The Markovian approach, which assumes exponentially distributed interinfection times, is dominant... more The Markovian approach, which assumes exponentially distributed interinfection times, is dominant in epidemic modeling. However, this assumption is unrealistic as an individual's infectiousness depends on its viral load and varies over time. In this paper, we present a Susceptible-Infected-Recovered-Vaccinated-Susceptible epidemic model incorporating non-Markovian infection processes. The model can be easily adapted to accurately capture the generation time distributions of emerging infectious diseases, which is essential for accurate epidemic prediction. We observe noticeable variations in the transient behavior under different infectiousness profiles and the same basic reproduction number R0. The theoretical analyses show that only R0 and the mean immunity period of the vaccinated individuals have an impact on the critical vaccination rate needed to achieve herd immunity. A vaccination level at the critical vaccination rate can ensure a very low incidence among the population in the case of future epidemics, regardless of the infectiousness profiles.
We propose a nonlinear model for the dynamics of urban burglaries which takes into account the de... more We propose a nonlinear model for the dynamics of urban burglaries which takes into account the deterring effect of the police. The model focuses on the timing of criminal activity rather than on the spatial spreading of burglaries and it is inspired in the age-dependent population dynamics. The structuring variables are the time elapsed between two consecutive offenses committed by a burglar or suffered by a house. The main ingredients of the model are the propensity of burglars to commit a crime and the rate at which houses are being burgled. These rates are taken as general as possible to allow different scenarios, including the widely used repeat victimization pattern. The dissuasive effect of the active police deployment is introduced by means of a memory term that depends on the number of the last committed burglaries. The asymptotic behavior of the model and the existence of a globally stable equilibrium are determined thanks to a suitable change of variables that involves a continuous rescaling of the time variable. This new approach provides some interesting analytic results on the equilibrium and the expected times between consecutive offenses. Numerical simulations are shown to illustrate these results.
After a presentation of the paper cited above at a workshop on Dynamic Networks at the Isaac Newt... more After a presentation of the paper cited above at a workshop on Dynamic Networks at the Isaac Newton Institute for Mathematical Sciences, Cambridge, Prof. Frank Ball in discussions explained two potential errors in our analysis. After further discussions, this was indeed confirmed. One mistake was an oversight, whereas the second one was more subtle. It turns out that the first mistake has impacts on the results of the paper, whereas the second one can be repaired and hence has no effect on the results. The oversight appears in Sect. 4.1 where the basic reproduction number R BA 0 for the SEIR-ω model is derived, and it only affects the case αω EI > 0. There the probability for an exposed but not yet infectious individual to rewire away from its infector, and reconnect to a new (susceptible) individual, is computed. The competing events are that the exposed individual rewires (at rate ω EI and only with probability α does the individual reconnect to a new individual), that the individual becomes infectious (when he/she stops rewiring), but also if the infector stops being infectious and recovers, The online version of the original article can be found under
The number of reported early syphilis cases in San Francisco has increased steadily since 2005. I... more The number of reported early syphilis cases in San Francisco has increased steadily since 2005. It is not yet clear what factors are responsible for such an increase. A recent analysis of the sexual contact network of men who have sex with men with syphilis in San Francisco has discovered a large connected component, members of which have a significantly higher chance of syphilis and HIV compared to nonmember individuals. This study investigates whether it is possible to exploit the existence of the largest connected component to design new notification strategies that can potentially contribute to reducing the number of cases. We develop a model capable of incorporating multiple types of notification strategies and compare the corresponding incidence of syphilis. Through extensive simulations, we show that notifying the community of the infection state of few central nodes appears to be the most effective approach, balancing the cost of notification and the reduction of syphilis incidence. Additionally, among the different measures of centrality, the eigenvector centrality reveals to be the best to reduce the incidence in the long term as long as the number of missing links (non-disclosed contacts) is not very large. Since 2001, San Francisco has experienced a sustained syphilis epidemic that has been nearly exclusively limited to men who have sex with men (MSM) 1. The epidemic, which