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The purpose of this work is the study of the qualitative behavior of the homogeneous in space solution of a delay differential equation arising from a model of infection dynamics. This study is mainly based on the monotone dynamical systems theory. Existence and smoothness of solutions are proved, and conditions of asymptotic stability of equilibriums in the sense of monotone dynamical systems are formulated. Then, sufficient conditions of global stability of the nonzero steady state are derived, for the two typical forms of the function f, specifying the efficiency of immune response-mediated virus elimination. Numerical simulations illustrate the analytical results. The obtained theoretical results have been applied, in a context of COVID-19 data calibration, to forecast the immunological behaviour of a real patient.

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We study a SIS epidemic model with an exponential demographic structure and a delay corresponding to the infectious period. The disease spread is described by a delay differential equation. Equilibriums and the basic reproduction number θ are identified. Using the monotone dynamical systems theory, local asymptotic stability of the two steady states is completely determined. Numerical simulations are carried out to illustrate the theoretical results.

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This work has two principal goals. First, we investigate the asymptotic behavior of a two-group epidemiological model and determine the expression of its basic reproduction number using the dynamical systems approach based on the spectral radius of the relative matrix. Second, we simulate the obtained analytical results using a new deep learning method that associates the ordinary differential equations governing the model to neural networks. A general disease-free equilibrium is considered and sufficient conditions of stability and convergence are formulated. A detailed description of the neural network model used in the simulation is provided. Moreover, the proposed deep learning simulation algorithm is compared to the simulation provided by “odeint”, a function from “SciPy” which is a Python library of mathematical routines.