Global mathematical analysis of a patchy epidemic model

Lahcen Boulaasair; Hassane Bouzahir; Mehmet Yavuz

doi:10.11121/ijocta.1558

Authors

Lahcen Boulaasair ISTI Lab, Ibn Zohr University, ENSA, PO Box 1136, Agadir, Morocco https://orcid.org/0000-0001-6495-5760
Hassane Bouzahir ISTI Lab, Ibn Zohr University, ENSA, PO Box 1136, Agadir, Morocco https://orcid.org/0000-0002-1254-967X
Mehmet Yavuz Necmettin Erbakan University, Department of Mathematics and Computer Sciences, Meram Yeniyol, 42090 Meram, Konya, Türkiye https://orcid.org/0000-0002-3966-6518

DOI:

https://doi.org/10.11121/ijocta.1558

Keywords:

Disease dynamics, Two-patch epidemic model, Stochastic analysis, Deterministic stability, Exponential disease extinction, Persistence of the disease, Numerical simulation

Abstract

The dissemination of a disease within a homogeneous population can typically be modeled and managed in a uniform fashion. Conversely, in non-homogeneous populations, it is essential to account for variations among subpopulations to achieve more precise predictive modeling and efficacious intervention strategies. In this study, we introduce and examine the comprehensive behavior of a deterministic two-patch epidemic model alongside its stochastic counterpart to assess disease dynamics between two heterogeneous populations inhabiting distinct regions. First, utilizing a specific Lyapunov function, we demonstrate that the disease-free equilibrium of the deterministic model is globally asymptotically stable. For the stochastic model, we establish that it is well-posed, meaning it possesses a unique positive solution with probability one. Subsequently, we ascertain the conditions necessary to ensure the total extinction of the disease across both regions. Furthermore, we explicitly determine a threshold condition under which the disease persists in both areas. Additionally, we discuss a scenario wherein the disease persists in one region while simultaneously becoming extinct in the other. The article concludes with a series of numerical simulations that corroborate the theoretical findings.

Downloads

Download data is not yet available.

Author Biographies

Lahcen Boulaasair, ISTI Lab, Ibn Zohr University, ENSA, PO Box 1136, Agadir, Morocco

Lahcen Boulaasair earned his Ph.D. from the National School of Applied Sciences of Agadir, under the mentorship of Professor Hassane Bouzahir. He is presently engaged as a researcher at the LISTI laboratory within the same institution. His research interests lie in the field of applied mathematics, with a focus on stochastic differential equations, partial differential equations and time-series.

Hassane Bouzahir, ISTI Lab, Ibn Zohr University, ENSA, PO Box 1136, Agadir, Morocco

Hassan Bouzahir is a Full Professor of Mathematics/Statistics at the National School of Applied Sciences, Ibn Zohr University, Agadir, Morocco. He earned his Ph.D. in Mathematics in April 2001 and has held notable positions such as Head of Departments and Director of Research Laboratories. With numerous publications, including research books, book chapters, and articles in prestigious journals and databases, his work spans various topics in Engineering and Mathematics. He has also contributed to IEEE conferences and delivered talks globally. Known for his exceptional teaching, he has mentored many graduate students and Ph.D. candidates. His international career includes professional activities in numerous countries, making him a respected figure in the global academic community.

Mehmet Yavuz, Necmettin Erbakan University, Department of Mathematics and Computer Sciences, Meram Yeniyol, 42090 Meram, Konya, Türkiye

Mehmet Yavuz received a BSc degree (2009) from Zonguldak Bulent Ecevit University, Türkiye, and both MSc (2012) and PhD (2016) degrees in applied mathematics from Balıkesir University, Türkiye. He visited the Department of Mathematics, University of Exeter, U.K., in the period 2019-2020 for postdoctoral research. He is currently an Associate Professor in the Mathematics and Computer Sciences Department at Necmettin Erbakan University, Türkiye. He has published more than 100 research papers in reputed journals and conference papers as well as 10 book chapters in international books. Moreover, he edited 4 international books published by esteemed publishers. He is currently concerned with fractional calculus, and its applications to the different fields of science, mathematical biology, nonlinear dynamics, and optimal control. He has 12 years of research and teaching experience in these fields. Apart from being an Associate Editor in several esteemed journals, he is the Editor-in-Chief of the journals “Mathematical Modelling and Numerical Simulation with Applications” and “Bulletin of Biomathematics”.

