The SIR epidemic model from a PDE point of view

Chalub, Fabio A. C. C.; Souza, Max O.

doi:10.1016/j.mcm.2010.05.036

Cited by 39 publications

(34 citation statements)

References 10 publications

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“…We can, in principle, use these transition probabilities to construct master equations and generating functions to formulate ODEs for the moments (Allen, 2010). However, as is usual, master equations are difficult to analyse and the formulated ODEs suffer from the identification of suitably reliable moment closure conditions; see Chalub andSouza (2011), Isham (1991), Krishnarajah et al (2005) and references therein for examples in epidemiological modelling. We will instead analyse the Markov process computationally.…”

Section: A Stochastic Modelmentioning

confidence: 99%

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Chen

Ward

Xie

2019

Theoretical Population Biology

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In-host mutation of a cross-species infectious disease to a form that is transmissible between humans has resulted with devastating global pandemics in the past. We use simple mathematical models to describe this process with the aim to better understand the emergence of an epidemic resulting from such a mutation and the extent of measures that are needed to control it. The feared outbreak of a human-human transmissible form of avian influenza leading to a global epidemic is the paradigm for this study. We extend the SIR approach to derive a deterministic and a stochastic formulation to describe the evolution of two classes of susceptible and infected states and a removed state, leading to a system of ordinary differential equations and a stochastic equivalent based on a Markov process. For the deterministic model, the contrasting timescale of the mutation process and disease infectiousness is exploited in two limits using asymptotic analysis in order to determine, in terms of the model parameters, necessary conditions for an epidemic to take place and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Furthermore, the basic reproduction number R is determined from asymptotic analysis of a distinguished limit. Comparisons between the deterministic and stochastic model demonstrate that stochasticity has little effect on most aspects of an epidemic, but does have significant impact on its onset particularly for smaller populations and lower mutation rates for representatively large populations. The deterministic model is extended to investigate a range of quarantine and vaccination programmes, whereby in the two asymptotic limits analysed, quantitative estimates on the outcomes and effectiveness of these control measures are established.

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Section: A Stochastic Modelmentioning

confidence: 99%

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Chen

Ward

Xie

2019

Theoretical Population Biology

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“…Equation (7) can be shown to be equivalent to (2) by observing that the latter is the equation for the projected characteristics of the former-see Chalub and Souza (2011) for a similar PDE derivation of the SIR model. The next step is to show that equation (5) can be consistently solved in the class of probability measures, and thus that its solution describes an approximation of the probability distribution of the discrete version, while maintaining its main features.…”

Section: Unified Modellingmentioning

confidence: 99%

Discrete and continuous SIS epidemic models: A unifying approach

Chalub

Souza

2014

Ecological Complexity

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The Susceptible-Infective-Susceptible (SIS) epidemiological scheme is the simplest description of the dynamics of a disease that is contact-transmitted, and that does not lead to immunity. Two by now classical approaches to such a description are: (i) the use of a mass-action compartamental model that leads to a single ordinary differential equation (SIS-ODE); (ii) the use of a discrete-time Markov chain model (SIS-DTMC). While the former can be seen as a mean-field approximation of the latter under certain conditions, it is also known that their dynamics can be significantly different, if the basic reproduction number is greater than one. The goal of this work is to introduce a continuous model, based on a partial differential equation (SIS-PDE), that retains the finite populations effects present in the SIS-DTMC model, and that allows the use of analytical techniques for its study. In particular, it will reduce itself to the SIS-ODE model in many circumstances. This is accomplished by deriving a diffusion-drift approximation to the probability density of the SIS-DTMC model. Such a diffusion is degenerated at the origin, and must conserve probability. These two features then lead to an interesting consequence: the biologically correct solution is a measure solution. We then provide a convenient representation of such a measure solution that allows the use of classical techniques for its computation, and that also provides a tool for obtaining information about several dynamical features of the model. In particular, we show that the SIS-ODE gives the most likely state, conditional on non-absorption. As a further application of * Corresponding Author Preprint submitted to ElsevierOctober 12, 2013 such representation, we show how to define the disease-outbreak probability in terms of the SIS-PDE model, and show that this definition can be used both for certain and uncertain initial presence of infected individuals. As a final application, we compute an approximation for the extinction time of the disease. In addition, we present many numerical examples that confirm the good approximation of the SIS-DTMC by the SIS-PDE.

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“…See also Chalub and Souza for an earlier connection with solutions of the replicator dynamics and Chalub and Souza for generalisations to higher dimensions. Similar equations, but with slightly different conditions, were also studied in Chalub and Souza—see below. See also Epstein and Mazzeo for a comprehensive treatment using classical tools.…”

Section: Introductionmentioning

confidence: 99%

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

Danilkina

Souza

Chalub

2018

Math Methods in App Sciences

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We consider partial differential equations of drift-diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure-valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0 + . The time evolution of these singular masses is also explicitly described and, as a by-product, uniqueness of this measure solution is obtained.

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The SIR epidemic model from a PDE point of view

Cited by 39 publications

References 10 publications

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Discrete and continuous SIS epidemic models: A unifying approach

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

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