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

  • Joko Harianto Universitas Cenderawasih
  • Katarina Lodia Tuturop Cenderawasih University, Indonesia
  • Venthy Angelika Cenderawasih University, Indonesia
Keywords: Measles Model; Population Density; Stability Analysis

Abstract

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 .

References

Soedarto, Penyakit Menular di Indonesia. Jakarta: CV. Sagung Seto, 2009.

R. Maksum, Imunologi dan Virologi, 1st ed. Jakarta: PT. ISFI, 2010.

W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics-I,” Bull. Math. Biol., 1991.

H. W. Hethcote, “Mathematics of Infectious Diseases,” SIAM Rev., 2000.

M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel, and B. M. Sahai, “A Vaccination Model for Transmission Dynamics of Influenza,” SIAM J. Appl. Dyn. Syst., 2004.

R. Peralta, C. Vargas-De-León, and P. Miramontes, “Global Stability Results in a SVIR Epidemic Model with Immunity Loss Rate Depending on the Vaccine-Age,” Abstr. Appl. Anal., 2015.

S. Islam, “Equilibriums and Stability of an SVIR Epidemic Model,” Best Int. J. Humanit. Arts, Med. Sci. (Best IJHAMS) , 2015.

J. Wang, M. Guo, and S. Liu, “SVIR Epidemic Model with Age Structure in Susceptibility, Vaccination Effects and Relapse,” IMA J. Appl. Math. (Institute Math. Its Appl., 2017.

M. A. Khan et al., “Stability Analysis of an SVIR Epidemic Model with non-Linear Saturated Incidence rate,” Appl. Math. Sci., 2015.

P. J. Witbooi, G. E. Muller, and G. J. Van Schalkwyk, “Vaccination Control in a Stochastic SVIR Epidemic Model,” Comput. Math. Methods Med., 2015.

E. Shim, “A Note on Epidemic Models with Infective Immigrants and Vaccination,” Mathematical Biosciences and Engineering. 2006.

A. d’Onofrio, P. Manfredi, and E. Salinelli, “Vaccinating Behaviour, Information, and the Dynamics of SIR Vaccine Preventable Diseases,” Theor. Popul. Biol., 2007.

X. Liu, Y. Takeuchi, and S. Iwami, “SVIR Epidemic Models with Vaccination Strategies,” J. Theor. Biol., 2008.

M. Tonaas, “Strategi Vaksinasi Kontinu pada Model Epidemik SVIR,” Bogor Agricultural University, 2011.

S. Melita Haryati, Kartono, “Kestabilan Model Susceptible Vaccinated Infected Recovered (SVIR) pada Penyebaran Penyakit Campak (Measles) (Studi Kasus di Kota Semarang),” J. Mat., vol. 17, no. 1, 2014.

S. Anis, M. Tonaas, and Suwandi, “Dinamika Model Epidemik SVIR Terhadap Penyebaran Penyakit Campak Dengan Strategi Vaksinasi Kontinu,” Mhs. Prodi Mat. FKIP Univ. Pengaraian, vol. 1, no. 1, 2015.

J. Harianto, “Local Stability Analysis of an SVIR Epidemic Model,” CAUCHY, 2017.

J. Harianto and T. Suparwati, “SVIR Epidemic Model with Non Constant Population,” CAUCHY, 2018.

R. Hidayati, “Analisis Kestabilan Model Epidemi Svir Dengan Tingkat Kejadian Tersaturasi,” Universitas Brawijaya, 2017.

A. Ibra and W. Purnami, “Model Susceptible Vaccinated Infected Recovered (SVIR) dan Penerapannya pada Penyakit Difteri di Indonesia,” in PRISMA, 2020, pp. 156–162.

L. Perko, Differential Equations and Dynamical Systems, 3th ed. New York: Springer, 2001.

L. Mirsky, F. R. Gantmacher, and K. A. Hirsch, “The Theory of Matrices,” Math. Gaz., 1961.

Published
2020-12-05
How to Cite
Harianto, J., Tuturop, K., & Angelika, V. (2020). Effect of Population Density on the Model of the Spread of Measles. NUMERICAL: Jurnal Matematika Dan Pendidikan Matematika, 4(2), 67-76. https://doi.org/10.25217/numerical.v4i2.831
Section
Articles