We study an individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity. Each individual is associated with a random infection-age dependent infectivity function. Individuals are infected through interactions across the locations with heterogeneous effects. The epidemic dynamics can be described using a time-space representation for the the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals. We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations. We then derive the PDE models from the limiting time-space integral equations, in particular, the density (with respect to the infection age) of the time-age-space integral equation for the number of infected individuals tracking the age of infection satisfies a linear PDE in time and age with an integral boundary condition. These integral equation and PDE limits can be regarded as dynamics on graphon under certain conditions.

We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and is represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. To each individual is associated a random infectivity function of its infection age, which is allowed to depend upon the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, and reflects the heterogeneities in individual characteristics and social connectivity.

We revisit the large population limit of our epidemic model with infection age dependent infectivity and progressive immunity waning, under the assumption that the supremum in t of the random infectivity function has a finite expectation, while the previous proofs assumed that this supremum admits a deterministic upper bound.

This paper addresses a generalized filtering framework in which the signal process X and observation process Y are both driven by correlated Brownian motions, and the coefficients of their governing stochastic differential equations depend jointly on (X, Y), with the exception of the diffusion coefficient of the observation process, which does not depend upon the signal. Unlike many prior works, the observation equation may have a degenerate (noninvertible or even zero) diffusion coefficient. In this framework, we derive filtering equations and prove of equivalence between the uniqueness of the nonlinear Kushner-Stratonovich equation and the linear Zakai equation. Finally we give a novel proof of uniqueness for the Zakai equation using a backward stochastic partial differential equation (BSPDE), overcoming the limitations of classical duality arguments. This approach successfully handles the randomness and anticipation introduced by the observation-dependent coefficients, which are not tractable under traditional deterministic PDE methods.

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection. In contrast to classical compartment models, after each infection, the infectivity is a random function of the time elapsed since one's infection. Similarly, recovered individuals become gradually susceptible after some time according to a random susceptibility function. We study the large population asymptotic behaviour of the model, by proving a functional law of large numbers (FLLN) and investigating the endemic equilibria properties of the limit. The limit depends on the law of the susceptibility random functions but only on the mean infectivity functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a special case of our FLLN limit.% for a particular set of infectivity and susceptibility random functions and initial conditions. For the endemic equilibria, if $ R_0 $ is lower than (or equal to) some threshold, the epidemic does not last forever and eventually disappears from the population, while if $ R_0 $ is larger than this threshold, the epidemic will not disappear and there exists an endemic equilibrium. The value of this threshold turns out to depend on the harmonic mean of the susceptibility a long time after an infection, a fact which was not previously known.

We study the stochastic SIR epidemic model with infection-age dependent infectivity for which a measure-valued process is used to describe the ages of infection for each individual. We establish a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the properly scaled measure-valued processes together with the other epidemic processes to describe the evolution dynamics. In the FLLN, assuming that the hazard rate function of the infection periods is bounded and the ages at time 0 of the infections of the initially infected individuals are bounded, we obtain a PDE limit for the LLN-scaled measure-valued process, for which we characterize its solution explicitly. The PDE is linear with a boundary condition given by the unique solution to a set of Volterra-type nonlinear integral equations. In the FCLT, we obtain an SPDE for the CLT-scaled measure-valued process, driven by two independent white noises coming from the infection and recovery processes. The SPDE is also linear and coupled with the solution to a system of stochastic Volterra-type linear integral equations driven by three independent Gaussian noises, one from the random infection functions in addition to the two white noises mentioned above. The solution to the SPDE can be also explicitly characterized, given this auxiliary process. The uniqueness of the SPDE solution is established under stronger assumptions on the distribution function of the infectious duration.

We establish a Law of Large Numbers and a Central Limit Theorem for a class of Crump Mode Jagers continuous time branching processes, where the birth rate is age dependent, and also random (different from one individual to the next), in the limit of a large number of ancestors. The only difficulty concerns the tightness in the Skorohod space D for the central limit theorem. We exploit a criterion for the CLT in D due to M. Hahn .

