The behaviour of measles transmission in three different populations

SIR Model can be employed to model the transmission of either fatal or non-fatal disease within a closed population based on certain assumptions. In this paper, the behaviour of non-fatal diseases transmission model is observed from three types of population, that is (i) increasing population, (ii) constant population, (iii) decreasing population. This paper acquired two equilibria, i.e, the disease-free equilibrium point and the endemic equilibrium point At the disease-free equilibrium, the behaviour of the model is stable when while at the endemic equilibrium, its behaviour is stable for any positive parameters and .


Introduction
Each year through surveillance activities reported more than 11,000 cases of measles suspects and results laboratory confirmation shows 12-39% of which are measles for sure (lab confirmed) while 16-43% is definite rubella [1]. From 2010 to 2015, it is estimated 23,164 cases of measles and 30,463 cases of rubella. The number of cases is estimated to be lower compared to actual figures in the field, given the number of unreported cases, especially from private health services and the completeness of surveillance reports that are still low.
The mathematical model becomes a powerful tool that eases the mathematization of the real-life problem using certain criteria and assumptions [2]. In the next step, the solution to the governing equation can be addressed, both analytical and numerical solutions.
One of the real-world problems is the transmission of a non-fatal disease i.e. measles, influenza, and the others. Over the years, childhood diseases are the most common form of infectious diseases. Since children are in particularly close contact with their peers, such diseases can spread quickly. The infected individuals will have immunity in a certain period. This transmission problem then converted to a mathematical SIR Model.
SIR Model which stands for Susceptible, Infective, Recovered originally developed to describe the transmission and extinction of an epidemic disease outbreak in the closed population. In this paper, we discussed the establishment of a non-fatal SIR model based on the assumptions. Once the model is formed, the analytical solutions and the equilibrium point is searched, which further interpreted, the behaviour of the disease and its existence, which is disease-free equilibrium and endemic equilibrium.
The stability of measles transmission model with and without vaccination at constant population was studied by Prawoto in 2017. In term of population with vaccination, there are and as the stationary points, while in population without vaccination, there are and as the stationary points [3]. Prawoto had analysed that the model is stable around hold by , and it would be stable around when . In the second population, the model would be stable around and when and respectively.

SIR Model
Non-fatal disease transmission, such as influenza, measles, etc, in a population, is assumed to have a fixed amount and a period of the outbreak. For example, at a certain time , in a population consisting of:  , susceptible: a subpopulation of those members consists of people who are susceptible to the disease.  , infective: a subpopulation of those members consists of people who have contracted the disease.  , recovered: people who have been cured of the disease. With , where is the number of total population, and the proportion of The compartment we put together is as follows: The mathematical model of the diagram above is. (1) is the number of vulnerable populations the disease is the number of the population is infected is a population recovering from illness is the rate of natality is the rate of mortality at is the rate of mortality in is the rate of mortality in is the rate of disease transmission is the rate of recovery To obtain the equilibrium of system [1], the right-hand side must equal zero. (2) Since the last equation has no effect on the two previous equations, then it can be ignored first. We obtained two equilibria which are:

And
To do the stability analysis of system [1], a linearization is needed because the system [1] is not a linear system. This process allows us to zoom in the behaviour of the model around the equilibria. For instance,

Then (4)
The Jacobian matrix is (5) For the equilibrium the Jacobian matrix is given: , is Identity Matrix of .
We obtained or .
To make the system is stable, it should be is negative due to , for then [4].
Taking and , we obtained the equilibrium . Simulation case 2 (the natality is equal to the mortality) (Fig. 2).
Taking and , we obtained the equilibrium Simulation case 3 (the natality is higher than the mortality) (Fig. 3) Taking and .
we obtained the equilibrium . The figure 1 to figure 3 of the simulation is given respectively as follows: . We acquired To make the system is stable, it should be or will be all negative when . When then are complex numbers and the real part is negative , then for all positive parameter and the system is stable. Simulation case 1 (the natality is smaller than the mortality) (Fig. 4) Taking and , we obtained the equilibrium . Simulation case 2 (the natality is equal to the mortality) (Fig. 5) Taking and , we obtained the equilibrium Simulation case 3 (the natality is higher than the mortality) (Fig. 6) Taking and , we obtained the equilibrium . The figure 4 to figure 6 of the simulation is given respectively as follows:

Conclusion
It is acquired two equilibria, that are: and . The first equilibrium is called a disease-free equilibrium because the infected population is 0. The second equilibrium is called endemic equilibrium because the population is infected with the positive. The stability of the disease-free equilibrium occurs if While the stability of the endemic equilibrium holds for all the positive parameter values and .