Z-control on dynamics of pollution-allergy model

Allergy due to toxic pollutants present in the environment has become a major public health threat affecting the lives of more than one billion people worldwide. Its prevalence and impact are on the rise due to urbanization and growing chemical industries, on the other hand emerging toxic allergens heavily impact the budgets of public health systems. In the current work, the dynamical model of allergy due to pollution is constructed and studied. Moreover, the basic reproduction number is formulated to calculate the threshold value of allergic diseases. The model is stabilized locally and globally, besides the conditions that emerged from stability theory are used to study the effects of various parameters. Z-type control mechanism has been used to optimize the stability of the model. Numerical simulations validate, that chaotic oscillated compartments are stabilized under the effect of Z-control which shows that allergic diseases due to the polluted environment can be controlled.