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Numerical Analysis of the SIR-SI Differential Equations With World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:7, 2013 Numerical Analysis of the SIRSI Differential Equations with Application to Dengue Disease Mapping in Kuala Lumpur, Malaysia N. A. Samat and D. F. Percy These involve approximating the continuous time scale as Abstract—The main aim of this study is to describe and discrete for these differential equations in order to calculate introduce a method of numerical analysis in obtaining approximate numerical solutions. The results of the numerical analysis solutions for the SIR-SI differential equations (susceptible-infective- provide us with useful information about the Susceptible, recovered for human populations; susceptible-infective for vector Infective and Recovered human populations, and the populations) that represent a model for dengue disease transmission. Firstly, we describe the ordinary differential equations for the SIR-SI Susceptible and Infective vector populations. We demonstrate disease transmission models. Then, we introduce the numerical these results using dengue data from Malaysia. analysis of solutions of this continuous time, discrete space SIR-SI model by simplifying the continuous time scale to a densely II. DIFFERENTIAL EQUATIONS FOR SIR-SI MODEL OF DENGUE populated, discrete time scale. This is followed by the application of DISEASE TRANSMISSION this numerical analysis of solutions of the SIR-SI differential equations to the estimation of relative risk using continuous time, In the study of dengue disease transmission model, the most discrete space dengue data of Kuala Lumpur, Malaysia. Finally, we common compartmental model used is as displayed in the present the results of the analysis, comparing and displaying the following Fig. 1. This model is adapted from [1] and [2] by results in graphs, table and maps. Results of the numerical analysis of [3]. solutions that we implemented offers a useful and potentially superior In this study, for i = ,2,1 …, M study regions, and model for estimating relative risks based on continuous time, discrete space data for vector borne infectious diseases specifically for dengue j = ,2,1 …,T time periods, each notation showed in Fig. 1 is disease. defined as follows. S (h) : total number of susceptible humans at time j Keywords—Dengue disease, disease mapping, numerical i, j (h) analysis, SIR-SI differential equations. I i, j : total number of infective humans at time j R (h) : total number of recovered humans at time j I. INTRODUCTION i, j S (v) : total number of susceptible mosquitoes at time j HIS paper discusses and explains about the development i, j (v) : total number of infective mosquitoes at time j Tof continuous time and discrete space stochastic SIR-SI I i, j models for dengue disease transmission. First, we describe µ (h) : birth and death rates of humans per day (assumed equal) relevant forms of ordinary differential equations, which are (v) : birth and death rates of mosquitoes per day (assumed commonly used in the analysis of variables that change µ continuously with respect to time. Here, we present ordinary equal) differential equations that specifically apply to infectious γ (h) : rate at which humans recover per day disease modeling, again in the particular context of dengue b : biting rate per day disease, based on the compartmental SIR-SI model presented m : number of alternative hosts available as the blood source in Fig. 1. A : constant recruitment rate for the mosquito vector This is followed by methods of numerical analysis to obtain β (h) : the transmission probability from mosquitoes to humans approximate solutions of these differential equations that β (v) : the transmission probability from humans to mosquitoes International Science Index, Mathematical and Computational Sciences Vol:7, No:7, 2013 waset.org/Publication/16320 represent a SIR-SI model of dengue disease transmission. (h) : the human population size for the study region i Ni (v) : the mosquito population size for the study region i Ni The authors acknowledge Universiti Pendidikan Sultan Idris and the Ministry of Higher Education in Malaysia for their financial support in respect of this study. Nor Azah Samat is with the Mathematics Department, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, 35900 Tanjong Malim, Perak, Malaysia (corresponding author to provide phone: +6015-48117415; fax: +6015-48117296; e-mail: [email protected]). D. F. Percy is with the Salford Business School, University of Salford, Greater Manchester, M5 4WT Salford, United Kingdom (phone: +440161- 2954710; e-mail: [email protected]). International Scholarly and Scientific Research & Innovation 7(7) 2013 1085 scholar.waset.org/1307-6892/16320 