Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis

Article history: Received 31 July 2017 Received in revised form 30 October 2017 Accepted 25 November 2017 Protection of children from vaccine preventable diseases, such as measles is among primary goal for health worker. Measles is a highly contagious disease that can spread in a population depending on the number of peoples susceptible or infected and also depending on their dynamics in the community. The model monitors the temporal dynamics of a childhood disease in the presence of preventive vaccine. We presented a nonlinear time fractional model of measles in order to understand the outbreaks of this epidemic disease. The Caputo fractional derivative operator of order α ∈ (0,1] is employed to obtain the system of fractional differential equations. The numerical solution of the time fractional model has been procured by employing Laplace Adomian decomposition method (LADM), qualitative and sensitivity analysis of the model was performed. Qualitative results shows that the model has endemic equilibrium which locally asymptotically stable for R0 > 1 and otherwise unstable. The convergence analysis and nonnegative solutions are verified for the proposed scheme. Simulation of different epidemiological classes at the effect of fractional parameter α revealed that most individuals undergoing treatment join the recovered class. This method proves to be very efficient techniques for solving epidemic model to control infectious disease.


Introduction
* Epidemiological study plays an important role to understand the impact of infectious disease in a community. In mathematical modeling, we investigate models by model building, perform estimation of parameters, check sensitivity of models by varying parameters and compute their numerical simulations (Panum, 1988). The research of this kind helps to understand the ratio of disease spread in the population and to control their parameters (Grenfell, 1992;Abubakar et al., 2012). These types of diseased models are often called infectious diseases (i.e., the disease which transferred from one person to another person). Measles, rubella, chicken pox, mumps, aids and gonorrhea syphilis are the examples of infectious disease (WHO, 2011). Rubeola virus is highly infectious illness which is also known as morbilli or measles. The virus can be found in the mucus of the throat, nose of an infected adult and child. Measles symptoms caused by Rubeola virus always included fever, coryza (runny nose), conjunctivitis and at least one of the three Cscough. Symptoms appear after the initial infection about 9-11 days (Ochoche and Gweryina, 2014). Complications of measles are fairly common but the patients have weak immune system are more likely to be worse such as those with HIV/AIDS or leukemia and those with vitamin deficiency. Healthy children over the age of 5 are less likely to have complications than adults over the age of 20. It is the first and worst eruptive fever occurs during childhood (Hethcote, 2000;Murray, 2002). It produces eye infection, bronchitis, laryngitis and vomiting, bronchitis is inflammation of the inner walls of airways and laryngitis is inflammation of the voice box.
Since in recent years fractional calculus has attracted great attentions from researchers and different aspects of the said subject is under consideration for research. This is due to the fact that fractional derivative is important tool to explain the dynamical behavior of various physical systems. The strength of this differential operator is their nonlocal characteristics which do not exist in the integer order differential operators. The distinguished features of fractional differential equations are that it outlines memory and transmitted properties of numerous mathematical models. As a fact, that fractional order models are more realistic and practical than the classical integer order models. Fractional order derivative produces greater degree of freedom in these models. Arbitrary order derivatives are powerful tools for the discretion of the dynamical behavior of various biomaterial and systems. The most iterating feature of these models is their global (nonlocal) characteristics which do not exist in the classical order models (Haq et al., 2017).
Laplace transform method is a useful technique in different field of biological science, engineering and applied mathematics. The coupling of ADM and Laplace transform leads to a powerful method known as Laplace Adomain decomposition method. With the help of Laplace transform, we convert a differential equation to an algebraic equation and the nonlinear terms are decomposed in terms of Adomain polynomials. The given numerical technique works powerfully for a system of deterministic as well as stochastic differential equations. More unambiguously, it can be used for classical as well as fractional order system of linear and nonlinear ordinary and partial differential equations. In this method, no perturbation or liberalization is required. Further it has no need of pre-defined step size like RK4. Also, this method does not depend upon on a parameter like needed for homotopy perturbation method (HPM) and homotopy analysis method (HAM). Although the solutions obtained via this method are the same as obtained by ADM, for detail see (Jafari et al., 2011a;2011b;Johnston et al., 2016). It is to be noted that LADM is powerful than standard ADM method (Haq et al., 2017). Kermack and McKendrick (1927) are inventor of the diseases models and played an important role in mathematical epidemiology to describe the transmission dynamics of measles formulate a deterministic and compartmental mathematical model. The population is consistently fraternization and replicates the demography of a typical emerging country, as it investigates an exponentially increasing dynamics. The total population (N) is divided into three class's s, i and r represented as susceptible, infected and recovered population respectively to describe in equations of the model (Akinboro et al., 2014).

Mathematical model
where is per capita removal rate, is transitivity, is per capita recovery rate. Since , and are interpreted rates, their ranges are 0 ≤ ≤ 1, 0 ≤ ≤ 1 and 0 ≤ ≤ 1 (Podlubny, 1999). The fractional order extension of this model has been first studied in (Podlubny, 1999;Arqub and El-Ajou, 2013). The aim of using fractional system of differential equations (FDEs) is naturally related to systems with memory effects which exist in most biological system that shows the realistic biphasic decline behavior of infection or diseases but at a slower rate. The new system is fractional differential equations (FDEs) which are described as follows.

