DYNAMICS OF COMPETING STRAINS WITH SATURATED INFECTIVITY AND MUTATION ON NETWORKS

2016 ◽  
Vol 24 (02n03) ◽  
pp. 257-273 ◽  
Author(s):  
QINGCHU WU ◽  
XINCHU FU
Keyword(s):  
Dynamical Systems ◽  
Complex Network ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Critical Values ◽  
Epidemic Threshold ◽  
Persistence Theory ◽  
The Stability

This paper deals with the dynamical behavior of a two-strain epidemic model in a complex network with saturated infectivity and mutation. The model is shown to exhibit the phenomena of strain coexistence where two epidemic strains co-exist and strain replacement where the strain with smaller basic reproduction number can become predominant. By using the stability and persistence theory of dynamical systems, we calculate the epidemic threshold, and show that above which at least one strain persists in a population, and also obtain the critical values to discriminate the strain coexistence and replacement. The results suggest that the newly mutated strain can spread in the population, even if it has a very small transmissivity.

THRESHOLD DYNAMICS IN A PERIODIC SVEIR EPIDEMIC MODEL

2011 ◽  
Vol 04 (04) ◽  
pp. 493-509 ◽  
Author(s):  
JINLIANG WANG ◽  
SHENGQIANG LIU ◽  
YASUHIRO TAKEUCHI
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Threshold Dynamics ◽  
Globally Asymptotically Stable ◽  
Infectious Risk ◽  
Persistence Theory ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we investigate the dynamical behavior of a class of periodic SVEIR epidemic model. Since the nonautonomous phenomenon often occurs as cyclic pattern, our model is then a periodic time-dependent system. It follows from persistence theory that the basic reproduction number is the threshold parameter above which the disease is uniformly persistent and below which disease-free periodic solution is globally asymptotically stable. The threshold dynamics extends the classic results for the corresponding autonomous model. Furthermore, we show that eradication policy on the basis of the basic reproduction number of the autonomous system may overestimate the infectious risk when the disease follows periodic behavior. The according simulation results are also given.

scholarly journals Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
The Stability ◽  
The Basic Reproduction Number

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

THEORETICAL INVESTIGATION OF THE IMPACT ON EPIDEMIC THRESHOLD OF TRAVEL BETWEEN COMMUNITIES OF RESIDENT POPULATIONS

2013 ◽  
Vol 21 (02) ◽  
pp. 1350010 ◽  
Author(s):  
KLOT PATANARAPEELERT ◽  
D. GARCIA LOPEZ ◽  
I-MING TANG ◽  
MARC A. DUBOIS
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Initial Phase ◽  
Reproduction Number ◽  
Travel Patterns ◽  
Epidemic Threshold ◽  
Contact Patterns ◽  
Resident Populations ◽  
The Impact ◽  
The Basic Reproduction Number

During the initial phase of an epidemic, individual displacements between different regions modify the contact patterns. Understanding mobility processes and their consequences is necessary to predict the subsequent spread of the disease in order to optimize control policies. The basic reproduction number is commonly used to determine the threshold between extinction and expansion of the disease. Once it is derived for an epidemic model that includes the travel of population between distinct localities, the dependence of the diseases dynamics upon travel rates becomes explicit. In this study, we examine the effects of travel on the epidemic threshold for a model of two communities. The travel rates are treated as varying subject to two scenarios. We show theoretically that if the transmission potentials within communities are moderate, the epidemic threshold can be modified by travel. The conditions for the presence of the threshold induced by travel is determined and the critical values of travel at which the basic reproduction number is equal to one are derived. We show further that these results can also be applied to a model of three communities under specific travel patterns and that the derived basic reproduction number has a form similar to that of the two communities problem.

scholarly journals Stability of the equilibria in a discrete-time sivs epidemic model with standard incidence

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2393-2408 ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Total Population ◽  
Reproduction Number ◽  
Flip Bifurcation ◽  
Sis Epidemic Model ◽  
Fold Bifurcation ◽  
The Stability ◽  
Conditions Of Stability

A discrete-time SIS epidemic model with vaccination is presented and studied. The model includes deaths due to disease and the total population size is variable. First, existence and positivity of the solutions are discussed and equilibria of the model and basic reproduction number are obtained. Next, the stability of the equilibria is studied and conditions of stability are obtained in terms of the basic reproduction number R0. Also, occurrence of the fold bifurcation, the flip bifurcation, and the Neimark-Sacker bifurcation is investigated at equilibria. In addition, obtained results are numerically discussed and some diagrams for bifurcations, Lyapunov exponents, and solutions of the model are presented.

