# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021126
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## Exact description of SIR-Bass epidemics on 1D lattices

 Department of Applied Mathematics, Tel Aviv University, Israel

* Corresponding author: samnordmann@gmail.com

Received  February 2021 Revised  June 2021 Early access September 2021

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.

Citation: Gadi Fibich, Samuel Nordmann. Exact description of SIR-Bass epidemics on 1D lattices. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021126
##### References:
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Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, Journal of Applied Probability, 7 (1970), 49-58.  doi: 10.2307/3212147.  Google Scholar [32] T. G. Kurtz, Limit theorems for sequences of jump markov processes approximating ordinary, Journal of Applied Probability, 8 (1971), 344-356.  doi: 10.2307/3211904.  Google Scholar [33] H. Matsuda, N. Ogita, A. Sasaki and K. Sato, Statistical mechanics of population: The lattice lotka-volterra model, Progress of Theoretical Physics, 88 (1992), 1035-1049.  doi: 10.1143/ptp/88.6.1035.  Google Scholar [34] D. Mollison, Epidemic Models: Their structure and Relation to Data, Cambridge University Press, 1995.   Google Scholar [35] L. Pellis, F. Ball, S. Bansal, K. Eames, T. House, V. Isham and P. Trapman, Eight challenges for network epidemic models, Epidemics, 10 (2015), 58-62.  doi: 10.1016/j.epidem.2014.07.003.  Google Scholar [36] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, Springer, (2007), 97–122.  Google Scholar [37] K. Sato, H. Matsuda and A. Sasaki, Pathogen invasion and host extinction in lattice structured populations, Journal of Mathematical Biology, 32 (1994), 251-268.  doi: 10.1007/BF00163881.  Google Scholar [38] K. J. Sharkey, I. Z. Kiss, R. R. Wilkinson and P. L. Simon, Exact equations for sir epidemics on tree graphs, Bulletin of Mathematical Biology, 77 (2015), 614-645.  doi: 10.1007/s11538-013-9923-5.  Google Scholar [39] P. L. Simon and I. Z. Kiss, From exact stochastic to mean-field ODE models: A new approach to prove convergence results, IMA Journal of Applied Mathematics, 78 (2013), 945-964.  doi: 10.1093/imamat/hxs001.  Google Scholar [40] S. A. Socolofsky and G. H. Jirka, Advective Diffusion Equation (lecture notes), 2004. Google Scholar

