseases, SIAM Rev. 42, Kermack, W. O. & McKendrick, A. G Contribution to the Mathematical

6 56 Sadik Olaniyi Maliki Theory of Epidemics, Proc Roy. Soc. Lond. A. A115, MathCAD 13 professional, 5 MathSoft Inc. Shabbir, G., Khan, H. & Sadiq, M. A A note on Exact solution of SIR and SIS epidemic models arxiv v1 [math.ca].