was declining a few years ago, is now experiencing a new resurgence, not only in San Francisco but also across the USA and Europe 2. Innovative prevention measures are needed to reduce syphilis morbidity among MSM, and thus avoid spreading to a larger population. Previous work on sexually transmitted diseases has shown that sexual contact networks can be very useful to tailor mitigation strategies 3. The San Francisco Department of Public Health (SFDPH) maintains legally mandated case-based surveillance for syphilis which includes collected sociodemographic, treatment, and contact tracing information on reported and investigated syphilis cases. This surveillance system allows for the description of sexual networks among reported cases that are investigated. Each pair of individuals who have had a sexual encounter at least once in a time span are connected to each other with a link. The number of early syphilis cases in San Francisco has increased steadily from 26.6/100,000 cases in 2007 to 157.1/100,000 cases in 2015. Network theory indicates the importance of core individuals in sustaining sexually transmitted infections (STI) epidemics. An algorithm was developed in ref. 4 to identify the sexual contact network in the MSM community of San Francisco from case data routinely collected by the SFDPH to better understand the epidemiology of recent syphilis cases and explore possible new approaches for disease control. A sexual contact network with 2,428 nodes and 2,046 links was created. Within this network, a total of 457 disconnected components were identified. Of these, 78% consisted of only 2 or 3 individuals. Eleven components of 10 or more clients were identified, including a large connected component of 953 individuals. Clients in this largest component were more likely to be HIV-positive (P < 0.001 from Chi-square) and to have had more cases of syphilis in the past (P < 0.0001 from ANOVA) than clients belonging to smaller connected components 4. A partner notification (PN) system is a process in which an infected individual (called an index case) notifies (directly or indirectly) his partners (neighbors in the sexual network) of his infectious state 5. It is hoped that partners of the index case will then seek evaluation and possible treatment (alert state). Partner notification is considered the cornerstone of sexually transmitted disease control, aiming at controlling transmission by (1) treating exposed partners, (2) preventing reinfection of the index cases, and (3) preventing infection of healthy partners.
This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which ... more This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate ω (and reconnect to non-infectious individuals with probability α or else simply drop the edge if α = 0), so-called preventive rewiring. The models are denoted SIR-ω and SEIR-ω, and we focus attention on the early stages of an outbreak, where we derive the expressions for the basic reproduction number R 0 and the expected degree of the infectious nodes E(D I) using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-ω and SEIR-ω epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same E(D I) for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of R 0 computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with α > 0 (perhaps unaware of being infected),
Density-Dependent Diffusion and Epidemics on Heterogeneous Metapopulations
Trends in Mathematics, 2015
Systems with many components (individuals or local populations as cities, or metropolitan areas, ... more Systems with many components (individuals or local populations as cities, or metropolitan areas, or regions, …) connected by non-trivial associations or relationships can be statistically described by means of the formalism of complex networks which is based on descriptors like degree distributions, degree-degree correlations, etc. In the last years, many researchers from different fields have been using different approaches to model processes taking place on complex networks.
The existence of a die-out threshold (different from the classic disease-invasion one) defining a... more The existence of a die-out threshold (different from the classic disease-invasion one) defining a region of slow extinction of an epidemic has been proved elsewhere for susceptible-awareinfectious-susceptible models without awareness decay, through bifurcation analysis. By means of an equivalent mean-field model defined on regular random networks, we interpret the dynamics of the system in this region and prove that the existence of bifurcation for of this second epidemic threshold crucially depends on the absence of awareness decay. We show that the continuum of equilibria that characterizes the slow die-out dynamics collapses into a unique equilibrium when a constant rate of awareness decay is assumed, no matter how small, and that the resulting bifurcation from the disease-free equilibrium is equivalent to that of standard epidemic models. We illustrate these findings with continuous-time stochastic simulations on regular random networks with different degrees. Finally, the behaviour of solutions with and without decay in awareness is compared around the second epidemic threshold for a small rate of awareness decay.
