Dynamics of epidemics: Impact of easing restrictions and control of infection spread

Abstract

During an infectious disease outbreak, mathematical models and computational simulations are essential tools to characterize the epidemic dynamics and aid in design public health policies. Using these tools, we provide an overview of the possible scenarios for the COVID-19 pandemic in the phase of easing restrictions used to reopen the economy and society. To investigate the dynamics of this outbreak, we consider a deterministic compartmental model (SEIR model) with an additional parameter to simulate the restrictions. In general, as a consequence of easing restrictions, we obtain scenarios characterized by high spikes of infections indicating significant acceleration of the spreading disease. Finally, we show how such undesirable scenarios could be avoided by a control strategy of successive partial easing restrictions, namely, we tailor a successive sequence of the additional parameter to prevent spikes in phases of low rate of transmissibility.

Keywords: COVID-19; Control of infection spread; Easing restrictions; SEIR model; Spikes of infections.

Fig. 1

(a) Time-series of $S$ (blue line), $E$ (yellow line), $I$ (red line), and $R$ (green line) for $\beta =3/2.9,$$\omega =1/5.2,$$\gamma =1/2.9,$ and $\sigma =0$. (b) Time-series of $I$ for $\sigma =0$ (red line), $\sigma =0.2$ (blue line), $\sigma =0.4$ (indigo line), $\sigma =0.5$ (green line), and $\sigma =0.6$ (black line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2

Time-series of $I$ showing the effect of relaxing totally the restrictions ($\sigma =0,$ red and blue lines) for (a) $\sigma =0.40,$ (b) $\sigma =0.57,$ (c) $\sigma =0.70,$ and (d) $\sigma =0.53$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3

Two-dimensional diagram of $\sigma$versus Time (moment) showing (a) the value of peaks after removing restrictions; (b) the difference between the numbers of infections at the peak and the moment of relaxing restriction; (c) the ratio between the new peak value and the flattened curve peak; (d) the value of the effective reproductive number ($R_t=R_{0}S/N$) at the moment of relaxing restriction. The crosses indicate the behavior shown in Fig. 2b.

Fig. 4

Time-series of $I$ showing the effect of relaxing the restrictions from $\sigma =0.57$ (black line) to (a) $\sigma =0.50$ (red and blue lines), (b) $\sigma =0.30$ (blue line), (c) $\sigma =0.30$ (blue line), and (d) $\sigma =0.20$ (blue line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5

Two-dimensional diagram of $\sigma$versus Time (moment) showing the ratio between the new peak value and the flattened curve peak for $\sigma =0.57$. The golden dashed line indicates the peak for initially applied restriction $\sigma =0.57$ and the cross in blue corresponds to the response indicated in Fig. 4c. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

Time-series of $I$ (infected) showing the effects of the interventions with the appearance of the second wave (black line).

Fig. 7

Two-dimensional diagram of $I$ (number of infections at the moment of intervention) versus$\sigma$ showing (a) the value of the first-wave peak, (b) the value of the second-wave peak for $\sigma =0,$ (c) magnification of previous figure, and (d) the value of the effective reproductive number ($R_t=R_{0}S/N$) at the moment of relaxing restriction. The cross corresponds to the curve shown in Fig. 6a.

Fig. 8

Time-series of $I$ showing some strategies of controlling infection spread until two years following by second waves (green line). $S$ indicate the number of susceptible people just before of the second wave outbreak. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9

The number of susceptibles (at two years) as a function of the moment of first intervention after the peak ($\Delta t=t-t_P$) for $\sigma =0.57$ applied since the very beginning (blue line), $\sigma =0.65$ since 7.5 infected people (red line), and $\sigma =0.65$ since 2.5 infected (green line). We consider that births and natural deaths are balanced. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10

Two-dimensional diagram showing the number of susceptibles at two years in terms of $I_{bp}$ (moment of the initial intervention using $\sigma =0.65$) and $\Delta t$ (the moment of first intervention after the peak). We consider that births and natural deaths are balanced.