Preliminaries
In this section, we give some fundamental results and definitions from fractional calculus. For detailed over view of the topic readers are referred to (Haq et al., 2017;Jafari et al., 2011a;2011b;Johnston et al., 2016).

Definition 1:
The Riemann-liouville fractional integration of order  is defined as The Riemann-Liouville derivative has certain disadvantages such that the fractional derivative of a constant is not zero. Therefore, we will make use of Caputo's definition owing to its convenience for initial conditions of the fractional differential equations.

Definition 3:
The definitions of Laplace transform of Caputo's derivative and Mittag-Leffler function in two arguments is written as

Mathematical analysis
The system is qualitatively analyzed by two ways i.e. disease Free Equilibrium and endemic Equilibrium. To evaluate the equilibrium point, we take Therefore the disease free equilibrium is The Equilibrium which is not disease free i.e., Endemic Equilibrium (EE). This state that the disease persists in a population and never dies out. If ≠ 0 then, Thus the endemic equilibrium state is given as: Theorem 3.1: The disease-free equilibrium 0 is locally asymptotically stable if 0 < 1 and is unstable if 0 > 1 Theorem 3.2: The endemic equilibrium state 1 = ( * , * , * ) of the model (4)- (5) is locally asymptotically stable if 0 > 1, otherwise unstable.

Reproductive number
In this system the threshold result of this equilibrium is 0 > 1, so this is in endemic state. Consider the Jacobean matrix as By using the relation | − | = 0 where = −1 and got the eigen value = + which represents the reproductive number 0 = +

Proof:
The uniqueness and existence for the solution of (4)-(6), in (0, ) can be obtained. Our aim is to show the domain 3 , ≥ 0 is positively invariant. Since The nonnegative solution satisfied the vector field points into + 3 .
Proof: For the system (4)-(6) the | − | = 0, then we get. < 0 or if is complex with negative real parts then system is stable.
Therefore ( ) < 0, since all the parameters are non-negative. So 1 is locally asymptotically stable. This proves the proposition.

Sensitivity analysis of
The sensitivity of 0 is as follows It can be seen that 0 is most sensitive to change in parameter, here, 0 is increasing with , and decreasing with , .

The Laplace-Adomian decomposition method
Consider the fractional-order epidemic model (4), (5) and (6) subject to the initial condition (7). The nonlinear term in this model is and , , are known constants. For 1 + 2 + 3 = 1 the fractional order model converts to the classical epidemic model. Applying the Laplace transform on both sides of (4), (5) and (6), we get by using the property of Laplace transform, we have by using the initial conditions (7), we get It should be assumed that method gives the solution as an infinite series The nonlinearity can be written as where is called the Adomian polynomials given as Substitute equations (26) and (27) in (23)-(25), we have the followings results Similarly, we have The purpose of the work is to analysis the mathematical behavior of the solution ( ), ( ), ( ) for the different values of . By taking the inverse Laplace transform on both sides of the equation (28), we get the values of 0 , 0 , 0 and used for further process. Putting the values of 0 , 0 , 0 and 0 into the equations (29), (30) and (31), get the values of ( ) = 0 + 1 + 2 + 3 + 4 … (32) ( ) = 0 + 1 + 2 + 3 + 4 ….

Numerical results and discussion
The numerical results of susceptible, infected and recovered population for = 1 , = 0.99 , = 0.95 where , , = 1, 2, 3 are established in Tables 2-4 by using LADM. For the reliable investigation, evaluation is made for different values of . From Figs. 1-3, we observe that fractional order SIR measles model has more degree of freedom as compared to ordinary derivatives. By taking noninteger values of fractional parameter, remarkable responses of the compartments of the proposed model are obtained. Another remarkable point to be considered that we used small interval of time because we have assumed comparatively small initial values. For large interval of time, the initial values to data are taken large so that the population may not be negative. For different values of solution converges to steady state and gives the better convergence by decreasing the fractional values of .

Convergence analysis
The obtained series solution is rapidly convergent and also converges uniformly to the exact solution. We use the classical techniques to verify the convergence of the series (35), (36) and (37) in (Shah et al., 2016). We check the condition of convergence of the method by using the idea of the following theorem (Abdelrazec and Pelinovsky, 2011;Naghipour and Manafian, 2015).

Conclusion
In this paper, we developed a scheme for analytical solution of epidemic fractional SIR model of measles by using Laplace Adomian decomposition method. The well-known epidemic model namely Susceptible-Infected-Recovered (SIR) is considered with and without demographic effects. The model represents population dynamics during the disease as a set of non-linear coupled ordinary differential equations. There is no exact solution available in the literature for this model up to the best of author's knowledge. It is observed that the infection rate and reproductive numbers play a key role for an epidemic to occur and the epidemic can be controlled by vaccination. It is also observed that to eliminate the disease, it is not necessary to vaccinate whole of the population. The efficiency and accuracy of the proposed scheme is provided by performing convergence analysis. The effect of fractional parameter on our obtained solutions is presented through Tables and graphs. It is worthy to observe that fractional derivatives show significant changes and memory effects as compared to ordinary derivatives.