Monte carlo simulations of a SIS-diffusion model in heterogeneous metapopulations
We present a model for the spread of infectious diseases in heterogeneous metapopulations. The eq... more We present a model for the spread of infectious diseases in heterogeneous metapopulations. The equations governing the epidemic dynamics have been recently proposed as an alternative formulation in a continuous‐time framework. In the model, the ...
In this paper we examine a prey-predator system with a characteristic of the predator subject to ... more In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by th e so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclud e the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion .
The first part of this paper is devoted to a complete description of the dynamics of a continuous... more The first part of this paper is devoted to a complete description of the dynamics of a continuously structured population model coupled with a dynamical resource. In the model, it is assumed that the energy each individual obtains from the resource is channeled between growth and reproduction in a proportion that depends on the individual's size. In the second part, an optimal allocation of this energy is obtained that turns out to be a convergence-stable ESS and is described by what is called a "bang-bang" strategy.
We present a study of the continuous-time equations governing the dynamics of a susceptible-infec... more We present a study of the continuous-time equations governing the dynamics of a susceptible-infectedsusceptible model on heterogeneous metapopulations. These equations have been recently proposed as an alternative formulation for the spread of infectious diseases in metapopulations in a continuous-time framework. Individual-based Monte Carlo simulations of epidemic spread in uncorrelated networks are also performed revealing a good agreement with analytical predictions under the assumption of simultaneous transmission or recovery and migration processes.
This paper deals with the mathematical description of the asymptotic behavior of the solutions of... more This paper deals with the mathematical description of the asymptotic behavior of the solutions of a couple of models for the dynamics of growing networks based on connecting, with a higher probability, nodes that have a neighbor in common. The first model, proposed by A. Vázquez, is nonlinear and, in general, the long-time behavior of the solutions differs from the one predicted by the linear reduction proposed in its original treatment. A second model is specifically derived from the rules defining an in silico model also proposed by Vázquez to simulate the growth of a network under the mechanism of connecting nearest neighbors. The two analytical models lead to very different predictions for the configuration of the network that are tested using the simulations of the in silico model.
A pair-approximation model for the spatial dynamics of a height-structured tree population is def... more A pair-approximation model for the spatial dynamics of a height-structured tree population is defined on a regular lattice where each site can be in one of three states: empty (gap site), occupied by an immature tree, and occupied by a mature tree. The nonlinearities are associated with resource competition effects of mature trees on immature ones (asymmetric competition) affecting the mortality of the latter but not their growth. The survival-extinction transition of the forest is expressed; the early dynamics of colonization are described in terms of local densities. Predictions of the pair approximation model are compared with results from numerical simulations of cellular automata.
Mathematical Models and Methods in Applied Sciences, 2006
In this paper we present a proof of existence and uniqueness of solution for a class of PDE model... more In this paper we present a proof of existence and uniqueness of solution for a class of PDE models of size structured populations with distributed state-at-birth and having nonlinearities defined by an infinite-dimensional environment. The latter means that each member of the population experiences an environment according to a sort of average of the population size depending on the individual size, rank or any other variable structuring the population. The proof of the local existence and uniqueness of solution as well as the continuous dependence on initial continuous is based on the general theory of quasi-linear evolution equations in nonreflexive Banach spaces, while the global existence proof is based on the integration of the local solution along characteristic curves.
This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete popu... more This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete population model with a general transition matrix. In the model, individuals are categorized into n dominance classes, newborns lie in the subordinate class, and it is considered as evolutionary trait a vector g of probabilities of transition among classes. For this trait, we obtain the evolutionary singular strategy and prove its neutral evolutionary stability. Finally, we obtain conditions for the invading potential of such a strategy, which is sufficient for the convergence stability of the latter. With the help of the previous results, we provide an explanation for the bimodal distribution of badges of status observed in the Siskin (Carduelis spinus). In the Siskin, as in several bird species, patches of pigmented plumage signal the dominance status of the bearer to opponents, and central to the discussion on the evolution of status signalling is the understanding of which should be the frequency distribution of badge sizes. Though some simple verbal models predicted a bimodal distribution, up to now most species display normal distributions and bimodality has only been described for the Siskin. In this paper, we give conditions leading to one of these two distributions in terms of the survival, fecundity and aggression rates in each dominance class.
