scholarly journals Dynamics of cholera epidemic models in fluctuating environments

2020 ◽  
Vol 21 (02) ◽  
pp. 2150011
Author(s):  
Tuan Anh Phan ◽  
Jianjun Paul Tian ◽  
Bixiang Wang
Keyword(s):  
Stochastic Model ◽  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Deterministic Models ◽  
Dynamical Behaviors ◽  
All Solutions ◽  
Number Formula ◽  
Asymptotical Analysis ◽  
The Basic Reproduction Number

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction value [Formula: see text] is defined in terms of the basic reproduction number [Formula: see text] for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. When [Formula: see text], the cholera infection will extinct within finite periods of time almost surely. When [Formula: see text], the cholera infection will persist most of time, and there exists a unique stationary ergodic distribution to which all solutions of the stochastic model will approach almost surely as noise intensities are bounded. When the basic reproduction number [Formula: see text] for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such that [Formula: see text], the cholera infection is suppressed by environmental noises. We carry out numerical simulations to illustrate our analysis, and to compare with the corresponding deterministic model. Biological implications are pointed out.

Dynamics of synthetic drug transmission models

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shitao Liu ◽  
Liang Zhang
Keyword(s):  
Stochastic Model ◽  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
History Of ◽  
Synthetic Drug ◽  
Transmission Models ◽  
The Basic Reproduction Number

Abstract The deterministic and stochastic synthetic drug transmission models with relapse are formulated. For the deterministic model, the basic reproduction number R 0 is derived. We show that if R 0 < 1, the drug-free equilibrium is globally asymptotically stable and if R 0 > 1, there exists a unique drug-addition equilibrium which is globally asymptotically stable. For the stochastic model, we show there exists a unique global positive solution of the stochastic model for any positive initial value. Then by constructing some stochastic Lyapunov functions, we show that the solution of the stochastic model is going around each of the steady states of the corresponding deterministic model under certain parametric conditions. The sensitive analysis of the basic reproduction number R 0 indicates that it is helpful to reduce the relapse rate of people who have a history of drug abuse in the control of synthetic drug spreading. Numerical simulations are carried out and approve our results.

Evolution of SARS-CoV-2 in the state of Alagoas-Brazil via an adaptive SIR model

2020 ◽  
pp. 2150040
Author(s):  
I. F. F. Dos Santos ◽  
G. M. A. Almeida ◽  
F. A. B. F. De Moura
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Sir Model ◽  
The State ◽  
Epidemic Dynamics ◽  
Historical Series ◽  
Input Parameters ◽  
Number Formula ◽  
Near Future ◽  
The Basic Reproduction Number

We investigate the spreading of SARS-CoV-2 in the state of Alagoas, northeast of Brazil, via an adaptive susceptible-infected-removed (SIR) model featuring dynamic recuperation and propagation rates. Input parameters are defined based on data made available by Alagoas Secretary of Health from April 19, 2020 on. We provide with the evolution of the basic reproduction number [Formula: see text] and reproduce the historical series of the number of confirmed cases with less than [Formula: see text] error. We offer predictions, from November 16 forward, over the epidemic situation in the near future and show that it will keep decelerating. Furthermore, the same model can be used to study the epidemic dynamics in other countries with great easiness and accuracy.

Global dynamics for a live-virus-blood model with distributed time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750067 ◽  
Author(s):  
Ding-Yu Zou ◽  
Shi-Fei Wang ◽  
Xue-Zhi Li
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Live Virus ◽  
Number Formula ◽  
Distributed Time Delay ◽  
The Basic Reproduction Number

In this paper, the global properties of a mathematical modeling of hepatitis C virus (HCV) with distributed time delays is studied. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states. It is shown that if the basic reproduction number [Formula: see text] is less than unity, then the uninfected steady state is globally asymptotically stable. If the basic reproduction number [Formula: see text] is larger than unity, then the infected steady state is globally asymptotically stable.

A MULTI-GROUP SVEIR EPIDEMIC MODEL WITH DISTRIBUTED DELAY AND VACCINATION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260001 ◽  
Author(s):  
JINLIANG WANG ◽  
YASUHIRO TAKEUCHI ◽  
SHENGQIANG LIU
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Distributed Delay ◽  
Necessary Condition ◽  
Global Dynamics ◽  
Vaccination Strategies ◽  
Graph Theoretical Approach ◽  
Number Formula ◽  
Next Generation Matrix ◽  
The Basic Reproduction Number

In this paper, based on a class of multi-group epidemic models of SEIR type with bilinear incidences, we introduce a vaccination compartment, leading to multi-group SVEIR model. We establish that the global dynamics are completely determined by the basic reproduction number [Formula: see text] which is defined by the spectral radius of the next generation matrix. Our proofs of global stability of the equilibria utilize a graph-theoretical approach to the method of Lyapunov functionals. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccines to obtain immunity or the possibility for them to be infected before acquiring immunity is neglected in each group, this condition will be satisfied and the disease can always be eradicated by suitable vaccination strategies. This may lead to over evaluation for the effect of vaccination.

scholarly journals Analysis of the Model on the Effect of Seasonal Factors on Malaria Transmission Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Victor Yiga ◽  
Hasifa Nampala ◽  
Julius Tumwiine
Keyword(s):  
Malaria Transmission ◽  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Current Control ◽  
Transmission Dynamics ◽  
Mosquito Vector ◽  
The Impact ◽  
The Basic Reproduction Number

Malaria is one of the world’s most prevalent epidemics. Current control and eradication efforts are being frustrated by rapid changes in climatic factors such as temperature and rainfall. This study is aimed at assessing the impact of temperature and rainfall abundance on the intensity of malaria transmission. A human host-mosquito vector deterministic model which incorporates temperature and rainfall dependent parameters is formulated. The model is analysed for steady states and their stability. The basic reproduction number is obtained using the next-generation method. It was established that the mosquito population depends on a threshold value θ , defined as the number of mosquitoes produced by a female Anopheles mosquito throughout its lifetime, which is governed by temperature and rainfall. The conditions for the stability of the equilibrium points are investigated, and it is shown that there exists a unique endemic equilibrium which is locally and globally asymptotically stable whenever the basic reproduction number exceeds unity. Numerical simulations show that both temperature and rainfall affect the transmission dynamics of malaria; however, temperature has more influence.