We prove the existence and uniqueness of a quasi-stationary distribution for three stochastic processes derived from the model of Muller’s ratchet. This model was invented with the aim of evaluating the limitations of an asexual reproduction mode in preventing the accumulation of deleterious mutations through natural selection alone. The main considered model is non-classical, as it is a stochastic diffusion evolving on an irregular set of infinite dimension with hard killing on a hyperplane. We are nonetheless able to prove exponential convergence in total variation to the quasi-stationary distribution even in this case. The parameters in this last convergence result are directly related to the core parameters of Muller’s ratchet. The speed of convergence to the quasi-stationary distribution is deduced both for the infinite dimensional model and for approximations with a large yet finite number of potential mutations. Likewise, we give uniform moment estimates of the empirical distribution of mutations in the population under quasi-stationarity.

We consider a spatial SIR epidemic model where the infectivity of infected individuals depends upon their age of infection, and infections are non local. The domain is an unbounded subset of ^d,and the individuals do not move. We extend our earlier result in , where the domain was bounded, and prove a law of large numbers as the size of the population tends to ∞.

In this work, we use a new approach to study the spread of an infectious disease. Indeed, we study a SIR epidemic model with variable infectivity, where the individuals are distributed over a compact subset D of ^d. We define empirical measures which describe the evolution of the state (susceptible, infectious, recovered) of the individuals in the various locations, and the total force of infection in the population. In our model, the individuals do not move. We establish a law of large numbers for these measures, as the population size tends to infinity.

We consider a space–time SI epidemic model with infection age dependent infectivity and non-local infections constructed on a grid of the torus Td=[0,1)d, where the individuals may migrate from node to node. The migration processes in either of the two states are assumed to be Markovian. We establish a functional law of large numbers by letting the initial approximate number of individuals on each node, N, to go to infinity and the mesh size of the grid, ε, to go to zero jointly. The limit is a system of parabolic PDE/integral equations. The constraint on the speed of convergence of the parameters N and ε is that Nεd→∞ as (N,ε)→(+∞,0).

This paper studies the distribution function of the time of extinction of a subcritical epidemic, when a large enough proportion of the population has been immunized and/or the infectivity of the infectious individuals has been reduced, so that the effective reproduction number is less than one. We do that for a SIR/SEIR model, where infectious individuals have an infection age dependent infectivity, as in the model introduced in the 1927 seminal paper of Kermack and McKendrick. Our main conclusion is that simplifying the model as an ODE SIR model, as it is largely done in the epidemics literature, introduces a biais toward shorter extinction time.

