Kolera adalah penyakit infeksi akut yang disebabkan oleh bakteri <em>Vibrio Cholerae</em> yang masuk ke dalam tubuh melalui makanan dan minuman yang dikonsumsi oleh penderita. Penelitian ini bertujuan untuk melihat penyebaran penyakit kolera dengan menggunakan model stokastik. Pada penelitian ini dikaji dua model yaitu, Pendekatan Stokastik Model Wang dan Modnak dengan populasi konstan dan pada populasi tidak konstan untuk menganalisis peluang wabah. Peluang wabah diperoleh dengan menggunakan <em>probability generating function</em> (pgf) keturunan proses bercabang. Simulasi dilakukan untuk melihat perilaku model populasi konstan dan tidak konstan.  Hasil dari penelitian ini adalah dalam model stokastik, ketika bilangan reproduksi dasar lebih besar dari satu  masih ada kemungkinan menuju peluang bebas penyakit. Selain itu, tidak ada perbedaan  antara model populasi konstan dan tidak konstan terhadap peluang bebas penyakit.

This article discusses SEIRS-SEI epidemic models on malaria with regard to human recovery rate. SEIRS-SEI in this model is an abbreviation of the population class used in the model, i.e Susceptible, Exposed, Infected, and Recovered populations in humans and Susceptible, Exposed, and Infected populations in mosquito. These epidemic models belong to mathematical models which clarify a phenomenon of epidemic transmission of malaria by observing the human recovery rate after being infected and susceptible. Human population falls into four classes, namely susceptible humans, exposed humans, infected humans, and recovered humans. Meanwhile, mosquito population serving as vectors of the disease is divided into three classes, including susceptible mosquitoes, exposed mosquitoes, and infected mosquitoes. Such models are termed SEIRS-SEI epidemic models. Analytical discussion covers model formation, existence and stability of equilibrium points, as well as numerical simulation to find out the influence of human recovery rate on population dynamics of both species. The results show that the fixed point without disease ( ) is stable in condition ℛ 0 < 1and unstable in condition ℛ 0 > 1. The simulation results show that the given treatment has an influence on the dynamics of the human population and mosquitoes. If the human recovery rate from the infected state becomes susceptible to increased, then the number of infected populations of both species will decrease. As a result, the disease will not spread and within a certain time will disappear from the population.

In this paper developed a mathematical model of the spread of dengue hemorrhagic fever (DHF) SIR type, where SIR is an abbreviation of susceptible (S), infected (I) and recovered (R). Results of analysis and simulation obtained two fixed points, namely the disease-free quilibrium and endemic equilibrium. Human population, mosquitoes and mosquito eggs stable around the disease-free quilibrium when ℛ 0 < 1 and stable around the endemic equilibrium point when ℛ 0 > 1. Increased of mosquitoes mortality rate can reduce the value of the basic reproduction number.

AbstrakPada artikel ini dijelaskan model persamaan diferensial nonlinear mangsa pemangsa Leslie Gower dengan waktu tunda pada mangsa dan pemangsa. Berdasarkan hasil analisis diperoleh empat titik tetap, satu di antaranya bersifat stabil dan tiga lainnya tidak stabil pada saat nilai = 0 (tanpa waktu tunda). Waktu tunda kritis ( 0 ) adalah nilai batas yang menyebabkan perubahan kestabilan. Simulasi numerik dibagi menjadi tiga kasus, yakni ketika nilai = 0 ( tanpa waktu tunda ) bersifat stabil, < 0 bersifat stabil dan saat nilai > 0 bersifat tidak stabil. Dari hasil simulasi saat nilai > 0 bersifat tidak stabil hal ini disebabkan karena terjadi bifurkasi pada model tersebut, titik tetap yang awalnya bersifat stabil menjadi tidak stabil.

In this paper, we study an SVEIR disease model of tuberculosis transmission dynamics in which the infected population is divided into two groups, namely infectious infected and noninfectious infected population. The equilibrium points and the basic reproduction number R 0 are determined. The stability analysis of the model was conducted by considering the basic reproduction number R 0. We show that if R 0 < 1, then the disease-free equilibrium is locally asymptotically stable. If R 0 > 1, then a unique endemic equilibrium is locally asymptotically stable.