Effect of Population Density on the Model of the Spread of Measles

Abstract


INTRODUCTION
Environmental factors can affect the spread of an illness so that they can affect public health. Dangerous diseases include infectious diseases. One of the problems of health in Indonesia is contagious diseases. Indonesia is a tropical island country because it is located in the equator area. Indonesia has a very high environmental temperature, so that this temperature is ideal for growing a microorganism [1]. These microorganisms are very small-sized viruses that can be viewed using microscope tools [2]. Measles disease is one of the problems in Indonesia. Measles disease is a contagious disease caused by Paramyxovirus. Efforts to prevent its spread can be made by giving vaccines to susceptible individuals infected with measles.
In the human body and healthy animal's inserted bacteria or pathogenic viruses that have been weakened is the meaning of vaccination. Vaccinations are expected to form an immune system to combat these pathogenic bacteria or viruses. The problem of disease spread phenomenon is often modeled in the form of mathematical equations, including SIR and SIRS models [3] [4]. Individuals who are faced can be assumed as new sub-populations in the SIR model. The addition of the V (Vaccination) sub-population of the SIR epidemic model indicates the number of individuals who have experienced the vaccination process, so one of the development models of the SIR epidemic model is the SVIR epidemic model. The population is divided into four subpopulations, S (Susceptible), V (Vaccination), I (Infected), and R (Recovered) [5]. The number of individuals susceptible to disease is denoted by S (Susceptible), the METHODS This study was conducted with a mathematical approach that refers to several references (literary studies). The stages in this study are as follows: Step 1 : Article collection and relevant information related to measles disease (literature study), Step 2 : Determination of assumptions as a reference for the restriction of problems and the process of drafting a measles epidemic model with various facts in a region.
Step 3 : Formulation of problems with mathematical descriptions of models and studies of the model.
Step 4 : Create the numerical simulation and its interpretation.

RESULTS AND DISCUSSION
Populations can be divided into four subpopulations, such as the susceptible subpopulation (S), sub-populations that are vaccinated (V), sub-population infected (I), and sub-population recovered (R). The size used for each of these sub-populations in this model is proportions. The proportion of each subpopulation will undoubtedly change or depend on time. In other words, the balance of each subpopulation is a function of time variables. Variable time is denoted by t. To solve this problem mathematically, letting S (t) is a proportion of the susceptible subpopulation, V (t) is a proportion of subpopulation that is vaccinated, I (t) is a proportion of sub-population infected, and R (t) is the proportion of sub-population recovered at the time t. It is assumed that S (t), V (t), I (t), and R (t) is differentiable of time. This model is formed by adding one parameter as a wide-sized area of the disease of measles populated by the population. These parameters are used to help analyze the population density dependence of the spread of measles disease dynamics. The phenomenon of spreading measles that is modeled need to be limited and defined by the following assumptions: 1. The population is not constant, and the community is not closed, 2. The birth rate is the same for each subpopulation, 3. Individuals who are born in each subpopulation enter into susceptible sub-population, 4. Birth rate equals death rate, 5. Types of disease is a deadly disease, 6. The disease can be permanently recovered, 7. The incubation period is relatively short, 8. The vaccine is given to someone suspect of measles, 9. Immunity to the disease is obtained over time, resulting in a permanent recovered, 10. A person who has been given a vaccine can be infected with measles before gaining immunity, 11. Contamination between infected people with susceptible people leads to the transmission of measles, 12. The distribution of homogeneous populations across the region is not broad.
From the assumptions outlined above, the following are given explanations of some of the notation used in the model. is positive for all , 9) = Natural birth rate with is positive, 10) = Transmission rate between infected sub-populations with susceptible sub-population, with is positive, 11) = Transmission rate between infected sub-populations with vaccinated sub-population, with is positive, 12) = Recovery rate of infectious measles, with is positive, 13) = Recovery rate after being given a vaccine, with is positive, 14) = Vaccine rate on susceptible sub-population, with is positive, 15) Death rate of Infectious measles, with is positive, 16) = area of human-populated, with is positive.
The following is given a transmission diagram of the SVIR epidemic model of measles disease. Based on the SVIR transfer diagram with the assumptions given, the following are given SVIR models in the form of a differential equation system. with initial conditions for and for all non-negative .

Equilibria and Basic Reproduction Number
Because the last equation does not affect the other equation, then the above system can be eliminated to: Furthermore, the local stability analysis of the equilibria of the SVIR model is analyzed in the neighbourhood of the equilibria. The initial step is to determine the equilibrium point of the system (2). It is mentioned that the dynamics of the SVIR model has two equilibrium points i.e. the non-endemic and endemic equilibrium point [13]. The non-endemic equilibrium point is a representation of a state without being infected by an infectious environment. While the endemic equilibrium point is a representation of some people in the environment still infected with disease. It means that the disease will even spread because there are still several infected people (I is positive) for in an environment. The equilibria of the system (2) Then we get: Where we have the roots of the equation (6), i. e.
Clearly that such that must hold. So, we get the endemic equilibrium i.e.

