Mathematical analysis of SLIPR infectious model without vaccination: A case study of measles outbreaks

Fuaada Mohd Siam(1), Hanis Nasir(2*),

(1) Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310, UTM Johor Bahru, Johor, Malaysia.
(2) Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030, Kuala Nerus, Terengganu.
(*) Corresponding Author


A five-compartment epidemiological model is analyzed to illustrate the dynamics of infectious diseases. In this model, the population is compartmented into susceptible, latent, infected, post-infection, and recovered. The model is a system of ordinary differential equations, where the stability is analyzed using Routh’s stability criterion. Two equilibrium points; disease-free and endemic equilibrium are stable but depends on the basic reproduction number. Derivation of the basic reproduction number is given using the next-generation method. This study ended by providing a case study of measles outbreaks before the effective implementation of the vaccine. The analysis of data fitting is done by using the Simulated Annealing minimization routine and the error value is 0.0406.


measles, infectious model, data fitting


Anagnost, J.J., & Desoer, C.A. (1991). An elementary proof of the Routh-Hurwitz stability criterion. Circuits, Systems and Signal Processing, 10(1), 101-114.

Bawa, M., Abdulrahman, S., Jimoh, O.R., & Adabara, N.U. (2013). Stability Analysis of the Disease–free equilibrium State for Lassa fever Disease. Journal of Science, Technology, Mathematics and Education (JOSTMED), 9(2), 115-123.

Brauer, F., (2005). The Kermack–McKendrick epidemic model revisited. Mathematical Biosciences, 198(2), 119-131.

Buchanan, R., & Bonthius, D.J. (2012), September. Measles virus and associated central nervous system sequelae. In Seminars in Pediatric Neurology (Vol. 19, No. 3, pp. 107-114). WB Saunders.

Busetti, F., 2003. Simulated annealing overview. Retrieved from

Chen, S.T. and Lam, S.K., 1985. Optimum age for measles immunization in Malaysia. Southeast Asian J Trop Med Public Health, 16, 493-9.

Diekmann, O., Heesterbeek, J.A.P., & Roberts, M.G., 2009. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), 873-885.

Eisenberg, J.N., Brookhart, M.A., Rice, G., Brown, M., & Colford Jr, J.M., 2002. Disease transmission models for public health decision making: analysis of epidemic and endemic conditions caused by waterborne pathogens. Environmental Health Perspectives, 110(8), 783-790.

Filia, A., Bella, A., Cadeddu, G., Milia, M.R., Del Manso, M., Rota, M.C., Magurano, F., Nicoletti, L., & Declich, S. (2015). Extensive nosocomial transmission of measles originating in cruise ship passenger, Sardinia, Italy, 2014. Emerging Infectious Diseases, 21(8), 1444.

Helps, C., Leask, J., Barclay, L., & Carter, S. (2019). Understanding non-vaccinating parents’ views to inform and improve clinical encounters: a qualitative study in an Australian community. BMJ Open, 9(5), p.e026299.

Jones, J.H. (2007). Notes on R0. Califonia: Department of Anthropological Sciences.

Kirkpatrick, S., Gelatt, C.D., & Vecchi, M.P. (1983). Optimization by simulated annealing. Science, 220(4598), 671-680.

Kusnin, F. (2017). ‘Immunisation Program in Malaysia’, Vaccinology 2017 III International Symposium for Asia Pacific Experts (Health Physician Disease Control Division Ministry of Health, Malaysia 2017). Retrieved from

Leung, A.K., Hon, K.L., Leong, K.F., & Sergi, C.M. (2018). Measles: a disease often forgotten but not gone. Hong Kong Med J, 24(5), 512-520.

May, R.M. (2004). Simple mathematical models with very complicated dynamics. In The Theory of Chaotic Attractors (pp. 85-93). Springer, New York, NY.

Oswald, W.B., Geisbert, T.W., Davis, K.J., Geisbert, J.B., Sullivan, N.J., Jahrling, P.B., Parren, P.W.I., & Burton, D.R. (2007). Neutralizing antibody fails to impact the course of Ebola virus infection in monkeys. PLoS Pathogens, 3(1), p.e 9.

Sivanandam, S.N., & Deepa, S.N. (2007). Linear system design using Routh Column polynomials. Songklanakarin Journal of Science & Technology, 29(6).

Thowsen, A. (1981). The Routh-Hurwitz method for stability determination of linear differential-difference systems. International Journal of Control, 33(5), 991-995.

Worldometers (2019). Retrieved from


Article Metrics

Abstract view : 2 times


  • There are currently no refbacks.

Copyright (c) 2021 Annals of Mathematical Modeling

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Annals of Mathematical Modeling is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at


View My Stats JAMM