International Journal of Nonlinear Analysis and Applications

‎A‎‎‎ two-area ‎epidemic ‎model ‎for ‎the ‎spread ‎of ‎COVID-19

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran

Abstract

In this paper, we model the spread of COVID-19 in a population of people travelling between two areas. New research implies that traveling of the asymptomatic infectious individuals, (i.e., infected individuals who have no symptoms of the disease and individuals with symptoms of the disease that are not detected by the healthcare system) can bring disease from one region to other regions even if the infectious individuals who are detected by the healthcare system, (i.e., confirmed cases), are inhibited from traveling among regions. We study the effect of travelling between two areas on the dynamics of COVID-19. Our model is formulated as a system of ordinary differential equations, with terms accounting for disease transmission, recovery, birth, death, and travel between two areas. We will give an explicit formula for calculating the basic reproduction number, $\mathcal{R}_{0}$, in the quarantine mode of two areas and explicit bounds on $\mathcal{R}_{0}$ for the case where the residents of both areas are in contact with each other. Our computations reveal the relationship between the basic reproduction number, a crucial quantity in epidemic control, and travel and return rates between areas. This suggests that it is essential to strengthen restrictions on passengers once infectious diseases appear.

Keywords

[1] F. Arrigoni and A. Pugliese, Limits of a multi-patch SIS epidemic model, J. Math. Bio. 45 (2002), 419–440.
[2] N.T.J. Bailey, Spatial models in the epidemiology of infectious diseases, Biological Growth and Spread, vol. 38 of Lecture Notes in Biomathematics, Springer-Verlag, 1980, pp. 233–261.
[3] F. Ball and O. Lyne, Epidemics among a population of households, C. Castillo- Chavez, with S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu (eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, vol. 126 of IMA Volumes in Mathematics and its Applications, 2002, pp. 115–142.
[4] D. Bernoulli, Essai d’une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l’inoculation pour la prevenir, Memoires de Mathematiques et de Physique, Academie Royale des Sciences, Paris, 1760, pp. 1–45.
[5] F. Brauer, Mathematical epidemiology: Past, present, and future, Infect. Disease Model. 2 (2017), no. 2, 113–127.
[6] G. Chowell, P.W. Fenimore, M.A. Castillo-Garsow, and C. Castillo-Chavez, SARS outbreak in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism, J. Theor. Biol. 224 (2003), 1–8.
[7] O. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, 2000.
[8] I.A. Hanski and M.E. Gilpin, Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, 1997.
[9] S.A. Levin, T.M. Powell, and J.H. Steele, Patch Dynamics, vol. 96 of Lecture Notes in Biomathematics, Springer[1]Verlag, 1993.
[10] I. Longini, A mathematical model for predicting the geographic spread of new infectious agents, Math. Biosci., 90 (1998), 367–383.
[11] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015.
[12] R. Memarbashi and S.M. Mahmoudi, A dynamic model for the COVID-19 with direct and indirect transmission pathways, Math. Metho. Appl. Sci. 44 (2021), 5873–5887.
[13] H. Minc, Nonnegative Matrices, Wiley Interscience, 1988.
[14] L. Rvachev and I. Longini, A mathematical model for the global spread of influenza, Math. Biosci. 75 (1985), no. 1, 3–22.
[15] L. Sattenspiel and K. Dietz, A structured epidemic model incorporating geographic mobility among areas, Math. Biosci. 128 (1995), 71–91.
[16] L. Sattenspiel and D.A. Herring, Structured epidemic models and the spread of influenza in the central Canadian subarctic, Human Bio. 70 (1998), 91–115.
[17] L. Sattenspiel and C. Simon, The spread and persistence of infectious diseases in structured populations, Math. Biosci. 90 (1988), 341–366.
[18] P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compart[1]mental models of disease transmission, Math. Biosci. 180 (2002), 29–48.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 16 July 2025
  • Receive Date: 12 August 2024
  • Revise Date: 31 January 2025
  • Accept Date: 01 February 2025