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Exact description of SIR-Bass epidemics on 1D lattices

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  • This paper is devoted to the study of a stochastic epidemiological model which is a variant of the SIR model to which we add an extra factor in the transition rate from susceptible to infected accounting for the inflow of infection due to immigration or environmental sources of infection. This factor yields the formation of new clusters of infections, without having to specify a priori and explicitly their date and place of appearance.

    We establish an exact deterministic description for such stochastic processes on 1D lattices (finite lines, semi-infinite lines, infinite lines) by showing that the probability of infection at a given point in space and time can be obtained as the solution of a deterministic ODE system on the lattice. Our results allow stochastic initial conditions and arbitrary spatio-temporal heterogeneities on the parameters.

    We then apply our results to some concrete situations and obtain useful qualitative results and explicit formulae on the macroscopic dynamics and also the local temporal behavior of each individual. In particular, we provide a fine analysis of some aspects of cluster formation through the study of patient-zero problems and the effects of time-varying point sources.

    Finally, we show that the space-discrete model gives rise to new space-continuous models, which are either ODEs or PDEs, depending on the rescaling regime assumed on the parameters.

    Mathematics Subject Classification: Primary: 92D30, 90B60.

    Citation:

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  • Figure 1.  Convergence to the (integrodifferential) ODE limitk (75) under Rescaling Assumptions 1. Snapshot at $ t=2 $ of $ [S_k(t)] $ on a segment $ k=0,\dots, \lfloor\frac{5}{ {\Delta x}}\rfloor $ so that $ x\in(0,5) $. Blue solid line represents the explicit solutionk (76) of the limiting ODEk (75); Green dotted line and orange dashed line represent the solution $ [S_k](t) $ of (73) for $ {\Delta x}=0.5 $ and $ {\Delta x}=0.1 $ respectively. Choice of parameters and initial conditions : $ p(x)=1-\frac{x}{5} $, $ q(x)= 5+x $, $ r(x)=2-\frac{2x}{5} $, $ [ S^0](x)=0.2-\frac{x}{25} $, $ [ R^0](x)=0.2+\frac{3x}{50} $

    Figure 2.  Convergence towards PDE limitk (79) under Rescaling Assumptions 2. Snapshot at $ t=2 $ of $ [S_k(t)] $ on a segment $ k=0,\dots, \lfloor\frac{10}{ {\Delta x}}\rfloor $, with $ {\Delta x}>0 $, so that $ x\in(0,10) $. Blue solid line represents the solution $ S(t,x) $ of the limiting PDEk (79); Green dotted line and orange dashed line represent the solution $ [S_k](t) $ of (78) for $ {\Delta x}=0.1 $ and $ {\Delta x}=0.01 $ respectively. Choice of parameters and initial conditions : $ \tilde p(x)=0.1+\frac{0.2x}{10} $, $ \tilde q(x)=1+\frac{x}{10} $, $ \tilde r(x)=0.3+\frac{0.5x}{10} $, $ [\tilde I^0](x)=0.2+\frac{0.5x}{10} $, $ [\tilde R^0](x)=0.5-\frac{0.3x}{10} $. The parameters $ p $, $ q $, $ r $ and initial conditions $ [S^0] $, $ [I^0] $, $ [R^0] $ are then defined by the rescalingk (77)

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