Stability of differential susceptibility and infectivity epidemic models

Bonzi, Bernard K.; Fall, A. A.; Iggidr, Abderrahman; Sallet, Gauthier

doi:10.1007/s00285-010-0327-y

Cited by 34 publications

(32 citation statements)

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“…Several authors have previously formulated and studied epidemiological models with heterogenous susceptibility, either in terms of a finite number of different susceptibility classes (Andersson and Britton 1998;Ball 1985;Bonzi et al 2010;Gart 1972;Hyman and Li 2005;Rodrigues et al 2009;Scalia-Tomba 1986) or as a continuous distribution of susceptibility (Coutinho et al 1999;Diekmann and Heesterbeek 2000;Dwyer et al 1997Dwyer et al , 2000Novozhilov 2008). Below, as we describe our results, we will mention some of the results obtained in these works, and their relations with the present investigation.…”

Section: Introductionsupporting

confidence: 57%

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The size of epidemics in populations with heterogeneous susceptibility

Katriel

2011

J. Math. Biol.

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We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.

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Section: Introductionsupporting

confidence: 57%

“…This paper deals with the case of epidemics. Several other studies model heterogeneous susceptibility when a continuous replenishment of susceptibles due to births or loss of immunity leads to an endemic equilibrium (Bonzi et al 2010;Hyman and Li 2005;Reluga et al 2008;Veliov 2005;White and Medley 1998).…”

Section: Introductionmentioning

confidence: 98%

The size of epidemics in populations with heterogeneous susceptibility

Katriel

2011

J. Math. Biol.

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“…We can easily see thatR = AĒ andĪ = LĒ, where A, L and K are as defined in (10). Thus, I ≫ 0 implies thatĒ ≫ 0 andR ≫ 0.…”

Section: Proofmentioning

confidence: 97%

Multi-patch and multi-group epidemic models: a new framework

Bichara

Iggidr

2017

J. Math. Biol.

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We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number [Formula: see text] is derived. The effects of heterogeneity in groups, patches and mobility patterns on [Formula: see text] and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number.

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“…To include the individual heterogeneity, concerning the mode of infection, it is natural to incorporate differential infectivity or susceptibility into epidemic models [29]. Thus, the total host population is divided into n + 2 compartments: a susceptible compartment S, the i-th stage I i of the disease progression [13,23], i = 1, 2, .…”

Section: Model Descriptionmentioning

confidence: 99%