The Identifiability of Mathematical Models in Epidemiology: Tuberculosis, HIV, COVID-19
Olga Krivorotko1,2, Sergey Kabanikhin1, Victoriya Petrakova3
1Sobolev Institute of Mathematics of SB RAS, Novosibirsk, Russia
2The Moscow Institute of Physics and Technology, Dolgoprudny, Russia
3Institute of Computational Modeling of SB RAS, Krasnoyarsk, Russia
Abstract. The paper is devoted to the short review and application of sensitivity-based identifiability approaches for analyzing mathematical models of epidemiology and related processes described by systems of differential equations and agent-based models. It is shown that for structural identifiability of basic SIR models (describe the dynamic of Susceptible, Infected and Removed groups based on nonlinear ordinary differential equations) of epidemic spread and linear compartmental models it is possible to use a priori information about the process. It is demonstrated that a model can be structurally identifiable but be practically non-identifiable due to incomplete data. The paper uses methods for analyzing the sensitivity of parameters to data variation, as well as analyzing the sensitivity of model states to parameter variation, based on linear and differential algebra, Bayesian, and Monte Carlo approaches. It was shown that in the SEIR-HCD model of COVID-19 propagation, described by a system of seven ordinary differential equations and based on the mass balance law, the parameter of humoral immunity acquisition is the least sensitive to changes in the number of diagnosed, critical and mortality cases of COVID-19. The spatial SEIR-HCD model of COVID-19 propagation demonstrated an increase the sensitivity of the partial immunity duration parameter over time, as well as a decrease in the limits of change in the infectivity and infection parameters. In the case of the SEIR-HCD mean-field model of COVID-19 propagation, the sensitivity of the system to the self-isolation index and the lack of sensitivity of the stochastic parameters of the system are shown. In the case of the agent-based COVID-19 propagation model, the change in the infectivity parameter was reduced by more than a factor of 2 compared to the statistics. A differential model of co-infection HIV and tuberculosis spread with multiple drug resistance was developed and its local identifiability was shown.
Key words: epidemiology, SIR, sensitivity-based identifiability analysis, identifiability, a priori information, Bayesian methods, Sobol method.