Spline Solutions to Dengue Disease Transmission Model

Transcription

1 VNSGU JOURNAL OF SCIENCE AND TECHNOLOGY Vol.6. No. 1 Jul ISSN : Accepted in : Oct 2017 Spline Solutions to Dengue Disease Transmission Model SHAH Keyuri J. Teaching Assistant Applied Science and Humanities Department Chhotubhai Gopalbhai Patel Inst. Of Tech. BardoliIndia keyu.shah277@gmail.com PANDYA Jigisha U. Assistant Professor Department of Mathematics Sarvajanik College of Engineering & Technology Surat India jigisha.pandya@scet.ac.in Abstract Mathematical models can be used to describe the transmission of disease namely HIV/AIDS Chikungunia Cancer Dengue etc. In this paper Numerical treatment of Dengue Disease Transmission model is discussed using S-I-R epidemiological model keeping some constraints. The objective of this paper is to present applicability of Bickley s spline collocation method to solve different types of epidemiological models. The numerical results obtained by this method are compared with the exact solution to show the efficiency of the method & this comparison shows that the collocation method is more convenient & effective tool for solving likewise problems. Keywords: Ordinary Differential System Spline function Bickley s method Dengue transmission 1. Introduction Numerical methods made enormous progress during 20 th century for the solution of odes. They have been applied to generate approximate solution of problem as equations are difficult to be solved analytically. Mathematical model is often used in various areas of infectious disease epidemiology. Mathematical modeling of dengue disease transmission in human and Vector populations has been done since the beginning of last century. Several studies on infection model within human have been done for various cases. Dengue disease is the most substantial mosquito-borne viral disease of human. It now a leading cause of childhood deaths and hospitalizations in many countries. Variations in environmental conditions especially seasonal climatic parameters effect to the transmission of dengue viruses and their principal mosquito vector Aedes aegypti. The epidemiological systems often exists a peculiar equilibrium the disease free equilibrium which corresponds to a steady state of the population without disease. The another one equilibrium is the endemic equilibrium state. It is the steady state solutions where the disease persists in the population. Dengue is an acute fever caused by a Flavi virus. The disease can occur in three forms: Dengue Fever () Dengue Hemorrhagic Fever () and Dengue Shock Syndrome (DSS). and DSS are gradually important public health problems in the Tropical and subtropical areas. Dengue disease caused by four distinct serotypes virus known as DEN -1 DEN-2 DEN-3 and DEN- 4 in which only DEN 2 and DEN 3 are mostly identified in humid country. A person infected by one of the four serotypes will never be infected again by the same serotype but he or she could be re infected by three other serotypes in about 12 weeks and then becomes more susceptible to developing. is an acute viral disease revealing with myalgias headache retro-orbital pain vomiting maculopapular rash leucopenia and thrombocytopenia. is characterized by four major clinical features: high fever hemorrhagic phenomena hepatomegaly and signs of impending

2 circulatory failure. The severity of disease in depends on the quantum of plasma leakage. The patients are presented with shock due to excessive plasma loss are labeled as dengue shock syndrome (DSS). A basic S-I-R (Susceptible-Infected-Recovered) model is used for representing and DSS transmission system. As mentioned in the literature [1] DSS are endemic cases reported every year. A total of cases of cases of and cases of DSS have been reported during twelve year review period. Between 1997 and 2008 the percentage of mortality and DSS cases reported 1.04% 40.83% and 58.13% respectively. Figure 1: The percentage of cases by clinical diagnosis between 1997 and 2008 Figure 2: The percentage of deaths by clinical diagnosis between 1997 and Introduction to Spline Functions and Survey of Advances in Spline Theory The spline functions along with their successive derivatives are continuous differentiable and more generally they are analytic functions. There is considerable evidence that in many circumstances a spline function is more acceptable as an approximate function than a polynomial involving a comparable number of parameters. As per our interest we regard the spline functions as an approximate tool to solve differential equations. The concept of mathematical splines was first introduced by Schoenberg who suggested their use for interpolation in many ways. The splines of even order interpolating the data at junctions appear in a very simple fashion and their existence criterion was developed by Ahlberg et al[2]and they established also the best approximation and convergence properties of splines.carl de Boor was also able to establish the idea of existence and uniqueness of particular bicubicsplines. In spline function theory many types of splines are found like generalized splines B-splines cardinal splines Lg-splines natural splines polynomial splines parabolic splines trigonometric splines. These functions were found as well as studied during the years 1964 to 1969.Callender presented the procedure for obtaining low order high accuracy spline approximations of solutions to initial value problem in ordinary differential equations.the error estimates of cubic splines were obtained for the first derivative by Kershaw[34] in 1972.In 1968 Bickley [5] brought forward a useful aspect of spline functions in light that can be employed to solve a linear two point boundary value problem approximately. The spline presented by him is expressed in terms of infinite series or in truncated powers. This work was supported by Fyfe[6] in 1969.In fact a substantial work regarding the error estimate in cubic spline approximations was presented there. Again Fyfe extended the use of same cubic spline function to the solution of a fourth order linear two point boundary value problem. The applicability of spline functions to non-linear differential equations took place due to Blue [9].It will be remarkable to mention that spline functions are found useful in the solution of partial differential equations and integral equations too. VNSGU Journal of Science and Technology V 6(1)

