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Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model

Part of: Equations of mathematical physics and other areas of application Stochastic analysis

Published online by Cambridge University Press:  16 July 2020

Dang H. Nguyen ,
Nhu N. Nguyen  and
George Yin
Dang H. Nguyen*
Affiliation:
University of Alabama
Nhu N. Nguyen*
Affiliation:
Wayne State University
George Yin*
Affiliation:
Wayne State University
*
*Postal address: University of Alabama, Tuscaloosa, AL35487, USA.
***Postal address: Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA.
***Postal address: Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA.
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Abstract

This work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, the existence and uniqueness of solutions of the underlying SPDEs are examined. Then, definitions of permanence and extinction are given, and certain sufficient conditions are provided for permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.

Keywords

SIR modelSPDEmild solutionpositivityextinctionpermanence

MSC classification

Primary: 60H15: Stochastic partial differential equations 92D25: Population dynamics (general) 92D30: Epidemiology
Secondary: 35Q92: PDEs in connection with biology and other natural sciences

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 613 - 636
Copyright
© Applied Probability Trust 2020

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References

Allen, L. J. S., Bolker, B. M., Lou, Y. andNevai, A. L. (2007). Asymptotic profiles of the steady states for an SIS epidemic patch model. SIAM J. Appl. Math. 67, 1283–1309.CrossRefGoogle Scholar
Allen, L. J. S., Bolker, B. M., Lou, Y. andNevai, A. L. (2008) Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model. Discrete Cont. Dyn. Syst. 21, 1–20.CrossRefGoogle Scholar
Arendt, W. (2004) Semigroups and evolution equations: Functional calculus, regularity and kernel estimates. In Handbook of Differential Equations: Evolutionary Differential Equations, eds C. M. Dafermos and E. Feireisl, North Holland, Amsterdam.Google Scholar
Ball, F. andSirl, D. (2012). An SIR epidemic model on a population with random network and household structure, and several types of individuals. Adv. Appl. Prob. 44, 63–86.CrossRefGoogle Scholar
Bao, J., Yin, G. andYuan, C. (2017). Two-time-scale stochastic partial differential equations driven by alpha-stable noises: Averaging principles. Bernoulli 23, 645–669.CrossRefGoogle Scholar
Britton, T. andLindholm, M. (2009). The early stage behaviour of a stochastic SIR epidemic with term-time forcing. J. Appl. Prob. 46, 975–992.CrossRefGoogle Scholar
Brzeźniak, Z. (1995). Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4, 1–45.CrossRefGoogle Scholar
Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach (Lect. Notes Math. Ser. 1762). Springer, Berlin.CrossRefGoogle Scholar
Chen, X., Chen, Z.-Q., Tran, K. andYin, G. (2019). Properties of switching jump diffusions: Maximum principles and Harnack inequalities. Bernoulli 25, 1045–1075.CrossRefGoogle Scholar
Curtain, R. F. andFalez, P. L. (1970). Itô’s Lemma in infinite dimensions. J. Math. Anal. Appl. 31, 434–448.CrossRefGoogle Scholar
Da Prato, G. andTubaro, L. (1985). Some results on semilinear stochastic differential equations in Hilbert spaces. Stochastics 15, 271–281.CrossRefGoogle Scholar
Da Prato, G. andZabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press.CrossRefGoogle Scholar
Davies, E. B. (1989). Heat Kernels and Spectral Theory (Camb. Univ. Tracts Math. 92). Cambridge University Press.Google Scholar
Dieu, N. T., Du, N. H. andNhu, N. N. (2019). Conditions for permanence and ergodicity of certain SIR epidemic models, Acta. Appl. Math. 160, 81–99.CrossRefGoogle Scholar
Dieu, N. T., Nguyen, D. H., Du, N. H. andYin, G. (2016). Classification of asymptotic behavior in a stochastic SIR model. SIAM J. Appl. Dynam. Sys. 15, 1062–1084.CrossRefGoogle Scholar