show all references

##### References:
 [1] R. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.   Google Scholar [2] H. Andersson, Limit theorems for a random graph epidemic model, The Annals of Applied Probability, 8 (1998), 1331-1349.  doi: 10.1214/aoap/1028903384.  Google Scholar [3] H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, volume 151., Springer-Verlag New York, 2000. doi: 10.1007/978-1-4612-1158-7.  Google Scholar [4] J. Badham and R. Stocker, The impact of network clustering and assortativity on epidemic behaviour, Theoretical Population Biology, 77 (2010), 71-75.  doi: 10.1016/j.tpb.2009.11.003.  Google Scholar [5] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffin & Company Ltd, Bucks, 1975.  Google Scholar [6] F. Ball and D. Sirl, An SIR epidemic model on a population with random network and household structure, and several types of individuals, Advances in Applied Probability, 44 (2012), 63-86.  doi: 10.1239/aap/1331216645.  Google Scholar [7] S. Bansal, B. T. Grenfell and L. A. Meyers, When individual behaviour matters: Homogeneous and network models in epidemiology, Journal of the Royal Society Interface, 4 (2007), 879-891.  doi: 10.1098/rsif.2007.1100.  Google Scholar [8] F. M. Bass, A new product growth for model consumer durables, Mathematical Models in Marketing, 132 (1969), 351–353. doi: 10.1007/978-3-642-51565-1_107.  Google Scholar [9] H. Berestycki, J.-P. Nadal and N. Rodríguez, A model of riot dynamics: Shocks, diffusion, and thresholds, Networks and Heterogeneous Media, 10 (2015), 443-475.  doi: 10.3934/nhm.2015.10.443.  Google Scholar [10] T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35.  doi: 10.1016/j.mbs.2010.01.006.  Google Scholar [11] L. Danon, A. P. Ford, T. House, C. P. Jewell, M. J. Keeling, G. O. Roberts, J. V. Ross and M. C. Vernon, Networks and the epidemiology of infectious disease, Interdisciplinary Perspectives on Infectious Diseases, 2011 (2011), Article ID 284909. doi: 10.1155/2011/284909.  Google Scholar [12] L. Decreusefond, J.-S. Dhersin, P. Moyal and V. C. Tran, Large graph limit for an SIR process in random network with heterogeneous connectivity, The Annals of Applied Probability, 22 (2012), 541-575.  doi: 10.1214/11-AAP773.  Google Scholar [13] O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Analysis: Theory, Methods & Applications, 1 (1977), 459-470.  doi: 10.1016/0362-546X(77)90011-6.  Google Scholar [14] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, Journal of Differential Equations, 33 (1979), 58-73.  doi: 10.1016/0022-0396(79)90080-9.  Google Scholar [15] K. Dietz, Epidemics and Rumours: A Survey, Journal of the Royal Statistical Society. Series A (General), 130 (1967), 505-528.  doi: 10.2307/2982521.  Google Scholar [16] J. Enright and R. R. Kao, Epidemics on dynamic networks, Epidemics, 24 (2018), 88-97.  doi: 10.1016/j.epidem.2018.04.003.  Google Scholar [17] G. Fibich, Bass-SIR model for diffusion of new products in social networks, Physical Review E, 94 (2016), 32305, 5pp. doi: 10.1103/PhysRevE. 94.032305.  Google Scholar [18] G. Fibich, Diffusion of new products with recovering consumers, Society for Industrial and Applied Mathematics, 77 (2017), 1230-1247.  doi: 10.1137/17M1112546.  Google Scholar [19] G. Fibich and R. Gibori, Aggregate diffusion dynamics in agent-based models with a spatial structure, Operations Reasearch, 58 (2010), 1450-1468.  doi: 10.1287/opre.1100.0818.  Google Scholar [20] G. Fibich and T. Levin, Network Effects in the Discrete Bass Model, Work in progress, 2021. Google Scholar [21] G. Fibich, T. Levin and O. Yakir, Boundary effects in the discrete Bass model, SIAM Journal on Applied Mathematics, 79 (2019), 914-937.  doi: 10.1137/18M1163646.  Google Scholar [22] M. Graziano and K. Gillingham, Spatial patterns of solar photovoltaic system adoption: The influence of neighbors and the built environmentz, Journal of Economic Geography, 15 (2015), 815-839.  doi: 10.1093/jeg/lbu036.  Google Scholar [23] B. T. Grenfell, O. N. Bjørnstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics, Nature, 414 (2001), 716-723.  doi: 10.1038/414716a.  Google Scholar [24] H. Heathcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [25] H. W. Hethcote, Three basic epidemiological models, Applied Mathematical Ecology (Trieste, 1986), 119–144, Biomathematics, 18, Springer, Berlin, 1989. doi: 10.1007/978-3-642-61317-3_5.  Google Scholar [26] T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of the Royal Society Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179.  Google Scholar [27] M. Keeling, The implications of network structure for epidemic dynamics, Theoretical Population Biology, 67 (2005), 1-8.  doi: 10.1016/j.tpb.2004.08.002.  Google Scholar [28] M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proc. R. Soc. Long. B, 266 (1999), 859-867.  doi: 10.1515/9781400841356.480.  Google Scholar [29] M. J. Keeling and K. T. Eames, Networks and epidemic models, Journal of the Royal Society Interface, 2 (2005), 295-307.  doi: 10.1098/rsif.2005.0051.  Google Scholar [30] W. Kermack and A. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society A, 115 1927. Google Scholar [31] T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, Journal of Applied Probability, 7 (1970), 49-58.  doi: 10.2307/3212147.  Google Scholar [32] T. G. Kurtz, Limit theorems for sequences of jump markov processes approximating ordinary, Journal of Applied Probability, 8 (1971), 344-356.  doi: 10.2307/3211904.  Google Scholar [33] H. Matsuda, N. Ogita, A. Sasaki and K. Sato, Statistical mechanics of population: The lattice lotka-volterra model, Progress of Theoretical Physics, 88 (1992), 1035-1049.  doi: 10.1143/ptp/88.6.1035.  Google Scholar [34] D. Mollison, Epidemic Models: Their structure and Relation to Data, Cambridge University Press, 1995.   Google Scholar [35] L. Pellis, F. Ball, S. Bansal, K. Eames, T. House, V. Isham and P. Trapman, Eight challenges for network epidemic models, Epidemics, 10 (2015), 58-62.  doi: 10.1016/j.epidem.2014.07.003.  Google Scholar [36] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, Springer, (2007), 97–122.  Google Scholar [37] K. Sato, H. Matsuda and A. Sasaki, Pathogen invasion and host extinction in lattice structured populations, Journal of Mathematical Biology, 32 (1994), 251-268.  doi: 10.1007/BF00163881.  Google Scholar [38] K. J. Sharkey, I. Z. Kiss, R. R. Wilkinson and P. L. Simon, Exact equations for sir epidemics on tree graphs, Bulletin of Mathematical Biology, 77 (2015), 614-645.  doi: 10.1007/s11538-013-9923-5.  Google Scholar [39] P. L. Simon and I. Z. Kiss, From exact stochastic to mean-field ODE models: A new approach to prove convergence results, IMA Journal of Applied Mathematics, 78 (2013), 945-964.  doi: 10.1093/imamat/hxs001.  Google Scholar [40] S. A. Socolofsky and G. H. Jirka, Advective Diffusion Equation (lecture notes), 2004. Google Scholar
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 : $p(x)=1-\frac{x}{5}$, $q(x)= 5+x$, $r(x)=2-\frac{2x}{5}$, $[ S^0](x)=0.2-\frac{x}{25}$, $[ R^0](x)=0.2+\frac{3x}{50}$
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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