The classical models of epidemics dynamics by Ross and McKendrick have to be revisited in order to incorporate elements coming from the demography (fecundity, mortality and migration) both of host and vector populations and from the diffusion and mutation of infectious agents. The classical approach is indeed dealing with populations supposed to be constant during the epidemic wave, but the presently observed pandemics show duration of their spread during years imposing to take into account the host and vector population changes as well as the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as vectors and infectious agents. Two examples are presented, one concerning the malaria in Mali and the other the plague at the middle-age.

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Revisiting the classical model by Ross and Kermack-McKendrick, the Susceptible–Infectious–Recovered (SIR) model used to formalize the COVID-19 epidemic, requires improvements which will be the subject of this article. The heterogeneity in the age of the populations concerned leads to considering models in age groups with specific susceptibilities, which makes the prediction problem more difficult. Basically, there are three age groups of interest which are, respectively, 0–19 years, 20–64 years, and >64 years, but in this article, we only consider two (20–64 years and >64 years) age groups because the group 0–19 years is widely seen as being less infected by the virus since this age group had a low infection rate throughout the pandemic era of this study, especially the countries under consideration. In this article, we proposed a new mathematical age-dependent (Susceptible–Infectious–Goneanewsusceptible–Recovered (SIGR)) model for the COVID-19 outbreak and performed some mathematical analyses by showing the positivity, boundedness, stability, existence, and uniqueness of the solution. We performed numerical simulations of the model with parameters from Kuwait, France, and Cameroon. We discuss the role of these different parameters used in the model; namely, vaccination on the epidemic dynamics. We open a new perspective of improving an age-dependent model and its application to observed data and parameters.

(1) Background: The estimation of daily reproduction numbers throughout the contagiousness period is rarely considered, and only their sum R0 is calculated to quantify the contagiousness level of an infectious disease. (2) Methods: We provide the equation of the discrete dynamics of the epidemic’s growth and obtain an estimation of the daily reproduction numbers by using a deconvolution technique on a series of new COVID-19 cases. (3) Results: We provide both simulation results and estimations for several countries and waves of the COVID-19 outbreak. (4) Discussion: We discuss the role of noise on the stability of the epidemic’s dynamics. (5) Conclusions: We consider the possibility of improving the estimation of the distribution of daily reproduction numbers during the contagiousness period by taking into account the heterogeneity due to several host age classes.

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The problem of stability in population dynamics concerns many domains of application in demography, biology, mechanics and mathematics. The problem is highly generic and independent of the population considered (human, animals, molecules,…). We give in this paper some examples of population dynamics concerning nucleic acids interacting through direct nucleic binding with small or cyclic RNAs acting on mRNAs or tRNAs as translation factors or through protein complexes expressed by genes and linked to DNA as transcription factors. The networks made of these interactions between nucleic acids (considered respectively as edges and nodes of their interaction graph) are complex, but exhibit simple emergent asymptotic behaviours, when time tends to infinity, called attractors. We show that the quantity called attractor entropy plays a crucial role in the study of the stability and robustness of such genetic networks.