Advanced Search
Article Contents
Article Contents

A martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies

This work is supported by NSF grant DMS-2027438, ARO MURI Grant W911NF1810208, NSF grant DMS-2027277, and Simons Math + X Investigator Award number 510776

Abstract Full Text(HTML) Figure(5) Related Papers Cited by
  • Deterministic compartmental models for infectious diseases give the mean behaviour of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. In this article, we consider the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derive a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects. Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data.

    Mathematics Subject Classification: Primary: 60G51; Secondary: 35Q91.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Plots of infected compartment fractions $ \textbf{i}_{N}(t) $ for the stochastic SIR and $ s(t),i(t) $, and $ r(t) $ for deterministic SIR. For both models, $ p_1 = 0.5 $, $ p_2 = 0.5 $, $ \gamma = 1 $, $ T = 23 $, $ {s}_0 = \textbf{s}_{0N} = 0.96 $, $ {i}_0 = \textbf{i}_{0N} = 0.04 $, $ {r}_0 = \textbf{r}_{0N} = 0 $, and In Figs. 1b and 1a, 1d and 1c, 1f and 1e, and 1h and 1g, $ \beta = 0.95, 1.1, 1.2, 1.3 $, respectively. For the SIR-IPC model displayed in Figs. 1b, 1d, 1f, and 1h, $ N = 10^{2.5} $, $ 10^{3} $, $ 10^{3.5}, 10 ^4 $ in every panel

    Figure 2.  Examples of the infinitesimal standard deviation for $ \sigma _{N }^{(l)} \left ( t \right ) $, $ l = 1, 2, 3 $. In both cases, $ p_1 = 0.5 $, $ p_2 = 0.5 $, $ \gamma = 1 $, $ T = 23 $, $ {s}_0 = \textbf{s}_{0N} = 0.96 $, $ {i}_0 = \textbf{i}_{0N} = 0.04 $, $ {r}_0 = \textbf{r}_{0N} = 0 $. In the cases with $ \mathcal R_0 = 0.95<1 $ in Figs 2a, 2c, and 2e, $ \beta = 0.95 $. And the blue, orange, green, and red lines show results with the same realization from simulations with the corresponding colors in Fig 1b, respectively. In the cases with $ \mathcal R_0 = 1.3>1 $ in Figs 2b, 2d, and 2f, $ \beta = 0.95 $. And the blue, orange, green, and red lines show results with the same realization from simulations with the corresponding colors in Fig 1h, respectively

    Figure 3.  Comparisons of the log–log plot of the theoretical and empirical scaling for $ \overline{ \mathcal V _{N}^{(l)} } $, $ l = 1, 2, 3 $, as in (38) - (40). The vertical bars are the error bars of the empirical scaling of the infinitesimal variance from the same realization as in Figs. 1b, and 1h. We set $ [T_1, T_2] $ as $ [0, 10], [1, 11], [2, 12],\cdots $ and $ [13, 23] $. The minimum and maximum values of $ y = \log\left (\overline{\mathcal {V}_{ N }^{(l)}}\right) $, $ l = 1, 2, 3 $, taken over all such intervals are set as the limits of the error bars. The straight lines with slope $ -1 $ show the theoretical scaling with the $ x $-intercepts as $ \log ( p_1 \beta { {i}_0} { s_0}) $, $ \log ( p_1 \beta { {i}_0} { s_0} + p_2 \gamma { {i}_0}) $, and $ \log (p_2 \gamma { {i}_0} ) $, for $ l = 1, 2, 3 $, respectively. In all the figures, $ p_1 = 0.5 $, $ p_2 = $, $ \gamma = 1 $, $ T = 23 $, $ {s}_0 = \textbf{s}_{0N} = 0.96 $, $ {i}_0 = \textbf{i}_{0N} = 0.04 $, and $ {r}_0 = \textbf{r}_{0N} = 0 $. In Figs. Figs. 3a, 3c, and 3e, $ \beta = 0.95 $. In Figs. 3a, 3c, and 3e, $ \beta = 1.3 $

    Figure 4.  Comparison of field data (solid black line) for daily confirmed case percentages with 30 realisations of the stochastic SIR model for Churchill County, NV and the Diamond Princess Cruise Ship

    Figure 5.   

