scholarly journals Positive Periodic Solutions of an Epidemic Model with Seasonality

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Gui-Quan Sun ◽  
Zhenguo Bai ◽  
Zi-Ke Zhang ◽  
Tao Zhou ◽  
Zhen Jin

An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated. For the autonomous case, we give the attractive regions of equilibria and perform some numerical simulations. Basic demographic reproduction numberRdis obtained. Moreover, only the basic reproduction numberR0cannot ensure the existence of the positive equilibrium, which needs additional conditionRd>R1. For the nonautonomous case, by introducing the basic reproduction number defined by the spectral radius, we study the uniform persistence and extinction of the disease. The results show that for the periodic system the basic reproduction number is more accurate than the average reproduction number.

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Wang ◽  
Shujing Gao ◽  
Yueli Luo ◽  
Dehui Xie

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction numberR0which is defined through the spectral radius of a linear integral operator. IfR0< 1, then the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.


2019 ◽  
Vol 12 (05) ◽  
pp. 1950051
Author(s):  
Xia Wang ◽  
Yuming Chen ◽  
Xinyu Song

In this paper, we propose and analyze a cholera model. The model incorporates both direct transmission (person-to-person transmission) and indirect transmission (contaminated environment-to-person transmission: hyper-infectivity and lower-infectivity). Moreover, we employ general nonlinear incidences and introduce infection age of infectious individuals and biological ages of pathogens in the environment. After considering the well-posedness of the system, we study the existence and local stability of steady states, which is determined by the basic reproduction number. To establish the attractivity of the infection steady state, we also get the uniform persistence and existence of compact global attractors. The main result is a threshold dynamics obtained by applying the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the infection-free steady state is globally asymptotically stable while when the basic reproduction number is larger than one, the infection steady state attracts each solution with nonzero infection force at some time point. The effect of multiple transmission modes on the disease dynamics is also discussed.


2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Yali Yang ◽  
Chenping Guo ◽  
Luju Liu ◽  
Tianhua Zhang ◽  
Weiping Liu

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.


CAUCHY ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 122-132
Author(s):  
Joko Harianto ◽  
Inda Puspita Sari

Discussion of local stability analysis of SVIR models in this article is included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability analysis that had not been discussed in some articles related to the SVIR model. The SVIR models discussed in this article involve logistics growth in the vaccinated compartment. The results obtained, i.e. if the basic reproduction number less than one and m is positive, then there is one equilibrium point i.e. E0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if the basic reproduction number more than one and b1 more than b, then there are two equilibrium points i.e. disease-free equilibrium point denoted by E0 and the endemic equilibrium point denoted by E1*. In this case the endemic equilibrium point E1* is locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results.


2021 ◽  
Vol 19 (1) ◽  
pp. 209-224
Author(s):  
Abdelheq Mezouaghi ◽  
◽  
Salih Djillali ◽  
Anwar Zeb ◽  
Kottakkaran Sooppy Nisar ◽  
...  

<abstract><p>In the case of an epidemic, the government (or population itself) can use protection for reducing the epidemic. This research investigates the global dynamics of a delayed epidemic model with partial susceptible protection. A threshold dynamics is obtained in terms of the basic reproduction number, where for $ R_0 &lt; 1 $ the infection will extinct from the population. But, for $ R_0 &gt; 1 $ it has been shown that the disease will persist, and the unique positive equilibrium is globally asymptotically stable. The principal purpose of this research is to determine a relation between the isolation rate and the basic reproduction number in such a way we can eliminate the infection from the population. Moreover, we will determine the minimal protection force to eliminate the infection for the population. A comparative analysis with the classical SIR model is provided. The results are supported by some numerical illustrations with their epidemiological relevance.</p></abstract>


2020 ◽  
Vol 28 (04) ◽  
pp. 927-944
Author(s):  
HUIJUAN LIU ◽  
FEI XU ◽  
JIA-FANG ZHANG

In this work, we construct an age-structured HIV-1 infection model to investigate the interplay between [Formula: see text] cells and viruses. In our model, we assume that the variations in the death rate of productively infected [Formula: see text] cells and the production rate of virus in infected cells are all age-dependent, and the target cells follow logistic growth. We perform mathematical analysis and prove the persistence of the semi-flow of the system. We calculate the basic reproduction number and prove the local and global stability of the steady states. We show that if the basic reproduction number is less than one, the disease-free equilibrium is globally asymptotically stable, and if the basic reproduction number is greater than one, the infected steady state is locally asymptotically stable.