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:7, 2013 µ ( h ) µ ( h ) µ ( h ) β ( h ) bI ( v ) µ ( h ) N ( h ) i , j γ ( h ) i ( h ) N i + m ( h ) ( h ) ( h ) S i , j I i , j R i , j Human Populations ( v ) ( v ) Vector I i , j S i , j Populations β ( v ) bI ( h ) i , j A = µ ( v ) N ( v ) ( h ) i N i + m ( v ) ( v ) µ µ Fig. 1 Compartmental SIR-SI model for dengue disease transmission For intervals over the continuous time scale indexed by t, III. NUMERICAL ANALYSIS OF SOLUTIONS OF THE SIR-SI the compartmental SIR-SI model for dengue disease DIFFERENTIAL EQUATIONS transmission shown in Fig. 1 can also be written In this study, we develop computer programming to perform mathematically as a system of ordinary differential equations, numerical analysis as a means of determining solutions of as time is the only independent variable in this context. complicated systems of nonlinear ordinary differential Therefore, the differential equations for dengue disease equations such as those in (1) to (5). Many researchers have transmission in human populations based on Fig. 1 are as used various types of computer software to help them with the follows: numerical analysis of differential equations in the context of disease modeling. For instance, [5] used Maple software to S′(h) t)( =µ(h)N(h) −µ(h)S(h) t)( −((β(h)b)/(N(h) +m))I(v) t)( S(h) t)( (1) solve the system of differential equations in her study on vector-host models for epidemics, whilst others have used Mathematica software to solve similar complicated differential (h) (h) (h) (v) (h) (h) (h) (h) (h) I′ t)( = ((β b)/(N + m))I t)( S t)( − µ I t)( − γ I t)( (2) equations. For the purpose of this research, we choose to use WinBUGS software for the analysis because it has a natural R′(h) (t) = γ (h) I (h) (t) − µ (h) R(h) (t) (3) iterative logic and readily enables us to include prior distributions for Bayesian analysis. WinBUGS software is a package designed to carry out Markov chain Monte Carlo Similarly, the differential equations for dengue disease (MCMC) computations for a wide variety of Bayesian models. transmission in vector populations are as follows: Details explanation about this free software is discussed in [6]. While, a discussion and application of Bayesian analysis of (v) (v) (v) (v) (v) (v) (h) (h) (v) (4) S′ (t) = µ N − µ S (t) − ((β b) /(N + m))I (t)S (t) disease mapping using this software can be found in [7]. In this section, we consider methods of numerical analysis (v) (v) (h) (h) (v) (v) (v) I ′ (t) = ((β b) /(N + m))I (t)S (t) − µ I (t) (5) for calculating solutions to our continuous time, discrete space SIR-SI model. In particular, we simplify the continuous time It is clear, because of the existence of products of functions, scale to a densely populated, discrete time scale. Although that this system of differential equations is nonlinear. As it fundamentally different in nature and scale, this approach involves five such equations, this system is very difficult to effectively reduces our continuous time, discrete space model solve analytically, whereas analytic solution of a system of to a similar form as the discrete time, discrete space SIR-SI linear ordinary differential equations would be relatively models as introduced by [3]. We now adopt this approach to straightforward. Many references point out that nonlinear solve the system of nonlinear differential equations explained International Science Index, Mathematical and Computational Sciences Vol:7, No:7, 2013 waset.org/Publication/16320 system such as this can usually be solved only by performing previously. numerical analysis with computer programming or by studying Therefore, for i = ,2,1 …, M study regions, and the asymptotic behavior of solutions as approximations to the j = ,2,1 …,T time periods, the deterministic continuous time, actual solutions (for example as explained in [4]). The next discrete space SIR-SI model for dengue disease transmission section discusses how to solve these equations using numerical in human populations reduces to this set of difference analysis with the help of computer software. equations: (h) (h) (h) (h) (h) (h) (v) (h) (6) Si, j = µ Ni + {1− µ − ((β b)/(Ni + m))I i, j−1}Si, j−1 International Scholarly and Scientific Research & Innovation 7(7) 2013 1086 scholar.waset.org/1307-6892/16320 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:7, 2013 (h) (h) (h) (h) (h) (h) (v) (h) involving geographical regions. Ii, j = 1( − µ −γ )Ii, j−1 + ((β b) /(Ni + m))Ii, j−1Si, j−1 (7) The continuous time, discrete space, stochastic SIR-SI model for dengue disease transmission that we analyzed here is R (h) = 1( − µ (h) )R(h) + γ (h) I (h) (8) i, j i, j−1 i, j−1 next used in Section IV for our development of a method to estimate relative risks.
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