An Age-Structured Diffusive Model for Epidemic Modelling: Lie Symmetries and Exact Solutions

• Roman Cherniha ^[2] ; Vasyl Davydovych ^[1]
  1. [1] National Academy of Sciences of Ukraine

     National Academy of Sciences of Ukraine

     Ucrania

     
  2. [2] National University of Kyiv-Mohyla Academy, University of Nottingham
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • A new age-structured diffusive model for the mathematical modelling of epidemics is suggested. The model can be considered as a generalization of two models suggested earlier for similar purposes. The Lie symmetry classification of the model is derived. It is shown that the model admits an infinite-dimensional Lie algebra of invariance. Using the Lie symmetries, exact solutions, in particular those of the travelling wave types and in terms of special functions, are constructed. Examples of application of exact solutions with the correctly-specified parameters for calculation of the total number of infected individuals during an epidemic are presented.

• Referencias bibliográficas
  • 1. Ablowitz, M., Zeppetella, A.: Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835–840 (1979)
  • 2. Arrigo, D.J.: Symmetry Analysis of Differential Equations. John Wiley & Sons Inc, Hoboken, N.J. (2015)
  • 3. Arrigo, D.J., Hill, J.M., Broadbridge, P.: Nonclassical symmetries reductions of the linear diffusion equation with a nonlinear source....
  • 4. Bentout, S., Chen, Y., Djilali, S.: Global Dynamics of an SEIR Model with Two Age Structures and a Nonlinear Incidence. Acta Appl. Math....
  • 5. Cherniha, R., Davydovych, V.: A mathematical model for the COVID-19 outbreak and its applications. Symmetry 12, 12 pp. (2020)
  • 6. Cherniha, R.M., Davydovych, V.V.: A reaction-diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications...
  • 7. Cherniha, R., Dutka, V., Davydovych, V.: A space distributed model and its application for modeling the COVID-19 pandemic in Ukraine. Symmetry...
  • 8. Cherniha, R., King, J.: Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. Stud....
  • 9. Cherniha, R., Serov, M., Pliukhin, O.: Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry. Exact Solutions...
  • 10. Clarkson, P.A., Mansfield, E.L.: Symmetry reductions and exact solutions of a class of nonlinear heat equations. Phys. D 70(70), 250–288...
  • 11. Davydovych, V., Dutka, V., Cherniha, R.: Reaction-diffusion equations in mathematical models arising in epidemiology. Symmetry 15, 2025...
  • 12. Ducrot, A.: Traveling wave solutions for a scalar age-structured equation. Discrete Contin. Dyn. Syst. B 7, 251–273 (2007)
  • 13. Gazeau, J.P., Winternitz, P.: Symmetries of variable coefficient Korteweg-de Vries equations. J. Math. Phys. 33(12), 4087–4102 (1992)
  • 14. Gilding, B.H., Kersner, R.: Travelling Waves in Nonlinear Reaction-Convection-Diffusion. Birkhauser Verlag, Basel (2004)
  • 15. Gopalsamy, K.: Age dependent population dispersion in linear habitats. Ecol. Mod. 3(2), 119–132 (1977)
  • 16. Gurtin, M.E., MacCamy, R.C.: Diffusion models for age-structured populations. Math. Biosci. 54, 49 (1981)
  • 17. Gurtin, M.E., MacCamy, R.C.: Product solutions and asymptotic behavior for age-dependent, dispersing populations. Math. Biosci. 62, 1577...
  • 18. Ibragimov, N.H., Torrisi, M., Valenti, A.: Preliminary group classification of equations vtt = f (x, vx )vx x + g(x, vx ). J....
  • 19. Jin, Y., Zhao, X.Q.: Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure. SIAM J. Math. Anal. 40(6),...
  • 20. Kawahara, T., Tanaka, M.: Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation. Phys. Lett. A 97, 311–314...
  • 21. Kingston, J.G.: On point transformations of evolution equations. J. Phys. A 24(14), L769–L774 (1991)
  • 22. Kingston, J.G., Sophocleous, C.: On form-preserving point transformations of partial differential equations. J. Phys. A 31(6), 1597–1619...
  • 23. Kubo, M., Langlais, M.: Periodic solutions for a population dynamics problem with age-dependence and spatial structure. J. Math. Biol....
  • 24. Lahno, V.I., Spichak, S.V., Stohnii, V.I.: Symmetry Analysis of Evolutionary Equations. Institute of Mathematics of NAS of Ukraine, Kyiv...
  • 25. Liu, Z., Wu, Y., Zhang, X.: Existence of periodic wave trains for an age-structured model with diffusion. Discrete Contin. Dyn. Syst....
  • 26. Lopez, E., Dimas, S., Bozhkov, Y.: Symmetries of Ricci flows. Adv. Nonlinear Anal. 12, 20230106 (2023)
  • 27. McCluskey, C.C.: Global stability for an SEI model of infectious disease with age structure and immigration of infecteds. Math. Biosci....
  • 28. Metz, J.A.J., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Springer, New York (1986)
  • 29. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)
  • 30. Nariboli, G.: Self-similar solutions of some nonlinear equations. Appl. Scientific Res. 22, 449–461 (1970)
  • 31. Trucco, E.: Mathematical models for cellular systems the von Foerster equation. Part I. Bull. Math. Biophys. 27, 285–304 (1965)
  • 32. Trucco, E.: Mathematical models for cellular systems. The von Foerster equation. Part II. Bull. Math. Biophys. 27, 449–471 (1965)
  • 33. Vaneeva, O.O., Popovych, R.O., Sophocleous, C.: Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion...
  • 34. Von Foerster, H.: Some remarks on changing populations. In: The Kinetics of Cellular Proliferation, 382–407 (1959)