The novel coronavirus (SARS-CoV-2), identified in China at the end of December 2019 and causing the disease COVID-19, has meanwhile led to outbreaks all over the globe, with about 571,700 confirmed cases and about 26,500 deaths as of March 28th, 2020. We present here the preliminary results of a mathematical study directed at informing on the possible application or lifting of control measures in Germany. The developed mathematical models allow to study the spread of COVID-19 among the population in Germany and to asses the impact of non-pharmaceutical interventions.

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We consider the qualitative behavior of a mathematical model for transmission dynamics with two nonlinear stages of contagion. The proposed model is inspired by phenomena occurring in epidemiology (spread of infectious diseases) or social dynamics (spread of opinions, behaviors, ideas), and described by a compartmental approach. Upon contact with a promoter (contagious individual), a naive (susceptible) person can either become promoter himself or become weakened, hence more vulnerable. Weakened individuals become contagious when they experience a second contact with members of the promoter group. After a certain time in the contagious compartment, individuals become inactive (are insusceptible and cannot spread) and are removed from the chain of transmission. We combine this two-stage contagion process with renewal of the naive population, modeled by means of transitions from the weakened or the inactive status to the susceptible compartment. This leads to rich dynamics, showing for instance coexistence and bistability of equilibria and periodic orbits. Properties of (nontrivial) equilibria are studied analytically. In addition, a numerical investigation of the parameter space reveals numerous bifurcations, showing that the dynamics of such a system can be more complex than those of classical epidemiological ODE models.