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>

Dynamical Behavior of a Malaria Model with Discrete Delay and Optimal Insecticide Control

2017 ◽  
Vol 12 (01) ◽  
pp. 19-38 ◽  
Author(s):  
Tuhin Kumar Kar ◽  
Soovoojeet Jana
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Control Variable ◽  
Discrete Delay ◽  
Insecticide Control

In this paper we have proposed and analyzed a simple three-dimensional mathematical model related to malaria disease. We consider three state variables associated with susceptible human population, infected human population and infected mosquitoes, respectively. A discrete delay parameter has been incorporated to take account of the time of incubation period with infected mosquitoes. We consider the effect of insecticide control, which is applied to the mosquitoes. Basic reproduction number is figured out for the proposed model and it is shown that when this threshold is less than unity then the system moves to the disease-free state whereas for higher values other than unity, the system would tend to an endemic state. On the other hand if we consider the system with delay, then there may exist some cases where the endemic equilibrium would be unstable although the numerical value of basic reproduction number may be greater than one. We formulate and solve the optimal control problem by considering insecticide as the control variable. Optimal control problem assures to obtain better result than the noncontrol situation. Numerical illustrations are provided in support of the theoretical results.

scholarly journals Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lian Duan ◽  
Lihong Huang ◽  
Chuangxia Huang
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Spatial Dynamics ◽  
Heterogeneous Environment ◽  
Threshold Parameter ◽  
Dispersal Rate ◽  
Siri Model ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{R}_{0} $\end{document}</tex-math></inline-formula> which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.</p>

scholarly journals Modeling Gender-Structured Wildlife Diseases with Harvesting: Chronic Wasting Disease as an Example

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Mo'tassem Al-Arydah ◽  
Robert J. Smith ◽  
Frithjof Lutscher
Keyword(s):  
Basic Reproduction Number ◽  
Chronic Wasting Disease ◽  
Reproduction Number ◽  
Economic Consequences ◽  
Wildlife Diseases ◽  
Wasting Disease ◽  
Effective Strategies ◽  
Optimum Interval ◽  
The Stability ◽  
The Basic Reproduction Number

Chronic wasting disease (CWD) is a prion infectious disease that affects members of the deer family in North America. Concerns about the economic consequences of the presence of CWD have led management agencies to seek effective strategies to control CWD distribution and prevalence. Current mathematical models are either based on complex simulations or overly simplified compartmental models. We develop a mathematical model that includes gender structure to describe CWD in a logistically growing population. The model includes harvesting as a management strategy for the disease. We determine the stability conditions of the disease-free equilibrium for the model and calculate the basic reproduction number. We find an optimum interval of harvesting: with too little harvesting, the disease persists, whereas too much harvesting results in extinction of the population. A sensitivity analysis shows that the disease threshold is more sensitive to female than male harvesting and that harvesting has the greatest effect on the basic reproduction number. However, while harvesting may be a way to control CWD, the range of admissible harvesting rates may be very narrow, depending on other parameters.

scholarly journals Simple MSEIR Model for Measles Transmission

2019 ◽  
pp. 1-11
Author(s):  
Mojeeb Al-Rahman EL-Nor Osman ◽  
Appiagyei Ebenezer ◽  
Isaac Kwasi Adu
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Recovery Rate ◽  
Birth Rate ◽  
Reproduction Number ◽  
Progression Rate ◽  
Asymptotically Stable ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria.  The basic reproduction number  is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out.  The disease was locally asymptotically stable if  and unstable if  . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results. Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.

scholarly journals PENGEMBANGAN MODEL EPIDEMIK SIRA UNTUK PENYEBARAN VIRUS PADA JARINGAN KOMPUTER

2019 ◽  
Vol 2 (1) ◽  
pp. 13
Author(s):  
Panca Putra Pemungkas ◽  
Sutrisno Sutrisno ◽  
Sunarsih Sunarsih
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Computer Network ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Virus Spread ◽  
Computer Based ◽  
Stability Issues ◽  
Spread Analysis

This paper is addressed to discuss the development of epidemic model of SIRA (Susceptible-Infected-Removed-Antidotal) for virus spread analysis purposes on a computer network. We have developed the existing model by adding a possibility of antidotal computer returned to susceptible computer. Based on the results, there are two virus-free equilibrium points and one endemic equilibrium point. These equilibrium points were analyzed for stability issues using basic reproduction number and Routh-Hurwitz Method.

Sign in / Sign up

Export Citation Format

Share Document