Chapter 2 Models of Sequences and Discrete Traits Evolution Étienne PARDOUX, Étienne PARDOUX I2M, CNRS, Aix-Marseille Université, FranceSearch for more papers by this author Étienne PARDOUX, Étienne PARDOUX I2M, CNRS, Aix-Marseille Université, FranceSearch for more papers by this author Gilles Didier, Gilles DidierSearch for more papers by this authorStéphane Guindon, Stéphane GuindonSearch for more papers by this author Book Author(s):Gilles Didier, Gilles DidierSearch for more papers by this authorStéphane Guindon, Stéphane GuindonSearch for more papers by this author First published: 12 April 2024 https://doi.org/10.1002/9781394284252.ch2 AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onEmailFacebookTwitterLinkedInRedditWechat Summary Models of sequence evolution are mainly used within the context of phylogenetic tree reconstruction. By combining the discrete-time chain with the sequence of sojourn times in the various states, the continuous-time process is reconstructed. Apart from a very small number of advanced research studies, all models of DNA sequence evolution assume that the evolutions of individual nucleotides in the sequence are independent, even if this assumption is not very realistic. The many studies conducted on genomes using observed data indicate that some sites are evolving more rapidly than others. The problem with general discrete distribution is the number of parameters to be estimated. Continuous-time Markov processes are used to model the evolution of discrete traits such as eye or coat color, the presence or absence of a given trait, the type of environment in which the species lives, the number of fingers, etc. References Felsenstein , J. ( 1981 ). Evolutionary trees from DNA sequences: A maximum likelihood approach . Journal of Molecular Evolution , 17 ( 6 ), 368 – 376 . 10.1007/BF01734359 CASPubMedWeb of Science®Google Scholar Felsenstein , J. and Churchill , G.A. ( 1996 ). A hidden Markov model approach to variation among sites in rate of evolution . Molecular Biology and Evolution , 13 ( 1 ), 93 – 104 . 10.1093/oxfordjournals.molbev.a025575 CASPubMedWeb of Science®Google Scholar Hasegawa , M. , Kishino , H. , Yano , T.-A. ( 1985 ). Dating of the human-ape splitting by a molecular clock of mitochondrial DNA . Journal of Molecular Evolution , 22 ( 2 ), 160 – 174 . 10.1007/BF02101694 CASPubMedWeb of Science®Google Scholar Jukes , T.H. and Cantor , C.R. ( 1969 ). Evolution of protein molecules . In Mammalian Protein Metabolism , H.N. Munro (ed.), Academic Press , New York . 10.1016/B978-1-4832-3211-9.50009-7 Google Scholar Kimura , M. ( 1980 ). A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences . Journal of Molecular Evolution , 16 ( 2 ), 111 – 120 . 10.1007/BF01731581 CASPubMedWeb of Science®Google Scholar Lewis , P.O. ( 2001 ). A likelihood approach to estimating phylogeny from discrete morphological character data . Systematic Biology , 50 ( 6 ), 913 – 925 . 10.1080/106351501753462876 CASPubMedWeb of Science®Google Scholar Pagel , M. ( 1994 ). Detecting correlated evolution on phylogenies: A general method for the comparative analysis of discrete characters. Proceedings of the Royal Society of London . Series B: Biological Sciences , 255 ( 1342 ), 37 – 45 . 10.1098/rspb.1994.0006 Web of Science®Google Scholar Paradis , E. , Claude , J. , Strimmer , K. ( 2004 ). APE: Analyses of phylogenetics and evolution in R language . Bioinformatics , 20 ( 2 ), 289 – 290 . 10.1093/bioinformatics/btg412 CASPubMedWeb of Science®Google Scholar Pardoux , E. ( 2007 ). Processus de Markov et applications. Algorithmes, réseaux, génome et finance . Dunod , Malakoff . Google Scholar Tavaré , S. ( 1986 ). Some probabilistic and statistical problems in the analysis of DNA sequences . Lectures on Mathematics in the Life Sciences , 17 ( 2 ), 57 – 86 . Google Scholar Wakeley , J. ( 1994 ). Substitution-rate variation among sites and the estimation of transition bias . Molecular Biology and Evolution , 11 ( 3 ), 436 – 442 . CASPubMedWeb of Science®Google Scholar Yang , Z. ( 1995 ). A space-time process model for the evolution of DNA sequences . Genetics , 139 ( 2 ), 993 – 1005 . 10.1093/genetics/139.2.993 CASPubMedWeb of Science®Google Scholar Yang , Z. ( 1996 ). Among-site rate variation and its impact on phylogenetic analyses . Trends in Ecology & Evolution , 11 ( 9 ), 367 – 372 . 10.1016/0169-5347(96)10041-0 CASPubMedWeb of Science®Google Scholar Models and Methods for Biological Evolution: Mathematical Models and Algorithms to Study Evolution ReferencesRelatedInformation

A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein-Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.

We study a stochastic epidemic model with multiple patches (locations), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any of the compartments. Each individual is associated with a random infectivity function which dictates the force of infectivity while the interactive infection process depends on the age of infection (elapsed time since infection). We prove a functional law of large number for the epidemic evolution dynamics including the aggregate infectivity process, the numbers of susceptible and recovered individuals as well as the number of infected individuals at each time that have been infected for a certain amount of time. From the limits, we derive a PDE model for the density of the number of infected individuals with respect to the infection age, which is a systems of linear PDE equations with a boundary condition that is determined by a set of integral equations.

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as “from distance” from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time since infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic Volterra-type integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.

We study a stochastic spatial epidemic model where the N individuals carry two features: a position and an infection state, interact and move in $${\mathbb {R}}^d$$ . In this Markovian model, the evolution of infection states are described with the help of the Poisson Point Processes , whereas the displacement of individuals are driven by mean field interactions, a (state dependence) diffusion and also a common noise, so that the spatial dynamic is a random process. We prove that when the number N of individual goes to infinity, the conditional propagation of chaos holds : conditionally to the common noise, the individuals are asymptotically independent and the stochastic dynamic converges to a “random” nonlinear McKean-Vlasov process. As a consequence, the associated empirical measure converges to a measure, which is solution of a stochastic mean-field PDE driven by the common noise.