  • [1] L. J. S. Allen, An introduction to stochastic epidemic models, In Mathematical Epidemiology, Lecture Notes in Math., 1945 (2008), 81–130. doi: 10.1007/978-3-540-78911-6_3.
    [2] L. J. S. AllenAn Introduction to Stochastic Processes with Applications to Biology, 2$^{nd}$ edition, CRC Press, Boca Raton, FL, 2011. 
    [3] L. J. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439–449, https://www.sciencedirect.com/science/article/pii/S0040580903001047. doi: 10.1016/S0040-5809(03)00104-7.
    [4] E. Almaraz and A. Gómez-Corral, On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2153-2176.  doi: 10.3934/dcdsb.2018229.
    [5] D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.
    [6] P. Azimi, Z. Keshavarz, J. G. Cedeno Laurent, B. Stephens and J. G. Allen, Mechanistic transmission modeling of COVID-19 on the diamond princess cruise ship demonstrates the importance of aerosol transmission, Proceedings of the National Academy of Sciences, 118 (2021). doi: 10.1073/pnas.2015482118.
    [7] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, 2$^{nd}$ edition, Hafner Press [Macmillan Publishing Co., Inc.], New York, 1975.
    [8] M. S. Bartlett, The critical community size for measles in the united states, Journal of the Royal Statistical Society: Series A (General), 123 (1960), 37-44.  doi: 10.2307/2343186.
    [9] F. K. BatistaA. M. del Rey and A. Queiruga-Dios, A review of seir-d agent-based model, Distributed Computing and Artificial Intelligence, 16th International Conference, Special Sessions, 1004 (2019), 133-140.  doi: 10.1007/978-3-030-23946-6_15.
    [10] T. Belin, A. Bertozzi, N. Chaudhary, T. Graves, J. Guterman, M. C. Jarashow, R. J. Lewis, J. Marion, F. Schoenberg, M. Shah, J. Tolles, E. Traub, K. Viele and F. Wu, Projections of Hospital-based Healthcare Demand due to COVID-19 in Los Angeles County May 24, 2021, 2021, http://file.lacounty.gov/SDSInter/dhs/1107440_COVID-19ProjectionPublicUpdateLewis05.24.21English.pdf.,
    [11] A. L. Bertozzi, E. Franco, G. Mohler, M. B. Short and D. Sledge, The challenges of modeling and forecasting the spread of COVID-19, Proc. Natl. Acad. Sci., 117 (2020), 16732–16738, https://www.pnas.org/content/117/29/16732. doi: 10.1073/pnas.2006520117.
    [12] B. M. Bibby and M. Sørensen, Martingale estimation functions for discretely observed diffusion processes, Bernoulli, 1 (1995), 17-39.  doi: 10.2307/3318679.
    [13] K. BichtelerStochastic Integration with Jumps, Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511549878.
    [14] E. Buckingham-JefferyV. Isham and T. House, Gaussian process approximations for fast inference from infectious disease data, Math. Biosci., 301 (2018), 111-120.  doi: 10.1016/j.mbs.2018.02.003.
    [15] Y. ChangJ. C. TzouM. J. Ward and J. C. Wei, Refined stability thresholds for localized spot patterns for the Brusselator model in $\mathbb{ R}^2$, European J. Appl. Math., 30 (2019), 791-828.  doi: 10.1017/S0956792518000426.
    [16] K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2$^{nd}$ edition, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4614-9587-1.
    [17] F. ComteV. Genon-Catalot and A. Samson, Nonparametric estimation for stochastic differential equations with random effects, Stochastic Process. Appl., 123 (2013), 2522-2551.  doi: 10.1016/j.spa.2013.04.009.
    [18] P. Cruises, Princess cruise lines (2020) Diamond Princess updates, 2021, https://www.princess.com/news/notices_and_advisories/notices/diamond-princess-update.html.
    [19] E. DongH. Du and L. Gardner, An interactive web-based dashboard to track COVID-19 in real time, The Lancet Infectious Diseases, 20 (2020), 533-534.  doi: 10.1016/S1473-3099(20)30120-1.
    [20] R. DurrettStochastic Calculus, A practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. 
    [21] R. Durrett, Essentials of Stochastic Processes, Springer Texts in Statistics, Springer-Verlag, New York, 1999.
    [22] R. Durrett, Probability Models for DNA Sequence Evolution, Probability and its Applications (New York), Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-6285-3.
    [23] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.
    [24] N. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai, K. Ainslie, M. Baguelin, S. Bhatia, A. Boonyasiri, Z. Cucunubá, G. Cuomo-Dannenburg, A. Dighe, I. Dorigatti, H. Fu, K. Gaythorpe, W. Green, A. Hamlet, W. Hinsley, L. C. Okell, S. v. Elsland, H. Thompson, R. Verity, E. Volz, H. Wang, Y. Wang, P. G. Walker, C. Walters, P. Winskill, C. Whittaker, C. A. Donnelly, S. Riley and A. C. Ghani, Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Report 9, Imperial College COVID-19 Response Team, Imperial College London, London, United Kingdom, 2020. doi: 10.25561/77482.
