Dynamics of an age-of-infection cholera model

Abstract

A new model for the dynamics of cholera is formulated that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is defined and proved to be a sharp threshold determining whether or not cholera dies out. Final size relations for cholera outbreaks are derived for simplified models when input and death are neglected.

Global dynamics of a coupled epidemic model

“…The coupled virus models with both transmissions being assumed to be mass-actions have been studied in [13,14,18], where the basic reproduction number for the coupled system is simply the sum of two basic reproduction numbers for the subsystems with only direct or indirect disease transmissions respectively. Similar results were also obtained for cholera models [19], pathogen models [2,3] and treatment models [24]. The coupled model proposed in [1] incorporated two virions with sensitive and resistant strains, respectively, and the basic reproduction number for the full system is the maximum of two basic reproduction numbers of viral strains.…”

“…We claim u 1 = 0. If not, we substitute y = u−βx 0 z/δ into equation (3) and find z = pu − (pβx 0 /δ + δ)z, which, together with the limit of u(t), implies that z → z 1 as t → ∞, where z 1 = pu 1 /(pβx 0 /δ + δ) > 0. Furthermore, y → y 1 as t → ∞, where y 1 = u 1 − βx 0 z 1 /δ = δu 1 /(pβx 0 /δ + δ) > 0.…”

“…However, for simplicity, we could assume that this quantity is a large constant w(t) ≡ W with respect to the infected poultry z(t). Therefore, the above equation can be approximated by (3) with p = β 4 W and δ = σ−β 3 . A similar argument works for mosquito-borne disease models where x, y, z, w correspond to the uninfected individuals, infected individuals, infected mosquitoes, and uninfected mosquitoes, respectively.…”

Dynamical analysis for a hepatitis B transmission model with immigration and infection age

“…Although dynamical analysis of epidemic models with age structures is particularly challenging, there has been recent progress in global analysis [13,22,2,7,15,17,18]. Age-structured models have also been developed to study the epidemiology of HBV infection [6,19,27,29,26].…”

“…By classical existence and uniqueness results for functional differential equations, there exists a unique solution for the integro-differential system (1) in which i(a, t) and c(a, t) are substituted for the expressions (2) and (3), respectively. For (2) and 3, it is easy to see that i(a, t) and c(a, t) remain nonnegative for any nonnegative initial value. Furthermore, if there exists a t * such that S(t * ) = 0 and S(t) > 0 for 0 < t < t * , then, from the S equation of (1), we have S (t * ) = Λ S − bω ∞ a1 v(a)c(a, t)da > 0, which implies that S(t) ≥ 0 for all t ≥ 0, noting that unsuccessfully immunized birth bω is included in Λ S .…”