With
and is the positive root of equation (6).
Basic reproduction number be a condition for the existence of the endemic equilibrium point of the system (2). Basic reproduction number can be obtained by specifying the positivity condition of the proportion of sub-population infected.
Reconsider equation (6); the endemic equilibrium point is a positive root because of the equibrium point is the proportion of sub-populations that in real life is positive. Positive roots are only hold when K>1. Thus, it was concluded that the existence of the endemic equilibrium point depends on the value of K, therefore, the K parameter can be defined as basic reproduction number, i. e.: Clearly that the existence of the non-endemic equilibrium points does not depend on . Reviewed from the can be concluded that if then there exists a unique equilibrium point of system (2), that isthe non-endemic equilibrium point denoted by . However, if , then there are two equilibrium point of system (2), that is and the endemic equilibrium point denoted by . We obtain that , with is positive root of the following

Stability Analysis
The result of local stability analysis of the equilibrium point of the system (2) is given in the following theorems. Theorem 1. Define 1) If the basic reproduction number less than one , then the non-endemic equilibrium point is locally asymptotically stable. However, if the basic reproduction number more than one , then the endemic equilibrium point is not stable. 2) The endemic equilibrium point islocally asymptotically stable, if the basic reproduction number more than one . Proof. 1) The following is Jacobian matrix (linearization method) of system (2) Hence Jacobian matrix in is: The characteristic equation of can be written as where: < 0 < 0 = Clearly that all Eigen value of are negative when . Consequently, is locally asymptotically stable [21]. However, when there exists positive Eigen value of hence is not stable.
1) Obviously, when , we obtained that hence Jacobian matrix around can be written as: The element of ekuivalent with and the element entri of ekuivalent with . We know that hence can be written as: The characteristic equation of , that is where: We have Hence according to Routh-Hurwitz criterion [22], dan are hold, then all the Eigen values of have negative real parts. So, is locally asymptotically stable [21]. Therefore, the proof is complete. According to the results that have been obtained, the interpretation is as follows: 1. If and the initial condition at the neighborhood of , then for , the solution from the system (1) will move towards the . This means that if , then for the number of susceptible, vaccinated, and infected individuals approaching , Then the disease does not spread, and the number of infected sub-populations decreases towards zero for an indefinite period. It means that measles disease tends to disappear in disease. It is called stable asymptotic local around the equilibrium point . 2. If and the initial condition at the neighborhood of , then for , the solution from the system (1) will move towards the . This means that if , then Measles disease still remains in the environment for an infinite time with the value S, V, I approaching . It is called stable asymptotic local around the equilibrium point .
Interpretation of analysis results is similar to the discussion in [13], [14], [15], [16], [17], [18], [19] and [20] that the primary reproduction number plays an important role to know the dynamics of the spread of disease. However, every primary reproduction number obtained varies depending on the parameters of the model formed. In this discussion, the population density factor influences the primary reproduction number. It can be seen from the presence of the population density parameter that appears on the primary reproduction number.

Numerical Results
The following solutions are provided from the system (1) with a numerical approach. The approach is numerically given in the form of graphs of sub-population susceptible, vaccinated, infected, and recovered versus time. All parameter values used are estimates. The stability of the equilibrium point of the system (1) is noted based on the value, which is and . In this simulation, three broadsize areas are used to review population density influence each subpopulation's proportion. The assumed area size is 20 km 2 , 200 km 2, and 2000 km 2 .
Assumed parameter values for the case , with The solution of the System (1) with the given parameters presented in Figure 2. For a total area of 20 km 2 , within 0 to 5 months the proportion of the infected sub-population increased above 0.1 then after 5 months decreased to 0. These results are shown in Figure 2 below. The graph in Figure 3 shows a total area of 200 km 2 , the proportion of sub-populations infected from the initial state of 0.1 decreased to 0. The chart also applies to an area of 2000 km 2 ; the infected population has decreased to 0. The three-area comparison of the region shows that the area is an influential factor in the dynamic spread of measles disease. The more dense the area then the proportion of the infected subpopulation will increase to a specific time limit. Further, the proportion of sub-population infected decreases until the spread will stop for an infinite time.

CONCLUSIONS
The analysis results of this model indicate that there are two case value, which is and . Each case of the value determines the local stability analysis of the equilibrium point of the model. Furthermore, the parameter is a necessary condition for the existence of equilibrium point and stability analysis of the model. When , there only a unique equilibrium point called the non-endemic equilibrium point denoted by . However, when there are two equilibrium point, namely and the endemic equilibrium point denoted by The results of local stability analysis show that when , then the non-endemic equilibrium point is locally asymptotically stable. This means that if hold, then measles disease does not spread, and the number of infected subpopulations decreases towards zero. This means that the spread of measles disease will stop. However, is locally asymptotically stable when . It This means that if , then measles disease is still in the environment for an infinite time. The proportion of each sub-population in this situation is dan .The area inhabited by the population is an influential factor in the dynamic spread of measles disease. The more dense the residential areas of the population then the proportion of infected sub populations will increase to a certain time limit. Furthermore, the proportion of sub-population is infected down until the spread will stop for infinite time. In the area above 200 km 2 , the proportion of the infected sub-population will continue to decline even the spread will stop for an infinite time.