Analysis of an improved epidemic model with stochastic disease transmission. (English) Zbl 1245.35136

Summary: This paper proposes an improved logistic epidemic model and carries out the complete parameters analysis of asymptotic behavior of infectious diseases. Some interesting details such as the threshold value of outbreak of epidemics and critical states of disease spread are derived. The mean and variance of proportion of infected population are given explicitly. The results show that our model is more reasonable and applicable to describe the real situation. Especially, \(P\frac12\) might be considered as the alarm for relate institutions to make effective policies to prevent and control some epidemics.


35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D30 Epidemiology
35Q84 Fokker-Planck equations
Full Text: DOI


[1] Gao, S.; Chen, L.; Teng, Z., Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Anal. Real World Appl., 9, 599-607 (2008) · Zbl 1144.34390
[2] Raimundo, S. M.; Yang, H.; Engel, A. B., Modelling the effects of temporary immune protection and vaccination against infectious diseases, Appl. Math. Comput., 189, 1723-1736 (2007) · Zbl 1117.92038
[3] Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42, 931-947 (2000) · Zbl 0967.34070
[4] Han, L.; Ma, Z.; Shi, T., An SIRS epidemic model of two competitive species, Math. Comput. Model., 37, 87-108 (2003) · Zbl 1022.92033
[5] Iannelli, M.; Kim, M.; Park, E., Asymptotic behavior for an SIS epidemic model and its approximation, Nonlinear Anal., 35, 797-814 (1999) · Zbl 0921.92029
[6] Tuckwell, H. C.; Williams, R. J., Some properties of a simple stochastic epidemic model of SIR type, Math. Biosci., 208, 76-97 (2007) · Zbl 1116.92061
[7] Nasell, I., Stochastic models of some endemic infections, Math. Biosci., 179, 1-19 (2002) · Zbl 0991.92026
[8] Ball, F.; Sirl, David; Trapman, Pieter, Analysis of a stochastic SIR epidemic on a random network incorporating household structure, Math. Biosci., 224, 53-73 (2010) · Zbl 1192.92037
[9] Ball, F.; Lyne, O., Optimal vaccination policies for stochastic epidemics among a population of households, Math. Biosci., 177, 333-354 (2002) · Zbl 0996.92032
[10] Britton, T., Stochastic epidemic models: a survey, Math. Biosci., 225, 24-35 (2010) · Zbl 1188.92031
[11] Roberts, M. G.; Saha, A. K., The asymptotic behaviour of a logistic epidemic model with stochastic disease transmission, Appl. Math. Lett., 12, 37-41 (1999) · Zbl 0932.92031
[12] Oksendsl, B., Stochastic Differential Equations (2005), Springer: Springer Berlin · Zbl 1071.60003
[13] Gardiner, C. W., Handbook of Stochastic Methods for Physics Chemistry and the Natural Sciences (1997), Springer: Springer New York · Zbl 0862.60050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.