Limiting profile for stationary solutions maximizing the total population of a diffusive logistic equation

2021 ◽  
Author(s):  
Jumpei Inoue
Keyword(s):  
Total Population ◽  
Stationary Solutions ◽  
Logistic Equation ◽  
Limiting Profile ◽  
Diffusive Logistic Equation

scholarly journals Impact of Regional Difference in Recovery Rate on the Total Population of Infected for a Diffusive SIS Model

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 888
Author(s):  
Jumpei Inoue ◽  
Kousuke Kuto
Keyword(s):  
Recovery Rate ◽  
Regional Difference ◽  
Total Population ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Logistic Equation ◽  
Sis Model ◽  
Reaction Diffusion Model ◽  
Diffusive Logistic Equation ◽  
Sis Epidemic

This paper is concerned with an SIS epidemic reaction-diffusion model. The purpose of this paper is to derive some effects of the spatial heterogeneity of the recovery rate on the total population of infected and the reproduction number. The proof is based on an application of our previous result on the unboundedness of the ratio of the species to the resource for a diffusive logistic equation. Our pure mathematical result can be epidemically interpreted as that a regional difference in the recovery rate can make the infected population grow in the case when the reproduction number is slightly larger than one.

Positive solutions for diffusive Logistic equation with refuge

2019 ◽  
Vol 9 (1) ◽  
pp. 1092-1101 ◽  
Author(s):  
Jian-Wen Sun
Keyword(s):  
Boundary Condition ◽  
Weight Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Value ◽  
Stationary Solutions ◽  
Logistic Equation ◽  
Diffusion Equations ◽  
Diffusion Operator ◽  
Diffusive Logistic Equation

Abstract In this paper, we study the stationary solutions of the Logistic equation $$\begin{array}{} \displaystyle u_t=\mathcal {D}[u]+\lambda u-[b(x)+\varepsilon]u^p \text{ in }{\it\Omega} \end{array}$$ with Dirichlet boundary condition, here 𝓓 is a diffusion operator and ε > 0, p > 1. The weight function b(x) is nonnegative and vanishes in a smooth subdomain Ω0 of Ω. We investigate the asymptotic profiles of positive stationary solutions with the critical value λ when ε is sufficiently small. We find that the profiles are different between nonlocal and classical diffusion equations.

Numerical solutions of diffusive logistic equation

2007 ◽  
Vol 31 (1) ◽  
pp. 112-118 ◽  
Author(s):  
G.A. Afrouzi ◽  
S. Khademloo
Keyword(s):  
Numerical Solutions ◽  
Logistic Equation ◽  
Diffusive Logistic Equation

scholarly journals On a Diffusive Logistic Equation

1998 ◽  
Vol 225 (1) ◽  
pp. 326-339 ◽  
Author(s):  
G.A Afrouzi ◽  
K.J Brown
Keyword(s):  
Logistic Equation ◽  
Diffusive Logistic Equation

scholarly journals A reductive analysis of a compartmental model for COVID-19: data assimilation and forecasting for the United Kingdom

2020 ◽  
Author(s):  
G. Ananthakrishna ◽  
Jagadish Kumar
Keyword(s):  
United Kingdom ◽  
Turning Point ◽  
Total Population ◽  
Deterministic Model ◽  
Logistic Equation ◽  
Original Model ◽  
Cumulative Number ◽  
Full Model ◽  
The United Kingdom ◽  
Model Equations

We introduce a deterministic model that partitions the total population into the susceptible, infected, quarantined, and those traced after exposure, the recovered and the deceased. We hypothesize ‘accessible population for transmission of the disease’ to be a small fraction of the total population, for instance when interventions are in force. This hypothesis, together with the structure of the set of coupled nonlinear ordinary differential equations for the populations, allows us to decouple the equations into just two equations. This further reduces to a logistic type of equation for the total infected population. The equation can be solved analytically and therefore allows for a clear interpretation of the growth and inhibiting factors in terms of the parameters in the full model. The validity of the ‘accessible population’ hypothesis and the efficacy of the reduced logistic model is demonstrated by the ease of fitting the United Kingdom data for the cumulative infected and daily new infected cases. The model can also be used to forecast further progression of the disease. In an effort to find optimized parameter values compatible with the United Kingdom coronavirus data, we first determine the relative importance of the various transition rates participating in the original model. Using this we show that the original model equations provide a very good fit with the United Kingdom data for the cumulative number of infections and the daily new cases. The fact that the model calculated daily new cases exhibits a turning point, suggests the beginning of a slow-down in the spread of infections. However, since the rate of slowing down beyond the turning point is small, the cumulative number of infections is likely to saturate to about 3.52 × 105 around late July, provided the lock-down conditions continue to prevail. Noting that the fit obtained from the reduced logistic equation is comparable to that with the full model equations, the underlying causes for the limited forecasting ability of the reduced logistic equation are elucidated. The model and the procedure adopted here are expected to be useful in fitting the data for other countries and in forecasting the progression of the disease.

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