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Discrete and Continuous Dynamical Systems - B

June 2020 , Volume 25 , Issue 6

Special issue on population dynamics in epidemiology and ecology

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Preface: Population dynamics in epidemiology and ecology
Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi and Glenn Webb
2020, 25(6): i-ii doi: 10.3934/dcdsb.2020125 +[Abstract](2440) +[HTML](565) +[PDF](81.19KB)
Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model
Elena Travaglia, Valentina La Morgia and Ezio Venturino
2020, 25(6): 2023-2040 doi: 10.3934/dcdsb.2019200 +[Abstract](2946) +[HTML](571) +[PDF](1200.63KB)

In Europe, the Eastern grey squirrel is an allochthonous species causing severe impacts on the native red squirrel. The invasive species establishes complex relationships with the native one, out-competing it through resource and disease-mediated competition. However, recent research shed light on the potential role of a predator, the pine marten, in reversing the outcome of the competition between squirrels. Here, we investigate this hypothesis developing a one predator-two prey ecoepidemic model, including disease (squirrel poxvirus) transmission. We assess the equilibria of the dynamical system and investigate their sensitivity to ecosystem parameters changes through numerical simulations.

Our analysis reveals that the system is more likely to evolve toward points where the red squirrels thrive than toward a disease-and-red-squirrels-free point. Although the disease is likely to remain endemic in the system, the introduction of the pine marten destabilizes previous equilibria, favouring the native squirrel and facilitating wildlife managers in their efforts to protect it. Nevertheless, the complete eradication of grey squirrels could be achieved only for specified values of the predation rates and pine marten carrying capacity. The active management of grey squirrel populations remains therefore necessary to try to eradicate the invader from the system.

Takens–Bogdanov singularity for age structured models
Zhihua Liu and Rong Yuan
2020, 25(6): 2041-2056 doi: 10.3934/dcdsb.2019201 +[Abstract](2124) +[HTML](393) +[PDF](339.74KB)

The main purpose of this article is to derive a easily feasible method for the determination of Takens–Bogdanov singularity in age structured models. We present a SIR epidemic model with age structure as an example to illustrate the theoretical results.

A two-group age of infection epidemic model with periodic behavioral changes
Mamadou L. Diagne, Ousmane Seydi and Aissata A. B. Sy
2020, 25(6): 2057-2092 doi: 10.3934/dcdsb.2019202 +[Abstract](2231) +[HTML](377) +[PDF](446.64KB)

In this paper we propose a two-group SIR age of infection epidemic model by incorporating periodical behavioral changes for both susceptible and infected individuals. Our model allows different incubation periods for the two groups. It is proved in this paper that the persistence and extinction of the disease are determined by a threshold condition given in term of the basic reproductive number \begin{document}$ R_0 $\end{document}. That is, the disease is uniformly persistent if \begin{document}$ R_0 >1 $\end{document} with the existence of a positive periodic solution, while the disease goes to extinction if \begin{document}$ R_0< 1 $\end{document} with the global asymptotic stability of the disease free periodic solution. The model we have proposed is general and can be applied to a wide class of diseases.

The operating diagram for a model of competition in a chemostat with an external lethal inhibitor
Bachir Bar and Tewfik Sari
2020, 25(6): 2093-2120 doi: 10.3934/dcdsb.2019203 +[Abstract](2207) +[HTML](403) +[PDF](1387.82KB)

The inhibition is an important phenomenon, which promotes the stable coexistence of species, in the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external lethal inhibitor. The model is a four-dimensional system of ordinary differential equations. We give a complete analysis for the existence and local stability of all steady states. We describe the bifurcation diagram which gives the behavior of the system with respect to the operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. This diagram, is very useful to understand the model from both the mathematical and biological points of view.

A discrete model of competing species sharing a parasite
Rafael Bravo De La Parra and Luis Sanz
2020, 25(6): 2121-2142 doi: 10.3934/dcdsb.2019204 +[Abstract](2027) +[HTML](332) +[PDF](481.86KB)

In this work we develop a discrete model of competing species affected by a common parasite. We analyze the influence of the fast development of the shared disease on the community dynamics. The model is presented under the form of a two time scales discrete system with four variables. Thus, it becomes analytically tractable with the help of the appropriate reduction method. The 2-dimensional reduced system, that has the same asymptotic behaviour as the full model, is a generalization of the Leslie-Gower competition model. It has the unfrequent property in this kind of models of including multiple equilibrium attractors of mixed type. The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number \begin{document}$ R_0 $\end{document}. In some cases, initial conditions decide among several exclusion or coexistence scenarios.

