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

This study is expected to contribute to the health sector, specifically to describe the dynamics of the measles spread through the models that have been analyzed. One of the factors that became the focus of this study was reviewing the influence of population density on measles spread. The initial step formulated the model and then determined the primary reproduction number  and analyzed the stability of the model equilibrium point. The results of the analysis of this model show that there are two conditions for the value of  which is a requirement that the existence of two model equilibrium points as well as local stability is needed, namely  and .  When , there exists a unique equilibrium point, called the non-endemic equilibrium point denoted by . Conversely, when , there are two equilibrium points, namely  and the endemic equilibrium point characterized by . The results of local stability analysis show that when , the equilibrium point  is stable asymptotic locally. It means that if  hold, then in a long time there will not be a spread of disease in the susceptible and vaccinated sub-population, or in other words, the outbreak of the disease will stop. Conversely, when  equilibrium point is stable asymptotic locally. It means that if , then measles disease is still in the environment for an infinite time with the condition of the proportions of each sub-population approach to , ,  and .