# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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, $P\frac{1}{2}$ might be considered as the alarm for relate institutions to make effective policies to prevent and control some epidemics.
##### MSC:
 35Q92 PDEs in connection with biology and other natural sciences 92D30 Epidemiology 35Q84 Fokker-Planck equations
##### References:
 [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 · doi:10.1016/j.nonrwa.2006.12.004 [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 · doi:10.1016/j.amc.2006.12.051 [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 · doi:10.1016/S0362-546X(99)00138-8 [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 · doi:10.1016/S0895-7177(03)80008-0 [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 · doi:10.1016/S0362-546X(97)00597-X [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 · doi:10.1016/j.mbs.2006.09.018 [7] Nasell, I.: Stochastic models of some endemic infections, Math. biosci. 179, 1-19 (2002) · Zbl 0991.92026 · doi:10.1016/S0025-5564(02)00098-6 [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 · doi:10.1016/j.mbs.2009.12.003 [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 · doi:10.1016/S0025-5564(01)00095-5 [10] Britton, T.: Stochastic epidemic models: a survey, Math. biosci. 225, 24-35 (2010) · Zbl 1188.92031 · doi:10.1016/j.mbs.2010.01.006 [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 · doi:10.1016/S0893-9659(98)00123-2 [12] Oksendsl, B.: Stochastic differential equations, (2005) [13] Gardiner, C. W.: Handbook of stochastic methods for physics chemistry and the natural sciences, (1997)