Nonstandard finite difference scheme for control of measles epidemiology

This paper is based on the analysis of SEIR measles models, which are used to study the integrating vaccination as a control strategy and taking the two stages of infectiousness and transmission dynamics of infectious diseases in a population. Measles is a higher contagious that can spread in a community population depending on the number of people susceptible or infected and also depending on their movement in a community. We construct an unconditionally convergent nonstandard finite difference (NSFD) scheme for SEIR measles model. NSFD preserve the positivity of all values of h. This method proved to be a very efficient technique for solving epidemic models. We obtained disease-free equilibrium (DFE) point, Endemic equilibrium (EE), reproduction number for the model. Moreover, the analysis of the epidemic models using nonstandard finite difference scheme reveals that the method provides a rapidly convergent series solution by little iteration and avoids the massive computational work. Numerical simulations show that the rate of infection is decreased with the passage of time and disease will die out in the community. The results are compared to the Differential Transformation Method to show this scheme is efficient and better accuracy for epidemic models.


Introduction
*In human population epidemiology studies play an important role to understand the disease in human population. The research of this kind help to understand the ratio of disease spread in the population and to control their parameters (Grenfell, 1992). 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. Rubella 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 Cs-cough. Symptoms appear after the initial infection about 9-11 days. Virus grows in lymphatic system and lungs after entering the body and. Blood vessels, central nervous system, urinary tract, imitates in the eyes are chance to be infected. According to the experts, the virus created itself within 1-3 weeks. Measles reinfection are very rare. Measles is an air borne disease so the ratio of 90 percent peoples who haven't got the immunity against measles cause to spread this disease from one person to another person if they live together at the same house (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, 2003). It produces infection of eyes, bronchitis, vomiting, bronchitis is inflammation of the inner walls of airways and laryngitis is inflammation of the voice box.
In recent study the differential transform method is the one of the most effective and efficient numerical techniques for solving system of linear and non-linear differential equations, integral and Integro differential equations. Till now this method solve various type of linear, non-linear differential equations as well as integral equations (Pukhov, 1978). DTM provides an efficient numerical and explicit solution with least calculations and high accuracy. DTM is very effective technique because it gives well approximately solution in a very small region (Pukhov, 1981;1982;. Now we discuss nonstandard finite difference scheme (NSFD) which works correctly for large time step size. In recent years, construction of NSFD discrete models has been tested for a wide range on nonlinear dynamical systems (Moaddy et al., 2011;Dimitrov and Kojouharov, 2005;Taylor et al., 1986).
The NSFD have been widely used for system of differential equations that are describing problems in mathematical biology and the other different areas. These Method showed the superiority in preserving the passivity (Jang, 2005;Gumel et al., 2005;Mounim and de Dormale, 2004;Mickens, 2003) when comparing all others well known numerical method of state variable of the system under study.
In this paper, we design and analyze a NSFD for mathematical model of the transmission of measles in the society. The dynamics of this model are studied using the qualitative theory of dynamical system. This method also preserves positivity of the solution which is one of the essential requirement when modeling epidemic disease. Comparison also made with other conventional approaches that are routinely used for such problems.

Material and method
To describe the transmission dynamics of measles we formulate a deterministic, compartmental, mathematical model. In order to describe the model equations, the total population (N) is divided into four classes: S, E, I, R represented as Susceptible, Exposed, infected and recovered population respectively. We divided our compartmental model into four compartment as shown in flow chart given in Fig. 1. The susceptible population which are not infected and are able to catch the disease and able to transmit others is in S class with infection contact rate βand increased by birth or immigration at a rate B with mortality rate μ (i.e., natural death rate). The compartment E shows the class of those peoples which are exposed but not able to transfer the disease to another individuals in the period of incubation. The class E is break through into infected class at a rate α, decreased by testing and measles therapy at a rate σ and moderated at a rate μ which is natural death rate. The compartment I which is class of infected peoples is decreased at the infection rate γ and decreased at the mortality rate μ. In this model we assumes that both recovered and infected peoples and recovered , exposed peoples have got the permanently immune from the disease where R is the compartment of recovered peoples reduced with mortality rate μ (Momoh et al., 2013).
Therefore, the flow chart of deterministic measles model is shown in Fig. 1.
We will assume the total population is constant for size N . ., + + + = .
The system is qualitatively analyzed by two ways i.e. disease Free Equilibrium and endemic Equilibrium.

