RESUMO
In this work, we propose a new non-standard finite-difference-method for the numerical solution of the time-continuous non-autonomous susceptible-infected-recovered model. For our time-discrete numerical solution algorithm, we prove preservation of non-negativity and show that the unique time-discrete solution converges linearly towards the time-continuous unique solution. In addition to that, we introduce a parameter identification algorithm for the susceptible-infected-recovered model. Finally, we provide two numerical examples to stress our theoretical findings.
RESUMO
Tuberculosis (TB) is the 10th leading cause of death worldwide, and since 2007 it has been the main cause of death from a single infectious agent, ranking above HIV/AIDS. The current COVID-19 is a pandemic which caused many deaths around the world. The danger is not only a coinfection as observed for TB and HIV for a long time, but that both TB and SARS-CoV-2 affect the respiratory organs and thus potentiate their effect or accelerate the critical course. A key public health priority during the emergence of a novel pathogen is the estimation of the clinical need to assure adequate medical treatment. This requires a correct adjustment to the critical case detection rate and the prediction of possible scenarios based on known patterns. The African continent faces constraining preconditions in regard to healthcare capacities and social welfare which may hinder required countermeasures. However, given the high TB prevalence rates, COVID-19 may show a particular severe course in respective African countries, e.g. South Africa. Using WHO's TB and public infrastructure data, we conservatively estimate that the symptomatic critical case rate, which affects the healthcare system, is between 8 and 12% due to the interaction of COVID-19 and TB, for a TB population of 0.52% in South Africa. This TB prevalence leads to a significant increase in the peak load of critical cases of COVID-19 patients and potentially exceeds current healthcare capacities.