Uniform persistence and flows near a closed positively invariant set

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R 0 . If R 0 ≤ 1, then the disease-free equilibrium P 0 is globally asymptotically stable, and the disease always dies out. If R 0 > 1, P 0 is unstable, a unique endemic equilibrium P * is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated.

Section: Existence and Uniqueness Of The Endemic Equilibriumsupporting

confidence: 67%

Global dynamics of a staged-progression model with amelioration for infectious diseases

Guo

Li²

2008

“…Using a uniform persistence result from [20] and an argument as in the proof of Proposition 3.3 of [38], it can be shown that, when R 0 > 1, instability of the DFE implies uniform persistence of model (1)-(4).…”

Section: Solutions Inmentioning

confidence: 99%

Reproduction numbers for infections with free-living pathogens growing in the environment

Bani-Yaghoub

Gautam

Shuai

et al. 2012

110

The basic reproduction number R 0 for a compartmental disease model is often calculated by the next generation matrix (NGM) approach. When the interactions within and between disease compartments are interpreted differently, the NGM approach may lead to different R 0 expressions. This is demonstrated by considering a susceptible-infectious-recovered-susceptible model with free-living pathogen (FLP) growing in the environment. Although the environment could play different roles in the disease transmission process, leading to different R 0 expressions, there is a unique type reproduction number when control strategies are applied to the host population. All R 0 expressions agree on the threshold value 1 and preserve their order of magnitude. However, using data for salmonellosis and cholera, it is shown that the estimated R 0 values are substantially different. This study highlights the utility and limitations of reproduction numbers to accurately quantify the effects of control strategies for infections with FLPs growing in the environment.

“…1/κ 2 × 11.9431 < 1, or κ > 3. 4, which means that we should increase vector mortality almost three-and-half times. Again assuming the extreme case, i.e.…”

Section: Discussion and Simulationsmentioning

confidence: 99%

“…We show that system (4) satisfies all the conditions of Theorem 4.3 in [4]. Choose X = R 3 and E = .…”

Section: Theorem 52: System (4) Is Uniformly Persistent In Int Ifmentioning

confidence: 92%

Stability analysis of pine wilt disease model by periodic use of insecticides

Awan

Ozair

Din³

et al. 2016

This work is related to qualitative behaviour of an epidemic model of pine wilt disease. More precisely, we proved that the reproductive number has sharp threshold properties. It has been shown that how vector population can be reduced by the periodic use of insecticides. Numerical simulations show that epidemic level of infected vectors becomes independent of saturation level by including the transmission through mating. ARTICLE HISTORY