Dynamical Behavior of a stochastic SIRS epidemic model

In this paper we study the Kernack - MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold $\lambda$ which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of $\lambda$, the system can globally tend towards an endemic case or a disease free case. The aim of this work is also to describe completely the omega-limit set of all positive solutions to the model. Moreover, the attraction of the omega-limit set and the stationary distribution of solutions will be pointed out.Comment: 16 page

Intervention-Based Stochastic Disease Eradication

Disease control is of paramount importance in public health with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Thus intervention controls, such as vaccination of susceptible individuals and/or treatment of infectives, are typically based on a deterministic schedule, such as periodically vaccinating susceptible children based on school calendars. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.Comment: 18 pages, 10 Figure

Ergodic stationary distribution of stochastic virus mutation model with time delay

The virus mutation can increase the complexity of the infectious disease. In this paper, the dynamical characteristics of the virus mutation model are discussed. First, we built a stochastic virus mutation model with time delay. Second, the existence and uniqueness of global positive solutions for the proposed model is proved. Third, based on the analysis of the ergodic stationary distribution for the model, we discuss the influence mechanism between the different factors. Finally, the numerical simulation verifies the theoretical results

The Field Theory Approach to Percolation Processes

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous non-equilibrium active to absorbing state phase transitions whose asymptotic features are governed respectively by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions d_c = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder.Comment: 54 pages, figures include

Modèles de dynamique des populations dans un environnement aléatoire

This thesis addresses some issues associated with population dynamics in random environment. Random environment is described by a Markov process with values in a finite space and which, involve certain forces on the choice of vital rates, will lead the population dynamics. When the dynamic is modeled by a birth and death process, we will answer the question : When almost surely extinction settled ? (Bacaër and Ed-Darraz, 2014). In (Ed-Darraz and Khaladi, 2015) we are interested to the final size of an epidemic in random environment. J Math Biol. 69 (1) :73-90 Ed-Darraz A, Khaladi M (2015) On the final epidemic size in random environnement, Math. Biosc 266 : 10-14.Les travaux réalisés dans cette thèse abordent certaines questions relatives à la dynamique des populations dans un environnement aléatoire. L'environnement aléatoire est décrit par un processus Markovien à valeurs dans un espace fini et qui, en appliquant certaines forces sur le choix des taux vitaux, dirigera la dynamique de la population. Lorsque la dynamique est modélisée par un processus de naissance et de mort, on répondra à la question : quand est-ce qu'on a une extinction presque sûre d'une population ? (Bacaër and EdDarraz, 2014). Lorsque la dynamique est déterministe, nous avons démontré un résultat bien connu pour la taille finale d'une épidémie (Ed-Darraz and Khaladi, 2015) Bacaër N, Ed-Darraz A (2014) On linear birth-and-death processes in a random environment. J Math Biol. 69 (1) :73-90 Ed-Darraz A, Khaladi M (2015) On the final epidemic size in random environnement, Math. Biosc 266 : 10-14

Threshold Dynamics of a Stochastic S

A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control

Adaptive Dynamics in Fluctuating Environments and its Application in Evolutionary Studies

