Modeling the interaction between prey and predator plays an important role in maintaining the balance of the ecological system. In this paper, a discrete-time mathematical model is constructed via a forward Euler scheme, and then studied the dynamics of the model analytically and numerically. The analytical results show that the model has two fixed points, namely the origin and the interior points. The possible dynamical behaviors are shown analytically and demonstrated numerically using some phase portraits. We show numerically that the model has limit-cycles on its interior. This guarantees that there exists a condition where both prey and predator maintain their existence periodically.

Pneumonia is an infection or acute inflammation located in the lung tissue and is caused by several microorganisms, such as bacteria, viruses, parasites, fungi and even exposure to chemicals or physical damage. In this article, we discuss the SEIPRS mathematical model on the spread of pneumonia. The SEIPRS mathematical model is formed from five interacting populations, namely the Susceptible population is healthy individuals but susceptible to pneumonia which is denoted by S, the Exposed population is latent individuals or exposed to pneumonia which is denoted by E, the Infected population is individuals infected with pneumonia which is denoted by I, and the treatment population is infected individuals who are given treatment denoted by P, and the recovered population is the recovered population denoted by R. In this article, the search for equilibrium points in the SEIPRS mathematical model and stability analysis is carried out. The analysis in this model produces two equilibrium points, namely the equilibrium point without disease at the condition R0<1, the endemic equilibrium point R0>1, and the basic reproduction number (R0) as the threshold value for the spread of disease. In this study, simulations were carried out with variations in parameter values ​​to see population dynamics. Population results show that increasing rates of vaccination and treatment can reduce the rate of spread of pneumonia.