References

Van Seventer, J.M., & Hochberg, N.S. (2016). Principles of Infectious Diseases: Transmission, Diagnosis, Prevention, and Control. International Encyclopedia of Public Health, 22-39. https://doi.org/10.1016/B978-0-12-803678-5.00516-6

Braun, D. (2016). Health Security: Abwarten ist keine Option; Epidemien erfordern die Zusammenarbeit von Sicherheits-und Gesundheitsexperten. DEU, 1-4.

Patrick, S. (2011). Weak links: fragile states, global threats, and international security. Oxford University Press.

Wang, C., Horby, P.W., Hayden, F.G., & Gao, G.F. (2020). A novel coronavirus outbreak of global health concern. Lancet (London, England), Vol.395, 470-473. https://doi.org/10.1016/S0140-6736(20)30185-9

Chriscaden, K.(2020). Impact of COVID-19 on people’s livelihoods, their health and our food systems. https://www.who.int/news/item/13-10-2020-impact-of-covid-19-on-people's-livelihoods-their-health-and-our-food-systems

Kermack, W.O., & Mckendrick, A.G. (1991). Contributions to the mathematical theory of epidemics I. Bulletin of Mathematical Biology, 53, 33-55. https://doi.org/10.1016/S0092-8240(05)80040-0

Yavuz, M., ur Rahman, M., Yildiz, M., & Joshi, H. (2024). Mathematical Modeling of Middle East Respiratory Syndrome Coronavirus with Bifurcation Analysis. Contemporary Mathematics, 5(3), 3997-4012. https://doi.org/10.37256/cm.5320245004

Ur Rahman, M., Yavuz, M., Arfan, M., & Sami, A. (2024). Theoretical and numerical investigation of a modified ABC fractional operator for the spread of polio under the effect of vaccination. AIMS Biophysics, 11(1), 97-120. https://doi.org/10.3934/biophy.2024007

ur Rahman, M., Arfan, M., & Baleanu, D. (2023). Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions. Bulletin of Biomathematics, 1(1), 1-23. https://doi.org/10.59292/bulletinbiomath.2023001

Li, M.Y., & Muldowney, J.S. (1995). Global stability for the SEIR model in epidemiology. Mathematical Biosciences, 125(2), 155-64. https://doi.org/10.1016/0025-5564(95)92756-5

Smith, H.L., Wang, L., & Li, M.Y. (2001). Global Dynamics of an SEIR Epidemic Model with Vertical Transmission. SIAM Journal on Applied Mathematics, 62, 58-69. https://doi.org/10.1137/S0036139999359860

Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., & Liu, W. (1989). Epidemiological models with age structure, proportionate mixing, and cross-immunity. Journal of Mathematical Biology, 27, 233-258. https://doi.org/10.1007/BF00275810

Shulgin, B., Stone, L., & Agur, Z. (1998). Pulse vaccination strategy in the SIR epidemic model. Bulletin of Mathematical Biology, 60, 1123-1148. https://doi.org/10.1016/S0092-8240(98)90005-2

d’Onofrio, A. (2005). On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Applied Mathematics Letters, 18, 729- 732. https://doi.org/10.1016/j.aml.2004.05.012

Shi, Z., Zhang, X., & Jiang, D. (2019). Dynamics of an avian influenza model with half-saturated incidence. Applied Mathematics and Computation, 355, 399-416. https://doi.org/10.1016/j.amc.2019.02.070

Calvo, J.G., Hern’andez, A., Porter, M.A., & Sanchez, F. (2019). A two-patch epidemic model with nonlinear reinfection. Revista de Matematica: Teoria y Aplicaciones, 27(1), 23-48. https://doi.org/10.15517/rmta.v27i1.39946

Moore, C. (1990). Unpredictability and undecidability in dynamical systems. Physical Review Letters, 64(20), 2354-2357. https://doi.org/10.1103/PhysRevLett.64.2354

Cao, Z., Feng, W., Wen, X., & Zu, L. (2019). Dynamical behavior of a stochastic SEI epidemic model with saturation incidence and logistic growth. Physica A: Statistical Mechanics and its Applications, 523, 894-907. https://doi.org/10.1016/j.physa.2019.04.228

Pang, Y., Han, Y., & Li, W. (2014). The threshold of a stochastic SIQS epidemic model. Advances in Difference Equations, 2014, 1-15. https://doi.org/10.1186/1687-1847-2014-320

Khan, T., Zaman, G., & El-Khatib, Y. (2021). Modeling the dynamics of novel coronavirus (COVID-19) via stochastic epidemic model. Results in Physics, 24, 104004. https://doi.org/10.1016/j.rinp.2021.104004

Joshi, H., & Yavuz, M. (2024). Numerical analysis of compound biochemical calcium oscillations process in hepatocyte cells. Advanced Biology, 8(4), 2300647. https://doi.org/10.1002/adbi.202300647

