# 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)
Impact of group mixing on disease dynamics. (English) Zbl 1200.92027
Summary: A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number ${ℛ}_{0}$ is identified. It is shown that if ${ℛ}_{0}<1$, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if ${ℛ}_{0}>1$, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.
##### MSC:
 92C60 Medical epidemiology 34D23 Global stability of ODE 37N25 Dynamical systems in biology 92D30 Epidemiology
##### References:
 [1] Brauer, F.; Den Driessche, P. Van; Wang, L.: Oscillations in a patchy environment disease model, Math. biosci. 215, 1 (2008) · Zbl 1176.34098 · doi:10.1016/j.mbs.2008.05.001 [2] Gantmacher, F. R.: Applications of the theory of matrices, (1959) · Zbl 0085.01001 [3] Hethcote, H. W.; Stech, H. W.; Den Driessche, P. Van: Nonlinear oscillations in epidemic models, SIAM J. Appl. math. 40, 1 (1981) · Zbl 0469.92012 · doi:10.1137/0140001 [4] Lajmanovich, A.; Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci. 28, 221 (1976) · Zbl 0344.92016 · doi:10.1016/0025-5564(76)90125-5 [5] Lloyd, A. L.; May, R. M.: Spatial heterogeneity in epidemic models, J. theor. Biol. 179, 1 (1996) [6] Miller, R. K.: Nonlinear Volterra integral equations, (1971) [7] Smith, H. L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical surveys and monographs 41 (1995) · Zbl 0821.34003 [8] Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6