In this work, we create and analyze a new model of an epidemic that incorporates indirect environmental transmission through a contaminated reservoir as well as direct human-to-human transmission. The environmental pathogen concentration is precisely predicted, and the population is separated into susceptible, exposed, infected, hospitalized, and recovered compartments. The transmission processes are described by saturated incidence functions to account for behavioral and environmental limitations. The model is investigated under three complementary mathematical frameworks. First, the deterministic integer-order system is analyzed to ascertain the global stability, boundedness, threshold dynamics, and positivity of the endemic and disease-free equilibria. Second, adequate conditions for disease extinction and persistence are determined by introducing stochastic perturbations to reflect random fluctuations resulting from environmental variability and unclear contact patterns. Third, the Caputo derivative is used to extend the model to a fractional-order formulation that takes into consideration memory effects found in biological systems. Volterra-type Lyapunov functionals are used to produce global stability results for the fractional system and to establish the existence and uniqueness of solutions. Numerical simulations are provided to illustrate the theoretical findings and to compare the effects of stochasticity and memory on disease dynamics. The results demonstrate that random perturbations can suppress outbreaks even when the basic reproduction number exceeds unity, while fractional-order dynamics significantly alter the speed and persistence of disease transmission.
Citation: Niaz Ali Shah, Wael Mahmoud Mohammad Salamehs, Isra Al-Shbeil, Ferdous M. O. Tawfiq. Modeling environmental and memory effects in epidemic transmission: Deterministic, stochastic, and fractional perspectives[J]. AIMS Mathematics, 2026, 11(8): 24490-24524. doi: 10.3934/math.2026988
In this work, we create and analyze a new model of an epidemic that incorporates indirect environmental transmission through a contaminated reservoir as well as direct human-to-human transmission. The environmental pathogen concentration is precisely predicted, and the population is separated into susceptible, exposed, infected, hospitalized, and recovered compartments. The transmission processes are described by saturated incidence functions to account for behavioral and environmental limitations. The model is investigated under three complementary mathematical frameworks. First, the deterministic integer-order system is analyzed to ascertain the global stability, boundedness, threshold dynamics, and positivity of the endemic and disease-free equilibria. Second, adequate conditions for disease extinction and persistence are determined by introducing stochastic perturbations to reflect random fluctuations resulting from environmental variability and unclear contact patterns. Third, the Caputo derivative is used to extend the model to a fractional-order formulation that takes into consideration memory effects found in biological systems. Volterra-type Lyapunov functionals are used to produce global stability results for the fractional system and to establish the existence and uniqueness of solutions. Numerical simulations are provided to illustrate the theoretical findings and to compare the effects of stochasticity and memory on disease dynamics. The results demonstrate that random perturbations can suppress outbreaks even when the basic reproduction number exceeds unity, while fractional-order dynamics significantly alter the speed and persistence of disease transmission.
| [1] |
G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo, et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855–860. https://doi.org/10.1038/s41591-020-0883-7 doi: 10.1038/s41591-020-0883-7
|
| [2] | A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funk, et al., Early dynamics of transmission and control of COVID-19: A mathematical modelling study, Lancet Infect. Dis., 20 (2020), 553–558. |
| [3] |
X. He, E. H. Y. Lau, P. Wu, X. Deng, J. Wang, X. Hao, et al., Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat. Med., 26 (2020), 672–675. https://doi.org/10.1038/s41591-020-0869-5 doi: 10.1038/s41591-020-0869-5
|
| [4] |
J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron, A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610–621. https://doi.org/10.1017/S0950268806007217 doi: 10.1017/S0950268806007217