In this paper, we present a general selection-mutation model of evolution on a one-dimensional co... more In this paper, we present a general selection-mutation model of evolution on a one-dimensional continuous fitness space. The formulation of our model includes both the classical diffusion approach to mutation process as well as an alternative approach based on an integral operator with a mutation kernel. We show that both approaches produce fundamentally equivalent results. To illustrate the suitability of our model, we focus its analytical study into its application to recent experimental studies of in vitro viral evolution. More specifically, these experiments were designed to test previous theoretical predictions regarding the effects of multiple infection dynamics (i.e., coinfection and superinfection) on the virulence of evolving viral populations. The results of these experiments, however, did not match with previous theory. By contrast, the model we present here helps to understand the underlying viral dynamics on these experiments and makes new testable predictions about the role of parameters such the time between successive infections and the growth rates of resident and invading populations.
Food webs are complex networks describing trophic interactions in ecological communities. Since R... more Food webs are complex networks describing trophic interactions in ecological communities. Since Robert May's seminal work on random structured food webs, the complexity-stability debate is a central issue in ecology: does network complexity increase or decrease food-web persistence? A multi-species predator-prey model incorporating adaptive predation shows that the action of ecological dynamics on the topology of a food web (whose initial configuration is generated either by the cascade model or by the niche model) render, when a significant fraction of adaptive predators is present, similar hyperbolic complexity-persistence relationships as those observed in empirical food webs. It is also shown that the apparent positive relation between complexity and persistence in food webs generated under the cascade model, which has been pointed out in previous papers, disappears when the final connectance is used instead of the initial one to explain species persistence.
The Markovian approach, which assumes exponentially distributed interinfection times, is dominant... more The Markovian approach, which assumes exponentially distributed interinfection times, is dominant in epidemic modeling. However, this assumption is unrealistic as an individual's infectiousness depends on its viral load and varies over time. In this paper, we present a Susceptible-Infected-Recovered-Vaccinated-Susceptible epidemic model incorporating non-Markovian infection processes. The model can be easily adapted to accurately capture the generation time distributions of emerging infectious diseases, which is essential for accurate epidemic prediction. We observe noticeable variations in the transient behavior under different infectiousness profiles and the same basic reproduction number R0. The theoretical analyses show that only R0 and the mean immunity period of the vaccinated individuals have an impact on the critical vaccination rate needed to achieve herd immunity. A vaccination level at the critical vaccination rate can ensure a very low incidence among the population in the case of future epidemics, regardless of the infectiousness profiles.
We propose a nonlinear model for the dynamics of urban burglaries which takes into account the de... more We propose a nonlinear model for the dynamics of urban burglaries which takes into account the deterring effect of the police. The model focuses on the timing of criminal activity rather than on the spatial spreading of burglaries and it is inspired in the age-dependent population dynamics. The structuring variables are the time elapsed between two consecutive offenses committed by a burglar or suffered by a house. The main ingredients of the model are the propensity of burglars to commit a crime and the rate at which houses are being burgled. These rates are taken as general as possible to allow different scenarios, including the widely used repeat victimization pattern. The dissuasive effect of the active police deployment is introduced by means of a memory term that depends on the number of the last committed burglaries. The asymptotic behavior of the model and the existence of a globally stable equilibrium are determined thanks to a suitable change of variables that involves a continuous rescaling of the time variable. This new approach provides some interesting analytic results on the equilibrium and the expected times between consecutive offenses. Numerical simulations are shown to illustrate these results.
