Mathematical Model for the Control of Infectious Disease

: We proposed a mathematical model of infectious disease dynamics. The model is a system of first order ordinary differential equations. The population is partitioned into three compartments of Susceptible ) ( t S , Infected ) ( t I and Recovered ) ( t R . Two equilibria states exist: the disease-free equilibrium which is locally asymptotically stable if R o < 1 and unstable if R o > 1. Numerical simulation of the model shows that an increase in vaccination leads to low disease prevalence in a population.

Mathematical models are widely used to examine, explain and predict the dynamics of infectious disease transmission and models of specific diseases of global importance have played important role in developing public health strategies for control and prevention of infectious disease. (Anderson RM and May RM. 1991), (Grassly NC, Fraser C. 2008) and (Keeling M, Rohani P. 2007).
Infectious disease models provide a mathematical representation of the dynamic transmission cycle involving interactions between infected and susceptible individuals that are generally expressed as a set of Ordinary differential equations (ODEs). (Keeling M, Rohani P. 2007). Infectious diseases includes malaria, tuberculosis, cholera, AIDS, bird flu, lassa fever, ebola and could be transmitted through direct or indirect contact with contaminated body fluid or surface most especially through sex, blood transfusion, breast feeding, etc. (Shah and Gupta, 2013). Some examples on the use of mathematical model for the analyzes of treatment and control of infectious disease can be found in ( Shah, N.H. and Gupta, J. 2013), (Seidu, B. and Makinde, O.D. (2014)., (Tripathi, A., et al 2007). The aim of this paper is to design and rigorously analyze a model that extends and complements the ones in the literatures by incorporating vaccination as control parameter

MATERIALS AND METHOD.
We formulate A non-linear deterministic model for the transmission dynamics of infectious diseases .The model subdivides the human population into three (3) compartments depending on the epidemiological status of individuals. The compartments are susceptible S, infected I and the recovered R. To indicate this mathematically, we have ( ) The following assumptions were considered to construct the model 1 All susceptible individuals can be infected through a direct contact with infectious individuals 2.
Individuals are only recruited into the susceptible class.

3.
Birth rate is not equal to death rate.

4.
We assume that susceptible individuals get infected with typhoid at a rate proportional to the susceptible population.

5.
All parameters are non-negative.

Disease-Free Equilibrium State (DFE):
The equilibrium state in the absence of infection is known as the disease-free equilibrium and is such that, y = 0,.
Therefore the Disease Free equilibrium is  Since all the parameters are positive then S > 0. Following the same method, we can easily verify that I(t) > 0 and R(t) > 0 for all t > 0. Hence, the solution S, I, R of the model (1) The Jacobian matrix of the system of equations is given by: The Jacobian Matrix of the system (1) -(3) at the disease free equilibrium point E0 is evaluated as follows:

RESULTS AND DISCUSSION
Numerical simulations were carried out to graphically illustrate the long term effect of protection on the dynamics of the infectious disease The Table below shows the set of parameter values and the state variables which were used. In order to support the analytical results, graphical representations showing the time graphs of different state variables are provided.  Figure 1 is the diagram showing the dynamics of the susceptible population. The Susceptible population decreases as time increases. This decrease may be possibly because of the high rate of recovery due to vaccination, the contact rate also has large impact on the spread of a disease through a population. The higher the rates of contact, the more rapid the spread of the disease. Fig. 2 shows the graph of infective against time. With high success of vaccination, there is low contact rate and low prevalence rate hence the infected population decreases over time. With low vaccination there is high contact rate and hence a high disease prevalence in the population. This rapid decline of the infected individuals may be due to early detection of the disease In Figure 3 it was observed that as the rate increases, the population of recovered individuals shows some rapid increase due to earlier treatment Conclusion: We conclude that effective control of infectious disease by vaccination will prevents rapid progression to infection especially in scarce resource setting where treatment is not readily available.
Vaccination is effective to prevent disease induced death rate, receiving proper treatment for those already infected among others. Death due to infectious disease decreases with an increase in vaccination because there will be a decrease in the number of infected individuals.