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Calculating analytical approximate solutions for non-linear infectious disease models is a difficult task. Such models often require computational tools to analyse analytical approximate methods which appear in some theoretical and practical applications in systems biology. They represent key critical elements and give some approximate solutions for such systems. The SIR epidemic disease model is given as the non-linear system of ODE’s. Then, we use a proper scaling to reduce the number of parameters. We suggest Elzaki transform method to find analytical approximate solutions for the simplified model. The technique plays an important role in calculating the analytical approximate solutions. The local and global dynamics of the model are also studied. An investigation of the behaviour at infinity was conducted via finding the lines and singular points at infinity. Model dynamic results are computed in numerical simulations using Pplane8 and SimBiology Toolbox for Mathlab. Results provide a good step forward for describing the model dynamics. More interestingly, the simplified model could be accurate, robust, and used by biologists for different purposes such as identifying critical model elements.

A system of kinetic equations can be applied to describe the behaviours of biochemical kinetics. The system is obtained from the reaction mechanisms by the law of mass action. The rate constant of any given elementary reaction, known as the proportionality constant, depends on the reaction condition (temperature, solvent, PH, etc.). The reaction conditions are generally held by biochemists to avoid dealing with higher-order complexities [

An arbitrary number

where

where

The readers can see further details about the chemical kinetics in [

The idea of sensitivity analysis has been used in dynamical analysis of biochemical kinetics and systems biology models. The concept of sensitivity analysis theory in application to chemical kinetic problems was given by Rabitz in [

The model input

The general form of the local sensitivity is given as a Jacobian matrix as follows

where the matrices

The initial conditions of the Equation (5) are determined by the input parameter

The idea of ELzaki transform was introduced by Tarig Elzaki for solving linear ordinary differential equations with constant coefficients. Then, this became more popular and works as a powerful tool in mathematics for solving some complicated equations, for example, solving high order ordinary differential equations with variable coefficients [

The SIR model, first published by Kermack and McKendrick in 1927 [

The main problem in this study is identifying the critical model parameters. Therefore, the aim in this work is to apply some mathematical tools to simplify and analyse the model and then identifying the model elements (variables and parameters). We proposes a number of steps of model analysis, which plays in reducing the number of elements and in calculating analytical approximate solutions of the SIR model. The proposed steps and their advantages are simply given. The first step is that a proper scaling is used in order to minimize the number of elements. This becomes a good step forward for simplifying the original model. Another step is calculating some analytical approximate solutions using Elzaki transformation. Furthermore, using local sensitivity method is another important step in this study. This helps us to identify critical model parameters of the reduced model. Finally, studying the behaviour at infinity of the reduced model provide us an understanding of global dynamics and drawing the global phase portrait of the system.

The SIR model may be diagrammed as in

where

with initial conditions

Equation (7) has a conservation law

where

The dimensionless SIR equations are then given by

where

Therefore, the conservation law (9) takes the form

In addition, the Equation (10) is then reduced to

with initial conditions

Many nonlinear systems of differential equations have not exact solutions. Calculating analytical approximate solutions for such systems is a difficult task. The SIR epidemic disease model is given as the non-linear system of ODE’s. Equation (12) can not be solved analytically. Therefore, calculating some analytical approximate solutions for the system provides more information about the behaviours of the model dynamics. Applying the idea of Elzaki transform, we can obtain some analytical approximate solutions of the Equation (12). Take Elzaki transform of Equation (12) to get:

Take the inverse Elzaki transform of Equation (13), the recursive relations are then given by

where

For

For

where,

For

where

In general, a series form of the model solution can be given

In this work, we propose some steps of model analysis and simplification for the SIR epidemic disease model. The suggested steps are used for minimizing the number of elements (variables and parameters), calculating analytical approximate solutions and identifying critical model parameters. The proposed steps in this study are presented below:

1) Define chemical mechanisms of SIR Model.

2) Define a kinetic model of biochemical reactions as a system of ODEs using the mass action law.

3) Eliminate some variables based on the stoichiometric conservation laws, and use proper scaling for the kinetic equations and determine the minimal number of parameters.

4) Apply Elzaki transform method to calculate analytical approximate solutions of the reduced model.

5) Simulate the reduced model dynamics for different parameter values using Pplane8 for Mathlab. Analyse the reduced models to identify the critical model parameters by the local sensitivity analysis in numerical simulations using the SimBiology Toolbox for Mathlab.

6) Study the global behaviours of the reduced model at infinity.

The above steps can be simply presented in the following flowchart:

The flowchart of proposed steps of model analysis and simplification, the steps are presented in the order of their application.

We use Pplane8 for Mathlab to study the stability analysis of the Equation (12) and to compute numerical simulations in

Furthermore, we identify the model interacting and numerical simulations in two and three dimensional planes, for different values of

In addition, we calculate the local sensitivity of state variables

they are less sensitive to the given parameter

This section is devoted to examine the global phase portrait for Equation (12) by studying the isoclines and the behaviours at infinity. The isoclines are the lines with equal slope. These lines are an important role in sketching the phase portrait. It is easy to know where the trajectories have vertical and horizontal tangent lines by finding the

isoclines for

Here, we study the direction of trajectories in the quadrants. In the first quadrant,

since

The second required one is the line at infinity. It is a projective line that is added to the affine plane. Finding them including singular points and studying the behaviours at infinity of the reduced Equation (12) is very important to an understanding its global dynamics. For this purpose, the two below nonlinear change of variables are used individually.

Salih in [

Applying the nonlinear change of variables (20) on the reduced Equation (12) and after rescaling of the variables, the new system is obtained

The above system has two singular points

with a positive

with two negative eigenvalues

The system below is obtained by applying the nonlinear change of variables Equation (21) on Equation (12) after rescaling of variables

System (25) has only two singular points

which it has one negative eigenvalue

which it has two positive eigenvalues

Mathematical presentations and numerical simulations of infectious disease models are crucial topics in systems biology. Describing the dynamics of such systems often requires some techniques of model analysis. We studied an epidemic disease model called SIR model with three species and two parameters. We proposed a number of steps of

model analysis. The suggested steps of model analysis here importantly play in reducing the number of elements and in calculating analytical approximate solutions of the SIR model. The simplified model helps to study the full model of SIR in different ways. Firstly, identifying the critical model parameters of the reduced model becomes much easier compared to the full model. Secondly, the mathematical representation of the reduced model can help to integrate experimental knowledge into a coherent picture. Furthermore, studying the behaviour at infinity of the reduced model helped us to understand its global dynamics and to draw the global phase portrait of the system. Finally, the reduced model could be accurate, robust, and applied by biologists for various purposes. The proposed techniques here of model analysis will be applied to a wide range of complex infectious disease models in systems biology.

We thank the Editor and the referee for their comments. We also thank both Raparin and Koya universities for providing us with wonderful facilities. This support is greatly appreciated.

Khoshnaw, S.H.A., Mohammad, N.A. and Salih, R.H. (2017) Identifying Critical Parameters in SIR Model for Spread of Disease. Open Journal of Modelling and Simulation, 5, 32-46. http://dx.doi.org/10.4236/ojmsi.2017.51003