After a presentation of the paper cited above at a workshop on Dynamic Networks at the Isaac Newt... more After a presentation of the paper cited above at a workshop on Dynamic Networks at the Isaac Newton Institute for Mathematical Sciences, Cambridge, Prof. Frank Ball in discussions explained two potential errors in our analysis. After further discussions, this was indeed confirmed. One mistake was an oversight, whereas the second one was more subtle. It turns out that the first mistake has impacts on the results of the paper, whereas the second one can be repaired and hence has no effect on the results. The oversight appears in Sect. 4.1 where the basic reproduction number R BA 0 for the SEIR-ω model is derived, and it only affects the case αω EI > 0. There the probability for an exposed but not yet infectious individual to rewire away from its infector, and reconnect to a new (susceptible) individual, is computed. The competing events are that the exposed individual rewires (at rate ω EI and only with probability α does the individual reconnect to a new individual), that the individual becomes infectious (when he/she stops rewiring), but also if the infector stops being infectious and recovers, The online version of the original article can be found under
The number of reported early syphilis cases in San Francisco has increased steadily since 2005. I... more The number of reported early syphilis cases in San Francisco has increased steadily since 2005. It is not yet clear what factors are responsible for such an increase. A recent analysis of the sexual contact network of men who have sex with men with syphilis in San Francisco has discovered a large connected component, members of which have a significantly higher chance of syphilis and HIV compared to nonmember individuals. This study investigates whether it is possible to exploit the existence of the largest connected component to design new notification strategies that can potentially contribute to reducing the number of cases. We develop a model capable of incorporating multiple types of notification strategies and compare the corresponding incidence of syphilis. Through extensive simulations, we show that notifying the community of the infection state of few central nodes appears to be the most effective approach, balancing the cost of notification and the reduction of syphilis incidence. Additionally, among the different measures of centrality, the eigenvector centrality reveals to be the best to reduce the incidence in the long term as long as the number of missing links (non-disclosed contacts) is not very large. Since 2001, San Francisco has experienced a sustained syphilis epidemic that has been nearly exclusively limited to men who have sex with men (MSM) 1. The epidemic, which was declining a few years ago, is now experiencing a new resurgence, not only in San Francisco but also across the USA and Europe 2. Innovative prevention measures are needed to reduce syphilis morbidity among MSM, and thus avoid spreading to a larger population. Previous work on sexually transmitted diseases has shown that sexual contact networks can be very useful to tailor mitigation strategies 3. The San Francisco Department of Public Health (SFDPH) maintains legally mandated case-based surveillance for syphilis which includes collected sociodemographic, treatment, and contact tracing information on reported and investigated syphilis cases. This surveillance system allows for the description of sexual networks among reported cases that are investigated. Each pair of individuals who have had a sexual encounter at least once in a time span are connected to each other with a link. The number of early syphilis cases in San Francisco has increased steadily from 26.6/100,000 cases in 2007 to 157.1/100,000 cases in 2015. Network theory indicates the importance of core individuals in sustaining sexually transmitted infections (STI) epidemics. An algorithm was developed in ref. 4 to identify the sexual contact network in the MSM community of San Francisco from case data routinely collected by the SFDPH to better understand the epidemiology of recent syphilis cases and explore possible new approaches for disease control. A sexual contact network with 2,428 nodes and 2,046 links was created. Within this network, a total of 457 disconnected components were identified. Of these, 78% consisted of only 2 or 3 individuals. Eleven components of 10 or more clients were identified, including a large connected component of 953 individuals. Clients in this largest component were more likely to be HIV-positive (P < 0.001 from Chi-square) and to have had more cases of syphilis in the past (P < 0.0001 from ANOVA) than clients belonging to smaller connected components 4. A partner notification (PN) system is a process in which an infected individual (called an index case) notifies (directly or indirectly) his partners (neighbors in the sexual network) of his infectious state 5. It is hoped that partners of the index case will then seek evaluation and possible treatment (alert state). Partner notification is considered the cornerstone of sexually transmitted disease control, aiming at controlling transmission by (1) treating exposed partners, (2) preventing reinfection of the index cases, and (3) preventing infection of healthy partners.
