یک مدل تصادفی برای حالت کلی اپیدمی چندمیزبانی براساس میانگین و واریانس

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشگاه لرستان، خرم آباد، ایران

چکیده

عوامل بیماری‌زا می‌توانند چندین گروه از یک زیست‌بوم را تحت تاثیر خود قرار دهند. در این مقاله، رفتاری تصادفی مبتنی بر میانگین و واریانس در چگونگی گسترش یک عامل بیماری‌زا در بیماری مشترک بین گونه‌های مختلف جانوری در حضور جمعیت‌های مستعد، در معرض، مبتلا و مصون بررسی می‌شود. در این مدل، دوره نهفتگی که یک دوره مهم در یک اپیدمی است، نیز درنظر گرفته می‌شود. این مدل تمام مدل‌های توصیف شده درباره اپیدمی از جمله مدل‌های SEIR ، SIR، SIS، SI و SIRS را پوشش می‌دهد. همچنین در آخر، با چند مثال عددی رفتار این مدل را مشاهده خواهیم نمود.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

A Stochastic Model for General Multi-Host Epidemic Models based on mean and variance

نویسندگان [English]

  • Amin Shahkarami
  • Bahman Ghazanfari
Department of Mathematics, Lorestan University, Khorramabad, Iran
چکیده [English]

Pathogenic factors can affect several groups of an epidemic. In this paper, by using the Brownian motion, an Ito stochastic differential equation is presented to express the stochastic behavior of an infected disease based on mean and variance. The model includes different species of population with the presence of suspected, exposed, infected and immune types. The model covers all the previous models including SI, SIS, SIR, SEIR, and SIRS. Also, the incubation, an important period in the epidemic, is considered. Finally, a few numerical examples of this model's behavior are presented.

کلیدواژه‌ها [English]

  • Deterministic model
  • Stochastic model
  • Basic reproduction number
  • Itô
  • integral
[1] Allen E.J., Stochastic differential equations and persistence time for two interacting populations, Dynamics of Continuous, Discrete and Impulsive Systems 5 (1999) 271-281.
[2] Berhazi B., El Fatini A., Lahrouzi A. and Settati A., A stochastic SIRS epidemic model with a general awareness-induced incidence, Physica A. 512 (2018) 968-980.
[3] Bjork T., Arbitrage Theory in Continuous Time, Oxford University Press. Third Edition, 2009.
[4] Cai Y., Kang Y., Banerjee M. and Wang M., A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations 259 (2015) 7463-7502.
[5] Dobson A., Population Dynamics of Pathogens with Multiple Host Species, supplement the American naturalist 164 (2004) 64–78.
[6] Driessche P.V.D. and Watmough J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29-48.
[7] Ergonul O. and Whitehouse C.A., Crimean-Congo Hemorrhagic Fever; A Global Perspective, Published by Springer, 2007.
[8] Gray A., Greenhalgh D., Mao X. and Pan J., The SIS epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications 394 (2012) 496-516.
[9] Higham D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001) 525-546.
[10] Kirupaharan N. and Allen L.J.S., Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependentmortality, Bulletin of Mathematical Biology 66 (2004) 841-864.
[11] Li C., Pei Y., Liang X. and Fang D., A stochastic toxoplasmosis spread model between cat and oocyst with jumps process, Communications in Mathematical Biology and Neuroscience (2016) 10-28.
[12] Lin Y. and Jiang D., Long-time behaviour of a perturbed SIR model by white noise, Discrete and Continuous Dynamical Systems 18 (2013) 1873-1887.
[13] Liu M., Bai C. and Wang K., Asymptotic stability of a twogroup stochastic SEIRmodel with infinite delays, Communications in Nonlinear Science and Numerical Simulation 19 (2014) 3444-3453.
[14] Liu Q., Jiang D., Shi N., Hayat T. and Alsaedi A., Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence, Physica A. Statistical Mechanics and Its Applications 462 (2016) 870-882.
[15] Mccormack R.K., Multi-host and multi-patch mathematical epidemic models for disease emergence with applications to Hantavirus in wild rodents, doctor of philosophy, Dean of the Graduate School, Texas Tech University, 2006.
[16] Mccormack R.K. and Allen L.J.S, Stochastic SIS and SIR multihost epidemic models, Hindawi Publishing Corporation, Proceedings of the Conference on Differential & Difference Equations and Applications (2006) 775-785.
[17] Miao A., Zhang J., Zhang T. and Aruna B.G.S., Threshold Dynamics of a Stochastic SIR model with Vertical Transmission and Vaccination, Computational and Mathematical Methods in Medicine (2017) 1-10.
[18] Tuckwell H.C. and Williams R.J., Some properties of a simple stochastic epidemic model of SIR type, Mathematical Biosciences 208 (2007) 76-97.
[19] Zhang X., Jiang D., Alsaedi A. and Hayat T., Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Applied Mathematics Letters 59 (2016) 87-93.
[20] Zhang X., Jiang D., Hayat T. and Ahmad B., Dynamics of a stochastic SIS model with double epidemic diseases driven by Léevy jumps, Physica A. Statistical Mechanics and Its Applications 471 (2017) 767-777.
[21] Zhao Y. and Jiang D., The threshold of a stochastic SIS epidemic model with vaccination, Applied Mathematics and Computation 243 (2014) 718-727.
[22] Zhao W., Li J., Zhang T., Meng X. and Zhang T., Persistence and ergodicity of plant disease model with Markov conversion and impulsive toxicant input, Communications in Nonlinear Science and Numerical Simulation 48 (2017) 70-84.
[23] Zhou X., Yuan S. and Zhao D., Threshold behavior of a stochastic SIS model with Léevy jumps, Applied Mathematics and Computation 275 (2016) 255-267.