Analysis of Fractional Order SIR Model

Analysis of Fractional Order SIR Model

A. George Maria Selvam 1, Britto Jacob. S3

1, 3 Sacred Heart College, Tirupattur-635 601, S. India.

D. Abraham Vianny2

2 Knowledge Institute of Technology, Kakapalayam 637 504, S. India

Abstract Fractional order SIR epidemic model is considered for dynamical analysis. The basic reproductive number is established and an analysis is carried out to study the stability of the equilibrium points. The time plots and phase portraits are provided for different sets of parameter values. Numerical simulations are presented to illustrate the stability analysis using Generalized Euler Method.

KeywordsFractional order, SIR model , Differential equations, Stability, Generalized Euler method.

1. FRACTIONAL DERIVATIVES AND INTEGRALS

   Fractional Calculus is a branch of mathematics that deals with the study of integrals and derivatives of non-integer orders, plays an outstanding role and have found several applications in large areas of research during the last decade. Behavior of many dynamical systems can be described and studied using the fractional order system. Fractional derivatives describe effects of memory. This section presents some important definitions of fractional calculus which arise as natural generalization of results from calculus.

   Definition 1. The Riemann Liouville fractional Integral of order 0 1 is defined as

   Definition 2. The Riemann Liouville fractional derivative is defined as

   Definition 3. The Caputo fractional derivative is defined as

   If and are integers such that , then order derivative of (using Eulers Gamma Function) is

2. GENERALIZED TAYLORS FORMULA AND EULER

   METHOD

   We introduce a generalization of Taylors formula that involves Caputo fractional derivatives. Suppose that

   , for , where .

   (1)

   In case of , the generalized Taylors formula (1) reduces to the classical Taylors formula.

   Zaid and Momani derived the generalized Eulers Method for the numerical solution of initial value problems with Caputo derivatives. The method is a generalization of the classical Eulers method. Consider the following general form of IVP;

   (2)

   for . The general formula for Generalized Eulers Method (GEM) is

   (3)

   If then the generalized Eulers method (3) reduces to the classical Eulers method.

3. MODEL DESCRIPTION

   Mathematical modeling is used to analyze, study the spread of infectious diseases and predict the outbreak and to formulate policies to control an epidemic. We obtain fractional SIR epidemic model by introducing fractional derivative of order in the classical SIR epidemic equations. In this paper, we study fractional order SIR epidemic model with vaccination and treatment. The total population is partitioned into three compartments which are Susceptible, Infected and Recovered with sizes denoted by and [4].

   Variable	Meaning
   	The number of Susceptible Individuals at time t
   	The number of Infectious Individuals at time t
   	The number of Recovered Individuals at time t
   Parameters	Meaning
   	Birth rate or Recruitment rate
   	Infectious period
   	Contact rate

   Then we have

   By introducing fractional order, and are the derivatives of and respectively, of arbitrary order in sense of Caputa and then the system leads to fractional equations given by,

   (4)

   Where and .

   The system of fractional differential equations above is reduced to an integer order system by setting . Also we have

   Since, can always be obtained by the equation

   We now consider the system of equations

   (5)

   with , , where

4. BASIC REPRODUCTIVE NUMBER

   The basic reproduction number is the average number of Secondary cases generated by a typical infective within a population with no immunity to the disease. It is denoted by [1, 2, 5]. If , then the disease dies out. If , the it implies that the disease spreads in the susceptible population.

   We determine by using the Next Generation Matrix (NGM) Approach. The Next Generation Matrix is

   given by . ,

   (6)

   At Disease Free Equilibrium (DFE), and substitute into the Next Generation Matrix, we get

   Since is the most dominant eigenvalue of Next Generation Matrix, then

   The DFE is locally stable if , whereas it is unstable if .

5. NON-NEGATIVE SOLUTIONS AND EQUILIBRIUM

   POINTS

   Given the equation of the population as, , and from the dynamics described by system of equations (5), the region and

   is positively invariant (non- negative solutions).

   To evaluate the equilibrium points, we consider

   The fractional order system (5) has two equilibria and .

6. STABILITY OF EQUILIBRIA Based on (5), the Jacobian matrix of the system is

   (7)

   At the Disease-Free Equilibrium (DFE), the Jacobian matrix

   (7) for is

   The eigen values are and . The Disease Free Equilibrium point of the system is asymptotically stable if , and .

   At the Endemic Equilibrium, the Jacobian matrix (7) for is

   The eigen values are and

   The Endemic Equilibrium point of the system is asymptotically stable if , then , .

7. NUMERICAL SIMULATIONS Numerical solution of the fractional order system is

   for

   Numerical techniques are used to analyze the qualitative properties of fractional order differential equations since the equations do not have analytic solutions in general.

   Example 1.Let us consider the parameter and the initial conditions (S(0), I(0), N(0)) = (0.95, 0.05, 1.00) the fractional derivative . For these parameter the corresponding eigen values are and for If and ,

   then the disease free equilibrium is locally asymptotically

   stable.

   Figure 1.

   Example 2. Let us consider the parameter and and the

   initial conditions (S(0), I(0), N(0)) = (0.95, 0.05, 1.00) the

   fractional derivative . For these parameter the corresponding eigen values are and

   for If

   and , then the endemic equilibrium is locally asymptotically stable.

   Example 3. Let us consider the parameter and and the

   initial conditions (S(0), I(0), N(0)) = (0.95, 0.05, 1.00) the

   fractional derivative order . For these parameter the corresponding eigen values are and for If

   and

   , then the endemic equilibrium is locally asymptotically stable.

   Figure 2.

   Figure 3.

   REFERENCES

   1. Fred Brauer, Mathematical Epidemiology, Springer, 2008.

   2. Matt J. Keeling and Pejman Rohani, Modeling Infectious Diseases, Princeton University Press, 2008.

   3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, INC 1993.

   4. Moustafa El-Shahcd and Ahmed Alsacdi, The Fractional SIRC Model and Influenza A, Mathematical Problems in Enginnering, Volume 2011, Article ID 480378, 9 pages, doi:10.1155/2011/480378.

   5. J. D. Murray, Mathematical Biology, An Introduction,Springer, 2002