A mathematical model of the immune response to an infectious disease is proposed, taking into account the impact of environmental pollution. The pollution factor is described by a generalized Hutchinson equation and influences the formation of the plasma cell cascade and the damage to the target organ. The conditions for the existence of a unique solution to the mathematical model and its non-negativity on the semi-axis $[0,\infty ) $ are established.

The stationary solutions of the model and the conditions for their existence are obtained. The stability of the stationary solution, which characterizes the state of absence of the infectious disease, is investigated. An estimate of the initial infection level is derived, under which the infectious disease does not occur.

[1] Stiefs D., Venturino E., Feude U. Evidence of chaos in eco-epidemic models. Mathematical Biosciences
and Engineering. 2009. Vol. 6, no. 4. P. 855–871.
[2] Galvani A. P. Epidemiology meets evolutionary ecology. Trends in Ecology and Evolution. 2003. Vol.
18, no. 3. P. 132–139.
[3] Marchuk G.I. Mathematical Modelling of Immune Response in Infectious Diseases. Springer-Science+Bizness Media. 1997. P. 350.
[4] Bodnar M., Fory´s U. A model of immune system where time-dependent immune reactivity. Nonlinear
Analysis: Theory, Methods & Applications.2009. Vol.70, no. 2. P. 1049-1058.
[5] Fory´s U. Hopf bifurcations in Marchuk’s model of immune reactions. Math. Comput. Modelling. 2001.
Vol. 34. P. 725–735.
[6] Baranovsky S. V., Bomba A. Y. Modification of the Model of Mixed Infection Dy-namics with Regard for the Diffusion Perturbations and Interactions Between Antigens. Journal of Mathematical Sciences. 2024. Vol. 282, no. 5. P. 870–884.
[7] Skvortsova M. Asymptotic properties of solutions in Marchuk’s basic model of dis-ease. Func. Dif. Eq. 2017. No 3-4. P. 127-135.
[8] Bihun Y., Ukrainets O. Mathematical modelling of the immune response to infectious diseases with the influence of environmental factors. Acta et Commentationes, Exact and Natural Sciences. 2024. Vol. 18, no. 2. P. 7–17.
[9] Fory´s U. Marchuk’s model of immune system dynamics with application to tumour growth. J. Theor. Med. 2002. Vol. 4, no. 1. P. 85–93.
[10] Smith H. An Introduction to Delay Differential Equations where Applications to the Life Sciences. Springer Science+Business Media. 2011. P. 172.
[11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Academic Press. 1993. P. 397.
[12] Hartman Ph. Ordinary Differential Equations. New York, London, Sydney, John Wiley Sons. 1964. P. 612.