2021 ◽  
Vol 16 ◽  
pp. 1-9
Author(s):  
Joko Harianto

This article discusses modifications to the SEIL model that involve logistical growth. This model is used to describe the dynamics of the spread of tuberculosis disease in the population. The existence of the model's equilibrium points and its local stability depends on the basic reproduction number. If the basic reproduction number is less than unity, then there is one equilibrium point that is locally asymptotically stable. The equilibrium point is a disease-free equilibrium point. If the basic reproduction number ranges from one to three, then there are two equilibrium points. The two equilibrium points are disease-free equilibrium and endemic equilibrium points. Furthermore, for this case, the endemic equilibrium point is locally asymptotically stable.


Author(s):  
Jude D. Kong ◽  
Edward W. Tekwa ◽  
Sarah A. Gignoux-Wolfsohn

AbstractObjectiveTo assess whether the basic reproduction number (R0) of COVID-19 is different across countries and what national-level demographic, social, and environmental factors characterize initial vulnerability to the virus.MethodsWe fit logistic growth curves to reported daily case numbers, up to the first epidemic peak. This fitting estimates R0. We then use a generalized additive model to discern the effects, and include 5 random effect covariates to account for potential differences in testing and reporting that can bias the estimated R0.FindingsWe found that the mean R0 is 1.70 (S.D. 0.57), with a range between 1.10 (Ghana) and 3.52 (South Korea). We identified four factors-population between 20-34 years old (youth), population residing in urban agglomerates over 1 million (city), social media use to organize offline action (social media), and GINI income inequality-as having strong relationships with R0. An intermediate level of youth and GINI inequality are associated with high R0, while high city population and high social media use are associated with high R0. Environmental and climate factors were not found to have strong relationships with R0.ConclusionStudies that aim to measure the effectiveness of interventions should account for the intrinsic differences between populations.


PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0252373
Author(s):  
Jude Dzevela Kong ◽  
Edward W. Tekwa ◽  
Sarah A. Gignoux-Wolfsohn

Objective To assess whether the basic reproduction number (R0) of COVID-19 is different across countries and what national-level demographic, social, and environmental factors other than interventions characterize initial vulnerability to the virus. Methods We fit logistic growth curves to reported daily case numbers, up to the first epidemic peak, for 58 countries for which 16 explanatory covariates are available. This fitting has been shown to robustly estimate R0 from the specified period. We then use a generalized additive model (GAM) to discern both linear and nonlinear effects, and include 5 random effect covariates to account for potential differences in testing and reporting that can bias the estimated R0. Findings We found that the mean R0 is 1.70 (S.D. 0.57), with a range between 1.10 (Ghana) and 3.52 (South Korea). We identified four factors—population between 20–34 years old (youth), population residing in urban agglomerates over 1 million (city), social media use to organize offline action (social media), and GINI income inequality—as having strong relationships with R0, across countries. An intermediate level of youth and GINI inequality are associated with high R0, (n-shape relationships), while high city population and high social media use are associated with high R0. Pollution, temperature, and humidity did not have strong relationships with R0 but were positive. Conclusion Countries have different characteristics that predispose them to greater intrinsic vulnerability to COVID-19. Studies that aim to measure the effectiveness of interventions across locations should account for these baseline differences in social and demographic characteristics.


2020 ◽  
Vol 1 (1) ◽  
pp. 65-84
Author(s):  
Mahnaz Alavinejad ◽  
Jemisa Sadiku ◽  
Jianhong Wu

For a variety of tick species, the resistance, behavioural and immunological response of hosts has been reported in the biological literature but its impact on tick population dynamics has not been mathematically formulated and analyzed using dynamical models reflecting the full biological stages of ticks. Here we develop and simulate a delay differential equation model, with a particular focus on resistance resulting in grooming behaviour. We calculate the basic reproduction number using the spectral analysis of delay differential equations with positive feedback, and establish the existence and uniqueness of a positive equilibrium when the basic reproduction number exceeds unit. We also conduct numerical and sensitivity analysis about the dependence of this positive equilibrium on the the parameter relevant to grooming behaviour. We numerically obtain the relationship between grooming behaviour and equilibrium value at different stages.


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