    [25] D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434.  doi: 10.1016/0021-9991(76)90041-3.
    [26] D. GomezM. J. Ward and J. Wei, The linear stability of symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt model, SIAM J. Appl. Dyn. Syst., 18 (2019), 729-768.  doi: 10.1137/18M1222338.
    [27] M. Z. Guo, G. C. Papanicolaou and S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm. Math. Phys., 118 (1988), 31–59, http://projecteuclid.org/euclid.cmp/1104161907. doi: 10.1007/BF01218476.
    [28] F. Hamba, Turbulent energy density in scale space for inhomogeneous turbulence, J. Fluid Mech., 842 (2018), 532-553.  doi: 10.1017/jfm.2018.155.
    [29] I. HameduddinC. MeneveauT. A. Zaki and D. F. Gayme, Geometric decomposition of the conformation tensor in viscoelastic turbulence, J. Fluid Mech., 842 (2018), 395-427.  doi: 10.1017/jfm.2018.118.
    [30] S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992.
    [31] P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Stochastic Processes, The Houghton Mifflin Series in Statistics, Houghton Mifflin Co., Boston, Mass., 1972.
    [32] S. J. IllingworthJ. P. Monty and I. Marusic, Estimating large-scale structures in wall turbulence using linear models, J. Fluid Mech., 842 (2018), 146-162.  doi: 10.1017/jfm.2018.129.
    [33] V. Isham, Assessing the variability of stochastic epidemics, Mathematical Biosciences, 107 (1991), 209-224.  doi: 10.1016/0025-5564(91)90005-4.
    [34] V. Isham, Stochastic models for epidemics with special reference to AIDS, Ann. Appl. Probab., 3 (1993), 1–27, http://links.jstor.org/sici?sici=1050-5164(199302)3:1<1:SMFEWS>2.0.CO;2-4&origin=MSN.
    [35] V. Isham, Stochastic models for epidemics, In Celebrating Statistics, Oxford Statist. Sci. Ser., Oxford Univ. Press, Oxford, 33 (2005), 27–54. doi: 10.1093/acprof:oso/9780198566540.003.0002.
    [36] J. Jiménez, Coherent structures in wall-bounded turbulence, J. Fluid Mech., 842 (2018), P1,100. doi: 10.1017/jfm.2018.144.
    [37] S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
    [38] W. O. Kermack, A. G. McKendrick and G. T. Walker, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115 (1927), 700–721, https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1927.0118.
    [39] C. KipnisS. Olla and S. R. S. Varadhan, Hydrodynamics and large deviation for simple exclusion processes, Comm. Pure Appl. Math., 42 (1989), 115-137.  doi: 10.1002/cpa.3160420202.
    [40] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-03752-2.
    [41] T. Kolokolnikov, M. Ward, J. Tzou and J. Wei, Stabilizing a homoclinic stripe, Philos. Trans. Roy. Soc. A, 376 (2018), 20180110, 13 pp. doi: 10.1098/rsta.2018.0110.
    [42] T. Kolokolnikov and J. Wei, Pattern formation in a reaction-diffusion system with space-dependent feed rate, SIAM Rev., 60 (2018), 626-645.  doi: 10.1137/17M1116027.
    [43] I. KrishnarajahA. CookG. Marion and G. Gibson, Novel moment closure approximations in stochastic epidemics, Bull. Math. Biol., 67 (2005), 855-873.  doi: 10.1016/j.bulm.2004.11.002.
    [44] H. Kunita, Lectures on Stochastic Flows and Applications, Springer-Verlag, Berlin, 1986.
    [45] Z. LiZ. TengD. Hong and X. Shi, Comparison of three sis epidemic models: Deterministic, stochastic and uncertain, Journal of Intelligent & Fuzzy Systems, 35 (2018), 5785-5796.  doi: 10.3233/JIFS-18007.
    [46] T. M. Liggett, Interacting Markov processes, In Biological Growth and Spread (Proc. Conf., Heidelberg, 1979), 38 (1980), 145–156.
    [47] T. M. Liggett, Interacting Particle Systems, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8542-4.
    [48] T. M. Liggett, Continuous Time Markov Processes, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/113.
    [49] X. LiuA. MubayiD. Reinhold and L. Zhu, Approximation methods for analyzing multiscale stochastic vector-borne epidemic models, Math. Biosci., 309 (2019), 42-65.  doi: 10.1016/j.mbs.2019.01.003.
    [50] A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theoretical Population Biology, 60 (2001), 59–71, http://www.sciencedirect.com/science/article/pii/S0040580901915254. doi: 10.1006/tpbi.2001.1525.
