From COVID-19 to post-COVID-19 condition: a modelling approach with mathematical formulation of follow-up strategies

The COVID-19 pandemic left behind another health burden, the post-COVID-19/long-COVID condition. We construct an extended SEI R-type model for COVID-19, which for the first time couples additional compartments that account for long-COVID conditions in infected, hospitalised, and recovered COVID-19 cases. In addition to exponentially distributed follow-up strategies, the key intervention to stop the progression to long-COVID condition is formulated through a general treatment function, encompassing linear, saturated, and logistic forms. The latter two formulations explicitly account for limited access to healthcare resources. From a mathematical standpoint, we derive a threshold quantity, , which, depending on the form of the treatment function, may coincide with or exceed the basic reproduction number, , and which guarantees the global asymptotic stability of the disease-free equilibrium (DFE) whenever it is less than one. In addition, we establish another threshold, , whose value, greater than unity, ensures the existence of at least one endemic equilibrium. Furthermore, we construct a nonstandard finite difference scheme that preserves the qualitative properties of the continuous model. Numerical simulations corroborate the analytical findings and reveal that, for certain treatment functions, reducing below unity alone may not suffice to eliminate the disease. Finally, the sensitivity analysis consistently emphasizes the critical importance of developing effective treatment protocols for long-COVID in order to reduce post-COVID incidence.