Joshi, H., Yavuz, M., & ¨ Ozdemir, N. (2024). Analysis of novel fractional order plastic waste model and its effects on air pollution with treatment mechanism. Journal of Applied Analysis & Computation, 14(6), 3078-3098. https://doi.org/10.11948/20230453

Adel, W., Elsonbaty, A., & Mahdy, A. (2024). On some recent advances in fractional order modeling in engineering and science. Computation and Modeling for Fractional Order Systems, 169-197. https://doi.org/10.1016/B978-0-44-315404-1.00016-3

Salih, R. I., Jawad, S., Dehingia, K., & Das, A. (2024). The effect of a psychological scare on the dynamics of the tumor-immune interaction with optimal control strategy. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(3), 276-293. https://doi.org/10.11121/ijocta.1520

Evirgen, F., Ozkose, F., Yavuz, M., & Ozdemir, N. (2023). Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioengineering, 10(3), 218-239. https://doi.org/10.3934/bioeng.2023015

El-Mesady, A., Ahmed, N., Elsonbaty, A., & Adel, W. (2023). Transmission dynamics and control measures of reaction-diffusion pine wilt disease model. The European Physical Journal Plus, 138(12), 1078. https://doi.org/10.1140/epjp/s13360-023-04705-8

Yavuz, M., Boulaasair, L., Bouzahir, H., Diop, M.A., & Benaid, B. (2024). The impact of two independent gaussian white noises on the behavior of a stochastic epidemic model. Journal of Applied Mathematics and Computational Mechanics, 23(1), 121-134. https://doi.org/10.17512/jamcm.2024.1.10

Izadi, M., El-Mesady, A., & Adel, W. (2024). A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion. Mathematical Modelling and Numerical Simulation with Applications, 4(1), 37-65. https://doi.org/10.53391/mmnsa.1408997

Ayaz, F., & Heredag, K. (2024). Fractional model for blood flow under MHD influence in porous and non-porous media. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(2), 156-167. https://doi.org/10.11121/ijocta.1497

Raeisi, E., Yavuz, M., Khosravifarsani, M., & Fadaei, Y. (2024). Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm. The European Physical Journal Plus, 139(4), 345. https://doi.org/10.1140/epjp/s13360-024-05111-4

Boulaasair, L., Bouzahir, H., Vargas, A.N., & Diop, M.A. (2022). Existence and uniqueness of solutions for stochastic urban-population growth model. Frontiers in Applied Mathematics and Statistics, 8, 960399. https://doi.org/10.3389/fams.2022.960399

Naik, P. A., Yavuz, M., Qureshi, S., Owolabi, K. M., Soomro, A., & Ganie, A. H. (2024). Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment. Computer Methods and Programs in Biomedicine, 254, 108306. https://doi.org/10.1016/j.cmpb.2024.108306

Evirgen, F., Ucar, E., Ucar, S., & Ozdemir, N. (2023). Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-73. https://doi.org/10.53391/mmnsa.1274004

Sabbar, Y. (2023). Asymptotic extinction and persistence of a perturbed epidemic model with different intervention measures and standard levy jumps. Bulletin of Biomathematics, 1(1), 58-77. https://doi.org/10.59292/bulletinbiomath.2023004

Kiouach, D., & Boulaasair, L. (2018). Stationary distribution and dynamic behaviour of a stochastic SIVR epidemic model with imperfect vaccine. Journal of Applied Mathematics, 2018(1), 1291402. https://doi.org/10.1155/2018/1291402

Elsonbaty, A., Alharbi, M., El-Mesady, A., & Adel, W. (2024). Dynamical analysis of a novel discrete fractional lumpy skin disease model. Partial Differential Equations in Applied Mathematics, 9, 100604. https://doi.org/10.1016/j.padiff.2023.100604

Boulaasair, L. (2023). Threshold properties of a stochastic epidemic model with a variable vaccination rate. Bulletin of Biomathematics, 1(2), 177-191. https://doi.org/10.59292/bulletinbiomath.2023009

Adel, W., Amer, Y.A., Youssef, E.S., & Mahdy, A.M. (2023). Mathematical analysis and simulations for a Caputo-Fabrizio fractional COVID-19 model. Partial Differential Equations in Applied Mathematics, 8, 100558. https://doi.org/10.1016/j.padiff.2023.100558

Fatima, B., Yavuz, M., Rahman, M.U., & Al- Duais, F.S. (2023). Modeling the epidemic trend of middle eastern respiratory syndrome coronavirus with optimal control. Mathematical Biosciences and Engineering, 20(7), 11847-11874. https://doi.org/10.3934/mbe.2023527