This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which ... more This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate ω (and reconnect to non-infectious individuals with probability α or else simply drop the edge if α = 0), so-called preventive rewiring. The models are denoted SIR-ω and SEIR-ω, and we focus attention on the early stages of an outbreak, where we derive the expressions for the basic reproduction number R 0 and the expected degree of the infectious nodes E(D I) using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-ω and SEIR-ω epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same E(D I) for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of R 0 computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with α > 0 (perhaps unaware of being infected),
Density-Dependent Diffusion and Epidemics on Heterogeneous Metapopulations
Trends in Mathematics, 2015
Systems with many components (individuals or local populations as cities, or metropolitan areas, ... more Systems with many components (individuals or local populations as cities, or metropolitan areas, or regions, …) connected by non-trivial associations or relationships can be statistically described by means of the formalism of complex networks which is based on descriptors like degree distributions, degree-degree correlations, etc. In the last years, many researchers from different fields have been using different approaches to model processes taking place on complex networks.
The existence of a die-out threshold (different from the classic disease-invasion one) defining a... more The existence of a die-out threshold (different from the classic disease-invasion one) defining a region of slow extinction of an epidemic has been proved elsewhere for susceptible-awareinfectious-susceptible models without awareness decay, through bifurcation analysis. By means of an equivalent mean-field model defined on regular random networks, we interpret the dynamics of the system in this region and prove that the existence of bifurcation for of this second epidemic threshold crucially depends on the absence of awareness decay. We show that the continuum of equilibria that characterizes the slow die-out dynamics collapses into a unique equilibrium when a constant rate of awareness decay is assumed, no matter how small, and that the resulting bifurcation from the disease-free equilibrium is equivalent to that of standard epidemic models. We illustrate these findings with continuous-time stochastic simulations on regular random networks with different degrees. Finally, the behaviour of solutions with and without decay in awareness is compared around the second epidemic threshold for a small rate of awareness decay.
Monte carlo simulations of a SIS-diffusion model in heterogeneous metapopulations
We present a model for the spread of infectious diseases in heterogeneous metapopulations. The eq... more We present a model for the spread of infectious diseases in heterogeneous metapopulations. The equations governing the epidemic dynamics have been recently proposed as an alternative formulation in a continuous‐time framework. In the model, the ...
In this paper we examine a prey-predator system with a characteristic of the predator subject to ... more In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by th e so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclud e the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion .
The first part of this paper is devoted to a complete description of the dynamics of a continuous... more The first part of this paper is devoted to a complete description of the dynamics of a continuously structured population model coupled with a dynamical resource. In the model, it is assumed that the energy each individual obtains from the resource is channeled between growth and reproduction in a proportion that depends on the individual's size. In the second part, an optimal allocation of this energy is obtained that turns out to be a convergence-stable ESS and is described by what is called a "bang-bang" strategy.
We present a study of the continuous-time equations governing the dynamics of a susceptible-infec... more We present a study of the continuous-time equations governing the dynamics of a susceptible-infectedsusceptible model on heterogeneous metapopulations. These equations have been recently proposed as an alternative formulation for the spread of infectious diseases in metapopulations in a continuous-time framework. Individual-based Monte Carlo simulations of epidemic spread in uncorrelated networks are also performed revealing a good agreement with analytical predictions under the assumption of simultaneous transmission or recovery and migration processes.
This paper deals with the mathematical description of the asymptotic behavior of the solutions of... more This paper deals with the mathematical description of the asymptotic behavior of the solutions of a couple of models for the dynamics of growing networks based on connecting, with a higher probability, nodes that have a neighbor in common. The first model, proposed by A. Vázquez, is nonlinear and, in general, the long-time behavior of the solutions differs from the one predicted by the linear reduction proposed in its original treatment. A second model is specifically derived from the rules defining an in silico model also proposed by Vázquez to simulate the growth of a network under the mechanism of connecting nearest neighbors. The two analytical models lead to very different predictions for the configuration of the network that are tested using the simulations of the in silico model.