Du, N. H., Nguyen, D. H. andYin, G. (2016). Conditions for permanence and ergodicity of certain stochastic predator–prey models. J. Appl. Prob. 53, 187–202.CrossRefGoogle Scholar
Du, N. H. andNhu, N. N. (2017). Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises. Appl. Math. Lett. 64, 223–230.CrossRefGoogle Scholar
Du, N. H. andNhu, N. N. (2018). Permanence and extinction for the stochastic SIR epidemic model. Submitted.Google Scholar
Ducrot, A. andGiletti, T. (2014). Convergence to a pulsating traveling wave for an epidemic reaction–diffusion system with non-diffusive susceptible population. J. Math. Biol. 69, 533–552.CrossRefGoogle Scholar
Gathy, M. andLefevre, C. (2009). From damage models to SIR epidemics and cascading failures. Adv. Appl. Prob. 41, 247–269.CrossRefGoogle Scholar
Hening, A. andNguyen, D. H. (2018). Stochastic Lotka–Volterra food chains. J. Math. Bio. 77, 135–163.CrossRefGoogle ScholarPubMed
Hening, A., Nguyen, D. H. andYin, G. (2018). Stochastic population growth in spatially heterogeneous environments: The density-dependent case. J. Math. Biol. 76, 697–754.CrossRefGoogle ScholarPubMed
Hieu, N. T., Du, N. H., Auger, P. andNguyen, D. H. (2015). Dynamical behavior of a stochastic SIRS epidemic model. Math. Model. Nat. Phenom. 10, 56–73.CrossRefGoogle Scholar
Kermack, W. O. andMcKendrick, A. G. (1927). Contributions to the mathematical theory of epidemics (part I). Proc. R. Soc. London A 115, 700–721.CrossRefGoogle Scholar
Kermack, W. O. andMcKendrick, A. G. (1932). Contributions to the mathematical theory of epidemics (part II). Proc. R. Soc. London A 138, 55–83.CrossRefGoogle Scholar
Kortchemski, I. (2015) A predator–prey SIR type dynamics on large complete graphs with three phase transitions. Stoch. Process. Appl. 125, 886–917.CrossRefGoogle Scholar
Liu, K. (2005). Stability of Infinite-Dimensional Stochastic Differential Equations with Applications. Chapman and Hall/CRC, New York.CrossRefGoogle Scholar
Nguyen, D. H. andYin, G. (2017). Coexistence and exclusion of stochastic competitive Lotka–Volterra models. J. Differential Equat. 262, 1192–1225.CrossRefGoogle Scholar
Nguyen, N. N. andYin, G. (2019). Stochastic partial differential equation SIS epidemic models: Modeling and analysis. Commun. Stoch. Anal. 13, 8.Google Scholar
Nguyen, N. N. andYin, G. (2020). Stochastic partial differential equation models for spatially dependent predator–prey equations. Discrete Cont. Dyn. Syst. Ser. B 25, 117–139.CrossRefGoogle Scholar
Ouhabaz, E. M. (2004). Analysis of Heat Equations on Domains (London Math. Soc. Monographs 31). Princeton University Press.Google Scholar
Peng, R. (2009). Asymptotic profiles of the positive steady state for an SIS epidemic reaction–diffusion model I. J. Differential Equat. 247, 1096–1119.CrossRefGoogle Scholar
Peng, R. andLiu, S. (2009). Global stability of the steady states of an SIS epidemic reaction–diffusion model. Nonlin. Anal. 71, 239–247.CrossRefGoogle Scholar
van Neeren. J. M. A. M., Veraar, M. C. andWeis, L. (2008). Stochastic evolution equations in UMD Banach spaces J. Funct. Anal. 255, 940–993.Google Scholar
van Neeren, J. M. A. M., Veraar, M. andWeis, L. (2012). Stochastic maximal LP-regularity. Ann. Prob. 40, 788–812.Google Scholar
Wang, W. andZhao, X. Q. (2012). Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11, 1652–1673.CrossRefGoogle Scholar
Wilkinson, R. R., Ball, F. G. andSharkey, K. J. (2016). The deterministic Kermack–McKendrick model bounds the general stochastic epidemic. J. Appl. Prob. 53, 1031–1040.CrossRefGoogle Scholar
Yagi, A. (2010). Abstract Parabolic Evolution Equations and their Applications. Springer, Berlin.CrossRefGoogle Scholar
Yin, G. andZhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. Springer, New York.CrossRefGoogle Scholar
Zhang, L., Wang, Z. C. andZhao, X. Q. (2015). Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period. J. Differential Equat. 258, 3011–3036.CrossRefGoogle Scholar