3 In this paper we propose a nonlinear mathematical model for the study of the transmission dynamics of Dengue Disease [17]. In the first section we present the introduction that guided the dengue disease spline functions and model s structure. In second section we present the model s equations and definition of the variables and parameters. In third section Numerical approach is applied to differential equations with some constrains by Bickley s spline collocation method [8] and finally the comparative study is described with graphical representation. 2. The Mathematical Model A basic S-I-R (Susceptible-Infected-Recovered) model is used for representing DSS transmission system in this study[1]. In this model we assume that the human and vector population have constant size. The model describes the dynamic of dengue in the two components of transmission viz. human hosts and vector. The total human population is denoted by it is partitioned into five classes the susceptible infectious with clinical diagnosis infectious with clinical diagnosis infectious with DSS clinical diagnosis and recovered which are denoted by respectively. The total vector population is denoted by and the vector population is divided into two classes the susceptible and infectious vector and they are denoted by respectively. The total human population size can be determined by: PH Sh I I IDSS Rh (2.1) The total vector population size P V can be determined by P V S V I V.The mathematical model for this transmission is shown below. The parameters of models are defined in Table 1. Figure 3: Transmission model of DSS system following population S-I-R structure in human Figure 4: Susceptible Infectious structure in vector population. Table - 1 Parameters involved in transmission of Dengue Notations Symbols Meaning Data Total human population size Total vector population size No. of susceptible vector population - No. of Infectious vector population - No. of susceptible human population - VNSGU Journal of Science and Technology V 6(1)

4 Notations Symbols Meaning Birth rate of human population Average biting rate of vector Natural death rate in human population Death rate in vector population Data Death rate of human population with 0.1 Death rate of human population with 0.2 Death rate of human population with DSS 0.8 Transmission probability from infectious vector to human & human becomes infectious with 0.3 Transmission probability from infectious vector to human & human becomes infectious with 0.5 Transmission probability from infectious vector to human & human becomes infectious with DSS 0 Transmission probability from human to vector & vector becomes infectious Constant rate from infectious vector to susceptible human & human becomes infectious by Constant rate from infectious vector to susceptible human & human becomes infectious by Constant rate from infectious vector to susceptible human & human becomes infectious by DSS Constant rate from infectious human to susceptible vector & vector becomes infectious vector M Constant recruitment rate of mosquitoes - Recover rate in human population According to above data the system of differential equation is as follows. ds h B P H ( ( ) I ) S h h V h VNSGU Journal of Science and Technology V 6(1)

5 di S I ( ' ) 1 V r I h di S I ( ' ) 2 V r I h di DSS S I ( r ' ) I 3 h V DSS DSS dr h r ' ( I I I ) DSS h R h ds V M ( V ( I I I )) S 4 DSS V di V ( I I I ) S I 4 DSS V v V The first five equations represent the susceptible infectious with infectious with infectious with DSS and recovered human population densities respectively. The sixth and seventh equations represent the susceptible and infectious vector population densities. Reducing the no. of parameters we introduce : S I h I I S I I I DSS h P P P DSS H H H P H R h S V I R S I V h P V P V H V P V From above we get following equations as dsh h ( h ( 1 2 3) IV PV ) Sh di ' 1PV ShIV 1I where1 r di P S I I where r di DSS ' 3PV ShIV 3I DSS where 3 DSS r div 4PH ( I I I DSS )(1 IV ) V IV ' 2 V h V 2 2 (2.1.1) For the biological interest the region of system (2.1.1) is restricted to (2.1.1) are positive. S I I I I : 0 S I I I I 1 h DSS V h DSS V and all of the parameters used in system VNSGU Journal of Science and Technology V 6(1)