Disease free equilibrium
Firstly we normalize the model by dividing by N for Eqs. 1-4.
When naturally, the disease die out then the solution of the above system asymptotically approaches a disease free population or equilibrium is of the form = , = 0 , = 0 . ( , , , ) = ( ,0,0,0)

Endemic equilibrium
It means that disease never dies out. When disease free equilibrium is unstable that is disease persist in the population then Endemic equilibrium takes the form:

Reproductive number
The reproductive number is: So this is disease free equilibrium.

Stability analysis
We evaluate the Jacobian matrix to examine the local stability of the disease free equilibrium.

Differential transform method
This method is transformation technique based on Taylor series expansion (Pukhov, 1978;1981;1982;) offers a convenient means for obtaining analytic solutions of differential equations. Zhou was the first man who introduced this technique in 1986 (Zhou, 1986), and is commonly used for the solution of electric circuit problems, it seems to be largely unknown to the research community. After that Pukhov (1986) worked in this technique and further developed these techniques for partial differential equations. This is a semi analytic technique and the solution of this technique is in the form of series also this method give closed form solution unlike other numerical methods.
here, ( ) = Original function and ( ) =Transformed function. The Inverse of differential Transformation is defined as: when 0 is taken as zero, then the above original function ( ) will be defined in the form of finite series and above function can be expressed in the form as: Using the fundamental result of DTM we obtained the following equation. From the above equation we can see that the basic idea of DTM is based on Taylor series.
The inverse differential transform of S(k) is defined as: when 0 is taken as zero, the given function y(x) is declared by a finite series and above equation can be written in the form . By solving the above equation for ( + 1), ( + 1), ( + 1) and ( + 1) up to order 5, we get the function of ( ), ( ), ( ) and ( ) respectively (20)

Nonstandard finite difference scheme (NSFD) for SEIR model
In this section, we design the NSFD scheme that replicates the dynamics of continuous model (1) The discrete method (13 -16) is constructed by using Mickens rules (Akinboro et al., 2014) which was formalized by Anguelov and Lubuma (2001).

Analysis of the scheme
Theorem 4: The NSFD scheme (5 -8) is a dynamical system on the biological feasible domain of the continuous model (13 -16).
Proof: First we prove the positivity (13-16). For this we take the explicit form.

Results and discussion
Case 1: For numerical results, we used the followings Table 1 values of parameters are considered from Momoh et al. (2013).    Graphs of S(t) by using NSFD For case 1 are shown in Figs. 8-10. Fig. 8 shows that the relation between susceptible and time in a year and Fig. 9 shows the relation between exposed and time in a year also Fig. 10 shows the relation between infected and time in a year. Graphs of S(t) by using NSFD For case 2 are shown in Figs. 11-13. Fig. 11 shows that the relation between susceptible and time in a year and Fig. 12 shows the relation between exposed and time in a year also Fig. 13 shows the relation between infected and time in a year.

Conclusion
Many researchers apply many numerical techniques to solve many mathematical models. To analyze SEIR measles model, we use two different techniques Differential transformation method and Non Standard finite difference scheme. After comparison our results we found that rather DTM is very efficient technique, is really worth, offers promoting approach and easy to implement also reliable for complicated systems of linear and nonlinear equations. Secondly we construct an unconditionally convergent nonstandard finite difference (NSFD) scheme for SEIR measles model. NSFD preserve the positivity of all values of h (step size).
The NSFD scheme is dynamically consistent, easy to implement and show a good accidence with analytical consequences obtained by dynamics analysis of the model. This method proved to be very efficient technique for solving epidemic models. To analyze SEIR measles model, we use two different cases of measles therapy rate and plot the S, E, I against time in years. In each case, we discussed two different cases for the different values of measles therapy rate to analyze for what reason the exposed peoples are effected. We conclude that, in high measles prevalence countries, testing diagnosis and exposed individuals at latent period therapy will have a much greater impact on the disease burden. By the isolation of the infections individuals from the other ones, the spread of disease into a population can be controlled as the therapy rate will be decreased. For this purpose, appropriate measures should be taken so that the interaction between infectious and susceptible children should be minimized. Moreover, the analysis of epidemic models using nonstandard finite difference scheme reveals that, the method provides rapidly convergent series solution by little iteration and avoids the massive computational work.