Bifurcation in Disease Dynamics with Latent Period of Infection and Media Awareness

2016 ◽  
Vol 26 (06) ◽  
pp. 1650097 ◽  
Author(s):  
Harkaran Singh ◽  
Joydip Dhar ◽  
Harbax Singh Bhatti
Keyword(s):  
Latent Period ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
State Variables ◽  
Sis Epidemic Model ◽  
Number Formula ◽  
Sensitive Parameters ◽  
Disease Free ◽  
The Basic Reproduction Number

In the present study, an SIS epidemic model with a latent period of infection and media awareness as control strategy is proposed. The asymptotic stability of the model is studied for both disease-free equilibrium and endemic equilibrium states with respect to the basic reproduction number [Formula: see text]. It is observed that the coefficient of media awareness [Formula: see text] does not affect [Formula: see text], but significantly affects the level of endemic equilibrium. Further, the specific conditions for the existence of Hopf bifurcation have been obtained for the endemic equilibrium state. We also performed the sensitivity analysis of the basic reproduction number and state variables at endemic steady state with respect to the model parameter and identified the respective sensitive parameters. Numerical simulations have been presented in support of our analytic findings.

Banana Xanthomonas wilt dynamics with mixed cultivars in a periodic environment

2019 ◽  
Vol 13 (01) ◽  
pp. 2050005
Author(s):  
Juliet N. Nakakawa ◽  
Joseph Y. T. Mugisha ◽  
Michael W. Shaw ◽  
Eldad Karamura
Keyword(s):  
Basic Reproduction Number ◽  
Rainy Season ◽  
Autonomous System ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Control Measures ◽  
Disease Dynamics ◽  
Periodic Environment ◽  
The Basic Reproduction Number ◽  
Banana Xanthomonas Wilt

In this paper, we study a deterministic model with non-autonomous system for mixed cultivars to assess the effect of cultivar susceptibility and seasonal variation on banana Xanthomonas wilt (BXW) disease dynamics. A special case of two cultivars classified as highly susceptible for inflorescence infection (ABB) and less susceptible (AAA) cultivar is considered. The basic reproduction number corresponding to the non-autonomous system is derived and numerically computed to determine disease dynamics. Results showed that the disease dies out whenever the periodic basic reproduction number is less than unity and a periodic solution is obtained when it is greater than one. Results further showed that for both cultivars, the basic reproduction number increases with increasing values of the transmission rates and declines exponentially with increasing values of roguing rates. The critical roguing rate of ABB-genome cultivar was higher than that of AAA-genome cultivars. The peaks in disease prevalence indicate the importance of effective implementation of controls during the rainy season. We conclude that highly susceptible cultivars play an important role in the spread of BXW and control measures should be effectively implemented during the rainy season if BXW is to be eradicated.

Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination

2015 ◽  
Vol 08 (06) ◽  
pp. 1550082 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Qaiser Badshah ◽  
Saeed Islam
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Preventive Vaccination ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

A TWO-STRAIN EPIDEMIC MODEL ON COMPLEX NETWORKS WITH DEMOGRAPHICS

2016 ◽  
Vol 24 (04) ◽  
pp. 577-609 ◽  
Author(s):  
YANRU YAO ◽  
JUPING ZHANG
Keyword(s):  
Basic Reproduction Number ◽  
Disease Transmission ◽  
Degree Distribution ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sis Model ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we develop a two-strain SIS model on heterogeneous networks with demographics for disease transmission. We calculate the basic reproduction number [Formula: see text] of infection for the model. We prove that if [Formula: see text], the disease-free equilibrium is globally asymptotically stable. If [Formula: see text], the conditions of the existence and global asymptotical stability of two boundary equilibria and the existence of endemic equilibria are established, respectively. Numerical simulations illustrate that the degree distribution of population varies with time before it reaches the stationary state. What is more, the basic reproduction number [Formula: see text] does not depend on the degree distribution like in the static network but depend on the demographic factors.

Stability of a Time Delayed SIR Epidemic Model Along with Nonlinear Incidence Rate and Holling Type-II Treatment Rate

2018 ◽  
Vol 15 (06) ◽  
pp. 1850055 ◽  
Author(s):  
Abhishek Kumar ◽  
Nilam
Keyword(s):  
Incidence Rate ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Type Ii ◽  
Treatment Rate ◽  
Nonlinear Incidence Rate ◽  
Number Formula ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we present a mathematical study of a deterministic model for the transmission and control of epidemics. The incidence rate of susceptible being infected is very crucial in the spread of disease. The delay in the incidence rate is proved fatal. In the present study, we propose an SIR mathematical model with the delay in the infected population. We are taking nonlinear incidence rate for epidemics along with Holling type II treatment rate for understanding the dynamics of the epidemics. Model stability has been done by the basic reproduction number [Formula: see text]. The model is locally asymptotically stable for disease-free equilibrium [Formula: see text] when the basic reproduction number [Formula: see text] is less than one ([Formula: see text]). We investigated the stability of the model for disease-free equilibrium at [Formula: see text] equals to one using center manifold theory. We also investigated the stability for endemic equilibrium [Formula: see text] at [Formula: see text]. Further, numerical simulations are presented to exemplify the analytical studies.

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