6 3. Numerical Analysis In this section numerical calculations are obtained by Bickley s method[8]. The formula for the method in in terms of its first derivatives is as follows ( xi x) ( x xi 1) ( x xi 1) ( xi x) ( xi x) [2( x xi 1) h] ( x xi 1) [2( xi x) h] s( x) mi 1 m 2 i y 2 i1 y 3 i 3 h h h h Differentiating (3.1) twice we get '' 2m 1 4m 6 s ( x ) i i i ( s s ) h h 2 i i 1 h..(3.1)...(3.2) Now considering IVP as with and we obtain y ( x ) f ( x s ) f ( x s ) f ( x s )...(3.3) '' ' ' i x i i y i i i i Equating (3.2) and (3.3) we get 2mi 1 4mi 6 ' ' 2 ( si si 1 ) fx ( xi si ) f y ( xi si ) f ( xi si )...(3.4) h h h From which can be computed. Substitution in (3.1) gives the spline solutions. The analytical solution for 1 st equation of system (2.1.1)is 2 nd equation is 3 rd equation is 4 th equ. is 5 th equ. is with initial guess.the graphical representation for all equations of system (2.1.1) is shown below. Figure 5: Comparison of analytical and Bickley method s Solution for equation (1) of system (2.1.1) Figure 6: Comparison of analytical and Bickley method s solution for equation (2) of system (2.1.1) VNSGU Journal of Science and Technology V 6(1)

7 Figure 7: Comparison of analytical and Bickley method s Solution for equation (3) of system (2.1.1) Figure 8: Comparison of analytical and Bickley method s solution for equation (4) of system (2.1.1) Figure 9: Comparison of analytical and Bickley method s Solution for equation (5) of system (2.1.1) 4. Discussion and Conclusion Numerical methods have been applied to generate approximate solution of the problems. In this paper we present the applicability of Bickley s spline collocation method to nonlinear IVP. The objective of this paper is to present a comparative study of Bickley s method and Analytical solution for Dengue Disease transmission model. This type of generalization is necessary for the applicability & reliability of spline collocation technique to explore wider field of differential equations. In this paper it is assumed that the human population is constant. The effect of non-constant human population is not taken into mathematical model. so on the further research the non-constant human population should also be considered and spline collocation method can be apply to solve whole system of differential equations. The present work projects the spline functions as efficient and effective tool for the solution of larger system of differential equations in order to accommodate as many parameters effecting the model and the results shows that Bickley s method gives fairly reliable results. Acknowledgement The authors would like to thank Prof. Doctor H.D. Veer Narmad South Gujarat University Surat Gujarat. VNSGU Journal of Science and Technology V 6(1)

8 References [1]. Kongnuy R.Naowanich E. Pongsumpun P. Analysis of a Dengue disease transmission with clinical diagnosis in Thailand International Journal of Mathematical Models & Methods in applied sciencesissue 3 Vol [2]. Ahlberg J. H. Nilson E. N. Walsh J. L. The theory of splines and their Applications Academic press Inc. New York 1967 [3]. Doctor H.D.Bulsari A.B. Kalthia N.L.Spline collocation Approach to Boundary value Problems International Journal for Numerical Methods in fluids Vol [4]. Doctor H.D. Spline Collocation Approach to Flow Problems Ph.D. ThesisVeer Narmad South Gujarat University 1985 [5]. Bickley W.G. Piecewise cubic interpolation and two point boundary value problems The computer Journal [6]. Fyfe D.J.The use of cubic spline in the solution of two point boundary value problems The computer Journal [7]. Kongnuy R.Pongsumpun P.Mathematical Modelingfor Dengue transmission with the effect Of season International science index International journal of Mathematical computational Physical & Quantum Engineering Vol 5 No [8]. Sastry S.S. Introductory Methods of Numerical Analysis 4th edition 2009 [9]. Blue J.L.Spline function methods for nonlinear boundary value problems communications of the ACM 12(6) VNSGU Journal of Science and Technology V 6(1)