Spreading speeds of rabies with territorial and diffusing rabid foxes
Khalaf M. Alanazi, Zdzislaw Jackiewicz and Horst R. Thieme
2020, 25(6): 2143-2183 doi: 10.3934/dcdsb.2019222 +[Abstract](2382) +[HTML](379) +[PDF](867.7KB)

A mathematical model is formulated for the fox rabies epidemic that swept through large areas of Europe during parts of the last century. Differently from other models, both territorial and diffusing rabid foxes are included, which leads to a system of partial differential, functional differential and differential-integral equations. The system is reduced to a scalar Volterra-Hammerstein integral equation to which the theory of spreading speeds pioneered by Aronson and Weinberger is applied. The spreading speed is given by an implicit formula which involves the space-time Laplace transform of the integral kernel. This formula can be exploited to find the dependence of the spreading speed on the model ingredients, in particular on those describing the interplay between diffusing and territorial rabid foxes and on the distribution of the latent period.

Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico
Pierre Magal, Glenn F. Webb and Yixiang Wu
2020, 25(6): 2185-2202 doi: 10.3934/dcdsb.2019223 +[Abstract](2881) +[HTML](463) +[PDF](3955.21KB)

A deterministic model is developed for the spatial spread of an epidemic disease in a geographical setting. The model is focused on outbreaks that arise from a small number of infected individuals in sub-regions of the geographical setting. The goal is to understand how spatial heterogeneity influences the transmission dynamics of susceptible and infected populations. The model consists of a system of partial differential equations with a diffusion term describing the spatial spread of an underlying microbial infectious agent. The model is applied to simulate the spatial spread of the 2016-2017 seasonal influenza epidemic in Puerto Rico. In this simulation, the reported case data from the Puerto Rican Department of Health are used to implement a numerical finite element scheme for the model. The model simulation explains the geographical evolution of this epidemic in Puerto Rico, consistent with the reported case data.

Density dependent replicator-mutator models in directed evolution
Matthieu Alfaro and Mario Veruete
2020, 25(6): 2203-2221 doi: 10.3934/dcdsb.2019224 +[Abstract](1499) +[HTML](315) +[PDF](725.64KB)

We analyze a replicator-mutator model arising in the context of directed evolution [24], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [14] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena.

Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system
Jean-Baptiste Burie, Ramsès Djidjou-Demasse and Arnaud Ducrot
2020, 25(6): 2223-2243 doi: 10.3934/dcdsb.2019225 +[Abstract](2147) +[HTML](348) +[PDF](471.51KB)

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. Using the variance of the dispersion in the phenotype trait space as a small parameter we provide a complete picture of the dynamical behaviour of the solutions of the problem. We show that the dynamics exhibits two main and long regimes – those durations are estimated – before the solution finally reaches its long time configuration, the endemic equilibrium. The analysis provided in this work rigorously explains and justifies the complex behaviour observed through numerical simulations of the system.

Global dynamics of an age-structured model with relapse
Mohammed Nor Frioui, Tarik Mohammed Touaoula and Bedreddine Ainseba
2020, 25(6): 2245-2270 doi: 10.3934/dcdsb.2019226 +[Abstract](2066) +[HTML](384) +[PDF](697.6KB)

The aim of this paper is to study a general class of $ SIRI $ age infection structured model where infectivity depends on the age since infection and where some individuals from the $ R $ class, also called quarantaine class in this work, can return to the infectiousness class after a while. Using classical technics we compute a basic reproductive number $ R_0 $ and show that the disease dies out when $ R_0 < 1 $ and persists if $ R_0 > 1 $. Some Lyapunov suitable functions are derived to prove global stability for the disease free equilibrium (DFE) when $ R_0 < 1 $ and for the endemic equilibrium (EE) when $ R_0 > 1 $. Using numerical results we show that the non homogeneous infectivity combined with the feedback to the infectiousness class of a part of the quarantaine population modifies drastically the behavior of the epidemic.

Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions
Zhihua Liu and Pierre Magal
2020, 25(6): 2271-2292 doi: 10.3934/dcdsb.2019227 +[Abstract](2057) +[HTML](329) +[PDF](396.46KB)

In this article we study a class of delay differential equations with infinite delay in weighted spaces of uniformly continuous functions. We focus on the integrated semigroup formulation of the problem and so doing we provide a spectral theory. As a consequence we obtain a local stability result and a Hopf bifurcation theorem for the semiflow generated by such a problem.

Optimal intervention strategies of a SI-HIV models with differential infectivity and time delays
Mboya Ba, P. Tchinda Mouofo, Mountaga Lam and Jean-Jules Tewa
2020, 25(6): 2293-2306 doi: 10.3934/dcdsb.2020035 +[Abstract](1249) +[HTML](167) +[PDF](594.36KB)

HIV infection is divided into stages of infection which are determined by the CD4 cells count progression. Through each stage, the time delay for the progression is important because the duration of HIV infection varies according to the infectious. Retarded optimal control theory is applied to a system of delays ordinary differential equations modeling the evolution of HIV with differential infectivity. Seeking to reduce the population of the infective individuals with low CD\begin{document}$ 4 $\end{document} cells, we use the ARV drug to control the fraction of infective individuals that is identified and will be put under treatment. We use optimal control theory to study our proposed system. Numerical simulations are provided to illustrate the effect of the Antiretroviral treatment (ART) taking into account the delays.

Dynamics of a metapopulation epidemic model with localized culling
Luca Bolzoni, Rossella Della Marca, Maria Groppi and Alessandra Gragnani
2020, 25(6): 2307-2330 doi: 10.3934/dcdsb.2020036 +[Abstract](1747) +[HTML](212) +[PDF](408.14KB)

A two–patches metapopulation mathematical model, describing the dynamics of Susceptibles and Infected in wildlife diseases, is presented. The two patches are identical in absence of control, and culling activities are performed in only one of them. Firstly, the dynamics of the system in absence of control is investigated. Then, two types of localized culling strategies (proactive end reactive) are considered. The proactive control is modeled by a constant culling effort, and for the ensuing model the disease free equilibrium is characterized and existence of the endemic equilibrium is discussed in terms of a suitable control reproduction number. The localized reactive control is modeled by a piecewise constant culling effort function, that introduces an extra–mortality when the number of infected individuals in the patch overcomes a given threshold. The reactive control is then analytically and numerically investigated in the frame of Filippov systems.

We find that localized culling may be ineffective in controlling diseases in wild populations when the infection affects host fecundity in addition to host mortality, even leading to unexpected increases in the number of infected individuals in the nearby areas.

An adaptative model for a multistage structured population under fluctuating environment
B. Ainseba and S. M. Bouguima
2020, 25(6): 2331-2349 doi: 10.3934/dcdsb.2020037 +[Abstract](1128) +[HTML](192) +[PDF](385.14KB)

We consider a modified version of a mathematical model describing the dynamics of the European Grapevine Moth, studied by Ainseba, Picart and Thiery. The improvment consists in including adaptation at the larval stage. We establish well-posedness of the model under suitable hypothesis.

A multi-stage SIR model for rumor spreading
Sun-Ho Choi, Hyowon Seo and Minha Yoo
2020, 25(6): 2351-2372 doi: 10.3934/dcdsb.2020124 +[Abstract](2916) +[HTML](203) +[PDF](1528.58KB)

We propose a multi-stage structured rumor spreading model that consists of ignorant, new spreader, old spreader, and stifler.We derive a mean field equation to obtain the multi-stage structured model on homogeneous networks. Since rumors spread from a few people, we consider a large population by setting the number of initial spread to one in total population \begin{document}$ n $\end{document} and limiting \begin{document}$ n $\end{document} to \begin{document}$ \infty $\end{document}. We investigate a threshold phenomenon of rumor outbreak in the sense of the large population limit by studying the driven multi-stage structured model. The main conclusion of this paper is that the proposed model has a threshold phenomenon in terms of a basic reproduction number which is similar to the SIR epidemic model. We present numerical simulations to show the developed theory numerically.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2021 CiteScore: 2.3




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