A pair-approximation model for the spatial dynamics of a height-structured tree population is def... more A pair-approximation model for the spatial dynamics of a height-structured tree population is defined on a regular lattice where each site can be in one of three states: empty (gap site), occupied by an immature tree, and occupied by a mature tree. The nonlinearities are associated with resource competition effects of mature trees on immature ones (asymmetric competition) affecting the mortality of the latter but not their growth. The survival-extinction transition of the forest is expressed; the early dynamics of colonization are described in terms of local densities. Predictions of the pair approximation model are compared with results from numerical simulations of cellular automata.
Mathematical Models and Methods in Applied Sciences, 2006
In this paper we present a proof of existence and uniqueness of solution for a class of PDE model... more In this paper we present a proof of existence and uniqueness of solution for a class of PDE models of size structured populations with distributed state-at-birth and having nonlinearities defined by an infinite-dimensional environment. The latter means that each member of the population experiences an environment according to a sort of average of the population size depending on the individual size, rank or any other variable structuring the population. The proof of the local existence and uniqueness of solution as well as the continuous dependence on initial continuous is based on the general theory of quasi-linear evolution equations in nonreflexive Banach spaces, while the global existence proof is based on the integration of the local solution along characteristic curves.
This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete popu... more This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete population model with a general transition matrix. In the model, individuals are categorized into n dominance classes, newborns lie in the subordinate class, and it is considered as evolutionary trait a vector g of probabilities of transition among classes. For this trait, we obtain the evolutionary singular strategy and prove its neutral evolutionary stability. Finally, we obtain conditions for the invading potential of such a strategy, which is sufficient for the convergence stability of the latter. With the help of the previous results, we provide an explanation for the bimodal distribution of badges of status observed in the Siskin (Carduelis spinus). In the Siskin, as in several bird species, patches of pigmented plumage signal the dominance status of the bearer to opponents, and central to the discussion on the evolution of status signalling is the understanding of which should be the frequency distribution of badge sizes. Though some simple verbal models predicted a bimodal distribution, up to now most species display normal distributions and bimodality has only been described for the Siskin. In this paper, we give conditions leading to one of these two distributions in terms of the survival, fecundity and aggression rates in each dominance class.
In this paper, we present a general selection-mutation model of evolution on a one-dimensional co... more In this paper, we present a general selection-mutation model of evolution on a one-dimensional continuous fitness space. The formulation of our model includes both the classical diffusion approach to mutation process as well as an alternative approach based on an integral operator with a mutation kernel. We show that both approaches produce fundamentally equivalent results. To illustrate the suitability of our model, we focus its analytical study into its application to recent experimental studies of in vitro viral evolution. More specifically, these experiments were designed to test previous theoretical predictions regarding the effects of multiple infection dynamics (i.e., coinfection and superinfection) on the virulence of evolving viral populations. The results of these experiments, however, did not match with previous theory. By contrast, the model we present here helps to understand the underlying viral dynamics on these experiments and makes new testable predictions about the role of parameters such the time between successive infections and the growth rates of resident and invading populations.
Food webs are complex networks describing trophic interactions in ecological communities. Since R... more Food webs are complex networks describing trophic interactions in ecological communities. Since Robert May's seminal work on random structured food webs, the complexity-stability debate is a central issue in ecology: does network complexity increase or decrease food-web persistence? A multi-species predator-prey model incorporating adaptive predation shows that the action of ecological dynamics on the topology of a food web (whose initial configuration is generated either by the cascade model or by the niche model) render, when a significant fraction of adaptive predators is present, similar hyperbolic complexity-persistence relationships as those observed in empirical food webs. It is also shown that the apparent positive relation between complexity and persistence in food webs generated under the cascade model, which has been pointed out in previous papers, disappears when the final connectance is used instead of the initial one to explain species persistence.
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Papers by Joan Saldaña