|
| [5] | P. Rohani, A. A. King, The role of environmental transmission in Ebola outbreaks, P. Roy. Soc. B, 286 (2019), 20191535. |
| [6] | F. Alvarez, C. Flores, Stochastic modeling of Ebola epidemics, Math. Biosc., 304 (2018), 1–10. |
| [7] | C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001). https://doi.org/10.1186/1471-2334-1-1 |
| [8] | J. H. Tien, D. J. D. Earn, Environmental transmission of infectious diseases, J. Theor. Biol., 413 (2017), 181–192. |
| [9] | A. A. King, E. L. Ionides, Inapparent infections and cholera dynamics, Nature, 554 (2018), 37–41. |
| [10] | B. Ahmad, J. J. Nieto, A study of a fractional-order cholera model, Chaos Soliton. Fract., 65 (2014), 203–211. |
| [11] |
J. Shaman, M. Kohn, Absolute humidity and the seasonal onset of influenza in the continental united states, PLoS Biol., 7 (2010), e1000316. https://doi.org/10.1371/journal.pbio.1000316 doi: 10.1371/journal.pbio.1000316
|
| [12] | N. Bacaër, Mathematical models for influenza pandemic, Springer, 2012. |
| [13] | Y. Chen, X. Li, Fractional stochastic modeling of influenza dynamics, Fractals, 30 (2022), 2250124. |
| [14] |
H. W Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
|
| [15] | M. J. Keeling, P. Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, 2017. |
| [16] | L. J. S. Allen, A primer on stochastic epidemic models, Springer, 2017. |
| [17] | A. Gray, D. Greenhalgh, Stochastic modeling of infectious diseases, Bull. Math. Biol., 83 (2021), 1–28. |
| [18] | D. Jiang, N. Shi, Stochastic epidemic models with environmental noise, Chaos Soliton. Fract., 155 (2022), 111720. |
| [19] | M. Liu, T. Zhang, Noise-induced effects in epidemic models, Nonlinear Dynam., 111 (2023), 987–1004. |
| [20] | X. Mao, Stochastic differential equation models of epidemic dynamics, J. Math. Anal. Appl., 495 (2021), 124755. |
| [21] | K. Diethelm, The analysis of fractional differential equations, Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2 |
| [22] | I. Area, J. J. Nieto, Fractional calculus in epidemiological modeling, Appl. Math. Comput., 375 (2020), 125091. |
| [23] | A. Firmino, D. F. M. Torres, Fractional-order epidemic models and their applications, Mathematics, 9 (2021), 543. |
| [24] | W. Gao, H. M. Baskonus, Memory effects in fractional epidemic models, Fractals, 31 (2023), 2350021. |
| [25] | X. Mao, Stochastic differential equations and applications, Woodhead Publishing, 2007. https://doi.org/10.1533/9780857099402 |
| [26] |
X. Li, C. Xu, Fractional stochastic epidemic models with memory effects, Chaos Soliton. Fract., 156 (2022), 111847. https://doi.org/10.1016/j.chaos.2022.111847 doi: 10.1016/j.chaos.2022.111847
|
| [27] | B. Ahmad, A. Alsaedi, Stochastic fractional-order epidemic models: Analysis and simulations, Math. Method. Appl. Sci., 46 (2023), 5361–5380. |
| [28] | L. Perko, Differential equations and dynamical systems, Springer, 2013. |
| [29] |
J. P. LaSalle, Some extensions of Lyapunov's second method, IRE T. Circuit Theor., 7 (1960), 520–527. https://doi.org/10.1109/TCT.1960.1086720 doi: 10.1109/TCT.1960.1086720
|
| [30] | O. Diekmann, H. Heesterbeek, T. Britton, Mathematical tools for understanding infectious disease dynamics, Princeton University Press, 2010. |
| [31] | I. Podlubny, Fractional differential equations, Academic Press, 1999. |
| [32] |
Z. Shuai, P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513–1532. https://doi.org/10.1137/120876642 doi: 10.1137/120876642
|
| [33] |
X. Mao, G. Marion, E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95–110. https://doi.org/10.1016/S0304-4149(01)00126-0 doi: 10.1016/S0304-4149(01)00126-0
|
| [34] | J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, Chichester, UK, 3 Eds., 2016. https://doi.org/10.1002/9781119121534 |
| [35] | R. L. Burden, J. D. Faires, A. M. Burden, Numerical analysis, Cengage Learning, Boston, MA, USA, 10th edition, 2015. |
| [36] | P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Springer-Verlag, Berlin, Germany, 1992. https://doi.org/10.1007/978-3-662-12616-5 |
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Niaz Ali Shah, Wael Mahmoud Mohammad Salamehs, Isra Al-Shbeil, Ferdous M. O. Tawfiq. Modeling environmental and memory effects in epidemic transmission: Deterministic, stochastic, and fractional perspectives[J]. AIMS Mathematics, 2026, 11(8): 24490-24524. doi: 10.3934/math.2026988
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