This article-based dissertation aims to understand by means of mathematical models how organisms evolutionarily respond to fluctuations in the environment. It uses the approach of adaptive dynamics to study the long-term evolution of phenotypic strategies in an environment that fluctuates in time because of biotic interactions and/or external factors. The dissertation demonstrates how this approach can reveal clear-cut explanations for complex environment-phenotype relationships by one general method-oriented article and two case studies in two additional articles. In the first article, I show that under the standard assumptions of adaptive dynamics, in particular mutation-limited evolution and small mutation steps, the generic dynamics of the resident-invader population in fluctuating environments can be fully characterized in terms of the behaviour near the boundaries of population state space, which in turn can be determined by the invasion criteria. This generalizes previous results for unstructured populations in a constant environment, which is important because it justifies the use and interpretation of various methods in the theory of adaptive dynamics for a significantly larger class of ecological situations that include fluctuating environments and structured populations. The two case studies are applications of the classification of invasion outcomes to explore the long-term evolutionary consequences of many successive invasion events. In the first case study, I investigate the evolution of the irreversible transition from a free-swimming state to an immobile sessile state as seen in many aquatic invertebrates. To this end, I study the adaptive dynamics of the settling rate of a hypothetical microorganism onto the wall of a chemostat with a fluctuating nutrient availability. The results show that different dilution rates and spatial competition mechanisms, as well as different frequencies of the nutrient fluctuations, have qualitatively different effects on the evolution of the settling rate as well as on species diversity. The model generates several hypotheses for further empirical studies. In the second case study, I investigate the evolution of the colonization rate in an extended competition-colonization model with ownership effects and stochastically varying mortality rate. I find that the strength of the trade-off, ownership effect and fluctuation intensity all have a non-monotonic effect on the emergence of species diversity via evolutionary branching. In particular, intermediate disturbance---as measured by the fluctuation intensity of the mortality rate---promotes evolutionary branching and hence the emergence of polymorphisms. This provides new evidence for the intermediate disturbance hypothesis. I also find that there can be multiple evolutionary attractors for polymorphic populations, each with its own basin of attraction. Consequently, random mutation-induced transition of coevolutionary trajectories between neighbouring basins of attraction makes the long-term evolutionary outcome uncertain. By means of these examples, the dissertation demonstrates that the approach of adaptive dynamics is a powerful tool for untangling the connection between environmental changes and adaptive strategies.Tämä artikkelipohjainen väitöskirja pyrkii ymmärtämään matemaattisten mallien avulla, miten organismit reagoivat evoluutioarvoisesti ympäristön vaihteluihin. Se käyttää adaptiivisen dynamiikan lähestymistapaa fenotyyppistrategioiden pitkän aikavälin kehityksen tutkimiseen ympäristössä, joka vaihtelee ajassa bioottisten vuorovaikutusten ja/tai ulkoisten tekijöiden vuoksi. Väitöskirja osoittaa, miten tämä lähestymistapa voi paljastaa selkeät selitykset monimutkaisille ympäristö fenotyyppisuhteille yhdellä yleisellä menetelmäkeskeisellä artikkelilla ja kahdella tapaustutkimuksella kahdessa ylimääräisessä artikkelissa. Ensimmäisessä artikkelissa osoitan, että mukautuvan dynamiikan, erityisesti mutaatiorajoitetun evoluution ja pienten mutaatiovaiheiden, oletusten mukaan asukas-hyökkääjän populaation yleinen dynamiikka vaihtelevissa ympäristöissä voidaan täysin luonnehtia käyttäytymisen suhteen väestön tilatilan rajat. Tämä puolestaan voidaan määrittää hyökkäyskriteereillä. Tämä yleistää strukturoimattomien populaatioiden aiemmat tulokset vakioympäristössä, mikä on tärkeää, koska se oikeuttaa erilaisten menetelmien käytön ja tulkinnan adaptiivisen dynamiikan teoriassa huomattavasti suuremmalle ekologisten tilanteiden luokalle, mukaan lukien vaihtelevat ympäristöt ja strukturoidut populaatiot. Nämä kaksi tapaustutkimusta ovat hyökkäystulosten luokittelun sovelluksia monien peräkkäisten hyökkäystapahtumien pitkän aikavälin evoluutiovaikutusten tutkimiseen. Ensimmäisessä tapaustutkimuksessa tutkin peruuttamattoman siirtymisen vapaan uinnin tilasta liikkumattomaan istumattomaan tilaan, kuten monissa vedessä elävissä selkärangattomissa. Tätä varten tutkin hypoteettisen mikro-organismin asettumisvauhdista mukautuvaa dynamiikkaa kemostaatin seinämään, jonka ravinteiden saatavuus vaihtelee. Tulokset osoittavat, että erilaisilla laimennusasteilla ja alueellisilla kilpailumekanismeilla sekä ravinnevaihteluiden eri taajuuksilla on laadullisesti erilaiset vaikutukset selvittelyasteen kehitykseen ja lajien monimuotoisuuteen. Malli tuottaa useita hypoteeseja empiirisiin lisätutkimuksiin. Toisessa tapaustutkimuksessa tutkin kolonisaatioasteen kehitystä laajennetussa kilpailu-kolonisaatiomallissa, jolla on omistusvaikutuksia ja stokastisesti vaihteleva kuolleisuusaste. Tulokset osoittavat, että kompromissin vahvuudella, omistusvaikutuksella ja vaihteluintensiteetillä on kaikilla ei-monotoninen vaikutus lajien monimuotoisuuden syntymiseen evoluutiohaarautumisen kautta. Erityisesti välituote häiriön (mitattuna kuolleisuuden vaihteluvoimakkuudella) edistävät evoluution haarautumista ja siten polymorfismien syntymistä. Tämä antaa uutta näyttöä välituotehäiriön hypoteesista. Huomaan myös, että polymorfisille populaatioille voi olla useita evoluutiovetovoimia, joista jokaisella on oma vetovoima-alue. Tästä johtuen satunnaisen mutaation aiheuttama koevoluutioreittien siirtyminen vierekkäisten vetovoima-alueiden välillä tekee pitkän aikavälin evoluutiotuloksen epävarmaksi. Näiden esimerkkien avulla väitöskirja osoittaa, että adaptiivisen dynamiikan lähestymistapa on tehokas työkalu ympäristömuutosten ja adaptiivisten strategioiden välisen yhteyden purkamiseen