    [51] S. McMiNN, R. Talbot and J. ENG, America's 200,000 COVID-19 deaths: Small cities and towns bear a growing share, https://www.npr.org, https://www.npr.org/sections/health-shots/2020/09/22/914578634/americas-200-000-COVID-19-deaths-small-cities-and-towns-bear-a-growing-share.
    [52] M. Métivier, Semimartingales, A course on stochastic processes. de Gruyter Studies in Mathematics, 2. Walter de Gruyter & Co., Berlin-New York, 1982.
    [53] M. Métivier and J. Pellaumail, Stochastic Integration, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980
    [54] L. Ministry of Health and J. Welfare, Ministry of health, labor and welfare, Japan (2020) about coronavirus disease 2019 (COVID-19), 2020., https://www.mhlw.go.jp/stf/newpage_09276.html.
    [55] F. Paquin-LefebvreW. Nagata and M. J. Ward, Pattern formation and oscillatory dynamics in a two-dimensional coupled bulk-surface reaction-diffusion system, SIAM J. Appl. Dyn. Syst., 18 (2019), 1334-1390.  doi: 10.1137/18M1213737.
    [56] S. Peszat and  J. ZabczykStochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.
    [57] B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes, Edward Arnold, London; Oxford University Press, New York, 1999.
    [58] P. E. Protter, Stochastic Integration and Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.
    [59] M.-A. Rizoiu, S. Mishra, Q. Kong, M. Carman and L. Xie, SIR-Hawkes: Linking epidemic models and hawkes processes to model diffusions in finite populations, International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, CHE, (2018), 419–428. doi: 10.1145/3178876.3186108.
    [60] J. Rocklöv, H. Sjödin and A. Wilder-Smith, COVID-19 outbreak on the diamond princess cruise ship: Estimating the epidemic potential and effectiveness of public health countermeasures, Journal of Travel Medicine, 27 (2020), taaa030.
    [61] E. Schumaker and M. Nichols, An american tragedy: Inside the towns hardest hit by coronavirus, abcNEWS, https://abcnews.go.com/Health/small-towns-face-COVID-19-pandemic-us-passes/story?id=74271392.
    [62] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006
    [63] W.-Y. Tan and S. Tang, A general markov model of the hiv epidemic in populations involving both sexual contact and iv drug use, Mathematical and Computer Modelling, 19 (1994), 83-132.  doi: 10.1016/0895-7177(94)90108-2.
    [64] B. Tóth and B. Valkó, Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws, J. Statist. Phys., 112 (2003), 497-521.  doi: 10.1023/A:1023867723546.
    [65] W. H. Tse and M. J. Ward, Asynchronous instabilities of crime hotspots for a 1-D reaction-diffusion model of urban crime with focused police patrol, SIAM J. Appl. Dyn. Syst., 17 (2018), 2018-2075.  doi: 10.1137/17M1162585.
    [66] J. C. TzouM. J. Ward and J. C. Wei, Anomalous scaling of Hopf bifurcation thresholds for the stability of localized spot patterns for reaction-diffusion systems in two dimensions, SIAM J. Appl. Dyn. Syst., 17 (2018), 982-1022.  doi: 10.1137/17M1137759.
    [67] M. Uchida, Approximate martingale estimating functions for stochastic differential equations with small noises, Stochastic Process. Appl., 118 (2008), 1706-1721.  doi: 10.1016/j.spa.2007.10.008.
    [68] S. R. S. Varadhan, Entropy methods in hydrodynamic scaling, In Proceedings of the International Congress of Mathematicians, 1 (1995), 196–208.
    [69] S. R. S. Varadhan, Lectures on hydrodynamic scaling, In Hydrodynamic Limits and Related Topics (Toronto, ON, 1998), Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 27 (2000), 3–40.
    [70] M. J. Ward, Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), 189-239.  doi: 10.1088/1361-6544/aabe4b.
    [71] G.-W. Weber, P. Taylan, Z.-K. Görgülü, H. A. Rahman and A. Bahar, Parameter estimation in stochastic differential equations, In Dynamics, Games and Science. II, 2 (2011), 703–733. doi: 10.1007/978-3-642-14788-3_51.
    [72] P. Yan, Distribution theory, stochastic processes and infectious disease modelling, In Mathematical Epidemiology, Lecture Notes in Math., 1945 (2008), 229–293. doi: 10.1007/978-3-540-78911-6_10.
    [73] P. Yan and G. Chowell, Beyond the initial phase: Compartment models for disease transmission, Quantitative Methods for Investigating Infectious Disease Outbreaks, 70 (2019), 135-182.  doi: 10.1007/978-3-030-21923-9_5.
  • 加载中
Open Access Under a Creative Commons license



Article Metrics

HTML views(449) PDF downloads(365) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint