On the Basic Reproduction Number of Reaction-Diffusion Epidemic Models

2019 ◽  
Vol 79 (1) ◽  
pp. 284-304 ◽  
Author(s):  
Pierre Magal ◽  
Glenn F. Webb ◽  
Yixiang Wu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Epidemic Models ◽  
The Basic Reproduction Number

scholarly journals Epidemic management with admissible and robust invariant sets

PLoS ONE ◽  
2021 ◽  
Vol 16 (9) ◽  
pp. e0257598
Author(s):  
Willem Esterhuizen ◽  
Jean Lévine ◽  
Stefan Streif
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Intervention Strategies ◽  
Epidemic Models ◽  
Invariant Sets ◽  
Social Distancing ◽  
The Basic Reproduction Number

We present a detailed set-based analysis of the well-known SIR and SEIR epidemic models subjected to hard caps on the proportion of infective individuals, and bounds on the allowable intervention strategies, such as social distancing, quarantining and vaccination. We describe the admissible and maximal robust positively invariant (MRPI) sets of these two models via the theory of barriers. We show how the sets may be used in the management of epidemics, for both perfect and imperfect/uncertain models, detailing how intervention strategies may be specified such that the hard infection cap is never breached, regardless of the basic reproduction number. The results are clarified with detailed examples.

scholarly journals Studies on the basic reproduction number in stochastic epidemic models with random perturbations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Andrés Ríos-Gutiérrez ◽  
Soledad Torres ◽  
Viswanathan Arunachalam
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Survival Function ◽  
Epidemic Models ◽  
Random Perturbations ◽  
Asymptotically Stable ◽  
Stochastic Epidemic Models ◽  
Stochastic Epidemic ◽  
Disease Free ◽  
The Basic Reproduction Number

AbstractIn this paper, we discuss the basic reproduction number of stochastic epidemic models with random perturbations. We define the basic reproduction number in epidemic models by using the integral of a function or survival function. We study the systems of stochastic differential equations for SIR, SIS, and SEIR models and their stability analysis. Some results on deterministic epidemic models are also obtained. We give the numerical conditions for which the disease-free equilibrium point is asymptotically stable.

scholarly journals Partial Differential Equations of an Epidemic Model with Spatial Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
El Mehdi Lotfi ◽  
Mehdi Maziane ◽  
Khalid Hattaf ◽  
Noura Yousfi
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

scholarly journals Stability in a Ross epidemic model with road diffusion

2021 ◽  
Vol 7 (2) ◽  
pp. 2840-2857
Author(s):  
Xiaomei Bao ◽  
◽  
Canrong Tian ◽  
Keyword(s):  
Infectious Disease ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Upper And Lower Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Asymptotically Stable ◽  
The Basic Reproduction Number

<abstract><p>Reaction-diffusion equations have been used to describe the dynamical behavior of epidemic models, where the spreading of infectious disease has the same speed in every direction. A natural question is how to describe the dynamical system when the spreading of infectious disease is directed diffusion. We introduce the road diffusion into a Ross epidemic model which describes the spread of infected Mosquitoes and humans. With the comparison principle the system is proved to have a unique global solution. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.</p></abstract>

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