Discrete-Time S-I-S Models With Simple and Complex Population Dynamics

Castillo‐Chavez, Carlos; Yakabu, Abdul-Aziz

doi:10.1007/978-1-4757-3667-0_9

Cited by 19 publications

(12 citation statements)

References 23 publications

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“…Allen has studied the discrete SI, SIS, SIR epidemic models and found that the SI and SIR models are similar in behavior to their continuous analogues under some natural restrictions, but the SIS model can have more diverse behaviour [23]. Castillo-Chavez and Yakubu have studied discrete time SIS models which exhibit bistability over a wide range of parameter values [24,25]. M@ndez and Fort investigated the dynamical evolution of discrete epidemic models by taking into account an intermediate population [26].…”

Section: The Disease-free Equilibrium and Its Stabilitymentioning

confidence: 99%

A discrete epidemic model for SARS transmission and control in China

Zhou

Brauer

2004

Mathematical and Computer Modelling

119

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Severe acute respiratory syndrome (SARS) is a rapidly spreading infectious disease which was transmitted in late 2002 and early 2003 to more than 28 countries through the medium of international travel. The evolution and spread of SARS has resulted in an international effort coordinated by the World Health Organization (WHO).We have formulated a discrete mathematical model to investigate the transmission of SARS and determined the basic reproductive number for this model to use as a threshold to determine the asymptotic behavior of the model. The dependence of the basic reproductive number on epidemic parameters has been studied. The parameters of the model have been estimated on the basis of statistical data and numerical simulations have been carried out to describe the transmission process for SARS in China. The simulation results matches the statistical data well and indicate that early quarantine and a high quarantine rate are crucial to the control of SARS.

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Section: The Disease-free Equilibrium and Its Stabilitymentioning

confidence: 99%

A discrete epidemic model for SARS transmission and control in China

Zhou

Brauer

2004

Mathematical and Computer Modelling

119

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“…Similar equations are obtained for patch replacing by . The dynamics of more general equations than these have been studied for a single population in [22]. Now, we introduce the diffusion of individuals between patches: assume that fractions and of susceptible and infected individuals from each population are exchanged at each step time.…”

Section: Model B: Dispersal In Two Patches With Sismentioning

confidence: 99%

Disease Spread in Coupled Populations: Minimizing Response Strategies Costs in Discrete Time Models

Alpízar

Gordillo

2013

Discrete Dynamics in Nature and Society

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Social distancing, vaccination, and medical treatments have been extensively studied and widely used to control the spread of infectious diseases. However, it is still a difficult task for health administrators to determine the optimal combination of these strategies when confronting disease outbreaks with limited resources, especially in the case of interconnected populations, where the flow of individuals is usually restricted with the hope of avoiding further contamination. We consider two coupled populations and examine them independently under two variants of well-known discrete time disease models. In both examples we compute approximations for the control levels necessary to minimize costs and quickly contain outbreaks. The main technique used is simulated annealing, a stochastic search optimization tool that, in contrast with traditional analytical methods, allows easy implementation to any number of patches with different kinds of couplings and internal dynamics.

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“…However, the disadvantage of this method is that it cannot guarantee the nonnegativity and boundedness of the solutions of the systems in some cases. As a result, some scholars establish discrete epidemic models by using the probability method [15][16][17]. Although the probability method can guarantee the nonnegativity and boundedness of the solutions, the model is more complex, and theoretical analysis is difficult.…”

Section: Introductionmentioning

confidence: 99%

“…Theoretical results of discrete epidemic models mainly focus on these aspects, including the computation of the basic reproduction number [8,18,19], the comparison between continuous-time epidemic models and discrete-time epidemic models [10,20], the local stability and global stability of the disease-free equilibrium and endemic equilibrium [6,7,9,11,15,16,21], the extinction, persistence, and permanence of a disease [12,[22][23][24][25], periodic systems [8,19,26], bifurcations and chaos phenomena [1,7,9,13,14,17,25,27,28], and so on. In particular, these papers about the bifurcation analysis of discrete epidemic model mainly involve the conditions on the existence of codimension-one bifurcations, such as fold bifurcation, flip bifurcation, Neimark-Sacker bifurcation [1,7,9,25,27,28], and so on, which were derived by using the center manifold theorem and bifurcation theory.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of a discrete SIR epidemic model with constant recovery

Cao

Wang

2020

Adv Differ Equ

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We state and study a discrete SIR epidemic model with bilinear incidence rate and constant recovery. We obtain conditions for the existence of the disease-free equilibrium and endemic equilibria. Theoretic analysis shows that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity, and the numerical simulations illustrate that it is asymptotically stable when the number is greater than unity. We also obtain conditions for the stability of the endemic equilibria. More attention is paid to the existence of the saddle-node bifurcation, the flip bifurcation, the 1 : 1 resonance, and the Neimark-Sacker bifurcation. We obtain sufficient conditions for those bifurcations. Our numerical simulations demonstrate our theoretical results and the complexity of the model.

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Discrete-Time S-I-S Models With Simple and Complex Population Dynamics

Cited by 19 publications

References 23 publications

A discrete epidemic model for SARS transmission and control in China

A discrete epidemic model for SARS transmission and control in China

Disease Spread in Coupled Populations: Minimizing Response Strategies Costs in Discrete Time Models

Bifurcation analysis of a discrete SIR epidemic model with constant recovery

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