Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in a patchy environment for small and large dispersal rates.We first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity matrix.Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity,respectively.We further establish the asymptotic behavior of the basic reproduction ratio in the case of small and large dispersal rates.Finally,we apply these results to a periodic Ross-Macdonald patch model.

Awareness as the Most Effective Measure to Mitigate the Spread of COVID-19 in Nigeria

A mathematical model consisting of a system of four nonlinear ordinary differential equations is constructed.Our aim is to study the dynamics of the spread of COVID-19 in Nigeria and to show the effectiveness of awareness and the need for relevant authorities to engage themselves more in enlightening people on the significance of the available control measures in mitigating the spread of the disease.Two equilibrium solutions;Disease free equilibrium and Endemic equilibrium solutions were calculated and their global stability analysis was carried out.Basic reproduction ratio(R0)was also obtained,in this research𝑅𝑅0=3.0784.Data obtained for Nigeria is used to conduct numerical simulations in order to support the analytic result and to show the significance of awareness in controlling the disease spread.From the simulation result,it was shown that to mitigate the spread of COVID-19 in Nigeria there is need for serious awareness programs to enlighten people on the available control measures;social distancing,self-isolation,use of personal protective equipment(such as face mask,hand globes,overall gown,etc.),regular hand washing using soap or sanitizer,avoiding having contact with person showing the symptoms and reporting any suspected case.

A Mathematical Model for the Control of Cholera Epidemic without Natural Recovery

In this research work, we present a mathematical model for the control of cholera outbreak without natural recovery. This follows a slight modification as compared to previous cholera models for the Nigerian case. Our model incorporates treatment, water hygiene as well as environmental sanitation. The model employs a system of nonlinear ordinary differential equations, which is analyzed in detail for its stability properties. We compute the basic reproduction ratio R0 for the various control parameters and discover that with proper combination of control measures, the spread of cholera could be minimized. Numerical simulation of the cholera model is done using MathCAD14, and the graphical profiles of the main variables are depicted. We conclude that improvement in treatment, water hygiene and the environmental sanitation is indeed effective in eradicating the cholera epidemic.

Estimating the basic reproductive ratio for the Ebola outbreak in Liberia and Sierra Leone

Background:Ebola virus disease has reemerged as a major public health crisis in Africa,with isolated cases also observed globally,during the current outbreak.Methods:To estimate the basic reproductive ratio R0,which is a measure of the severity of the outbreak,we developed a SEIR(susceptible-exposed-infected-recovered)type deterministic model,and used data from the Centers for Disease Control and Prevention(CDC),for the Ebola outbreak in Liberia and Sierra Leone.Two different data sets are available:one with raw reported data and one with corrected data(as the CDC suspects under-reporting).Results:Using a deterministic ordinary differential equation transmission model for Ebola epidemic,the basic reproductive ratio R0 for Liberia resulted to be 1.757 and 1.9 for corrected and uncorrected case data,respectively.For Sierra Leone,R0 resulted to be 1.492 and 1.362 for corrected and uncorrected case data,respectively.In each of the two cases we considered,the estimate for the basic reproductive ratio was initially greater than unity leading to an epidemic outbreak.Conclusion:We obtained robust estimates for the value of R0 associated with the 2014 Ebola outbreak,and showed that there is close agreement between our estimates of R0.Analysis of our model also showed that effective isolation is required,with the contact rate in isolation less than one quarter of that for the infected non-isolated population,and that the fraction of high-risk individuals must be brought to less than 10%of the overall susceptible population,in order to bring the value of R0 to less than 1,and hence control the outbreak.

Modelling the Effect of Self-Immunity and the Impacts of Asymptomatic and Symptomatic Individuals on COVID-19 Outbreak

COVID-19 is one of themost highly infectious diseases ever emerged and caused by newly discovered severe acute respiratory syndrome coronavirus 2(SARS-CoV-2).It has already led the entire world to health and economic crisis.It has invaded the whole universe all most every way.The present study demonstrates with a nine mutually exclusive compartmental model on transmission dynamics of this pandemic disease(COVID-19),with special focus on the transmissibility of symptomatic and asymptomatic infection from susceptible individuals.Herein,the compartmental model has been investigated with mathematical analysis and computer simulations in order to understand the dynamics of COVID-19 transmission.Initially,mathematical analysis of the model has been carried out in broadly by illustrating some well-known methods including exactness,equilibrium and stability analysis in terms of basic reproduction number.We investigate the sensitivity of the model with respect to the variation of the parameters’values.Furthermore,computer simulations are performed to illustrate the results.Our analysis reveals that the death rate from coronavirus disease increases as the infection rate increases,whereas infection rate extensively decreases with the increase of quarantined individuals.The quarantined individuals also lead to increase the concentration of recovered individuals.However,the infection rate of COVID-19 increases more surprisingly as the rate of asymptomatic individuals increases than that of the symptomatic individuals.Moreover,the infection rate decreases significantly due to increase of self-immunity rate.

A mathematical model for a disease outbreak considering waningimmunity class with nonlinear incidence and recovery rates

In the spread of infectious diseases,intervention levels play a crucial role in shaping interactions between healthy and infected individuals,leading to a nonlinear transmission process.Additionally,the availability of medical resources limits the recovery rate of infected patients,adding further nonlinear dynamics to the healing process.Our research introduces novelty by combining nonlinear incidence and recovery rates alongside waning immunity in an epidemic model.We present a modified SIRW-type model,examining the epidemic problem with these factors.Through analysis,we explore conditions for non-endemic and co-existing cases based on the basic reproduction ratio.The local stability of equilibria is verified using the Routh-Hurwitz criteria,while global stability is assessed using Lyapunov functions for each equilibrium.Furthermore,we investigate bifurcations around both non-endemic and co-existing equilibria.Numerically,we give some simulations to support our analytical findings.

Stochastic and spatio-temporal analysis of the Middle East Respiratory Syndrome outbreak in South Korea, 2015

South Korea was free of the Middle East Respiratory Syndrome(MERS)until 2015.The MERS outbreak in South Korea during 2015 was the largest outbreak of the Coronavirus outside the Middle East.The major characteristic of this outbreak is inter-or intra-hospital transmission.This recent MERS outbreak in South Korea is examined and assessed in this paper.The main objectives of the study is to characterize the pattern of the MERS outbreak in South Korea based on a basic reproductive ratio,the probability of ultimate extinction of the disease,and the spatio-temporal proximity of occurrence between patients.The survival function method and stochastic branching process model are adapted to calculate the basic reproductive ratio and the probability of ultimate extinction of the disease.We further investigate the occurrence pattern of the outbreak using a spatio-temporal autocorrelation function.

Analysis of a hepatitis B viral infection model with immune response delay

In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte （CTL） cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.