Spatial spread of epidemic with Allee effect

The spatial spread of an epidemic is investigated in the case of a bistable dynamics, where the effective transmission rate depends on the fraction of infected and the state of no epidemic is linearly stable. The front propagation phenomenon is investigated both numerically and theoretically, by an analysis in a four-dimensional phase plane. A good agreement between numerical and theoretical results has been found both for the front profiles and for the speed of invasion. We discovered a novel phenomenon of front stoppage: In some regime of parameters, the front solution ceases to exist, and the propagating pulse of infection decays despite the initial outbreak.

Figure 1

Spatial profiles of the fraction of infected individuals (the upper panel) and the fraction of susceptible individuals (the lower panel) at different time instances. The curves were computed by numerically solving the system of Eqs. (1). After a short transient, a front propagation is established. In both panels, the profiles are shown at time instances of $\overline{t}=0$ (red dashed-dotted curves), $\overline{t}=200$ (black curves), $\overline{t}=400$ (blue curves), and $\overline{t}=600$ (green curves), where $\overline{t}=\alpha t$ is the rescaled time. The parameters are $\text{Ibar}=0.02$, ${\overline{r}}_{\text{min}}=0.14$, ${\overline{r}}_{\text{max}}=2.8$, $D=1$, where ${\overline{r}}_{\text{min}}=r_{\text{min}}/\alpha$ and ${\overline{r}}_{\text{max}}=r_{\text{max}}/\alpha$.

Figure 2

Front speed computed numerically by solving the system of Eqs. (1) (the blue dotted line) for $\text{Ibar}=0.025$ (the rest of the parameters are the same as in Fig. 1). The plateau value of the front speed (shown by the red dashed line) is computed theoretically by finding the desired trajectory in a $4\text{D}$ phase space (see text).

Figure 3

The trajectory in the phase space of $I,v$ (the upper panel) and $S,u$ (the lower panel) corresponding to the front profiles $I\left(\xi \right)$ and $S\left(\xi \right)$. The theoretical analysis of the four-dimensional phase space shows two different solutions (two different trajectories in the phase space) for the front propagation problem formulated in Eqs. (3). One of these trajectories is chosen by the dynamics (blue solid curves), i.e., realized as a late time solution of Eqs. (1) (see Fig. 4); another trajectory is unstable and not chosen by the dynamics (black dashed curves). The parameters are the same as in Fig. 1.

Figure 4

Front profiles computed both by solving numerically the full system of Eqs. (1) and from the theoretical analysis in the phase space, where the system of Eqs. (3) is solved by using the so-called “shooting” procedure. The numerical solution of Eqs. (1) at late times ($\overline{t}=600$, see Fig. 1) shown by the black solid lines agrees well with the theoretical profiles [solution of Eqs. (3), the red dashed lines] both for the fraction of infected (the upper panel) and the fraction of susceptible (the lower panel). The parameters are the same as in Fig. 1.

Figure 5

Front speed (the upper panel) and $S_{\text{final}}$ (the lower panel) computed theoretically as a function of Ibar. The left curves correspond to ${\overline{r}}_{\text{max}}=2.0$, while the curves on the right correspond to ${\overline{r}}_{\text{max}}=2.8$. Each curve consists of two branches. The blue dashed-dotted lines correspond to the stable branches chosen by the dynamics, while the black dotted lines correspond to the unrealized (unstable) branches (see Fig. 3). There is no front propagation beyond a certain value of Ibar (the threshold $I={\text{Ibar}}_c$ is shown by the red dashed line). The other parameters are the same as in Fig. 1.

Figure 6

Fraction of infected profiles at times $\overline{t}=0$ (red dashed-dotted curve), $\overline{t}=50$, $\overline{t}=100$, $\overline{t}=150$, and $\overline{t}=164$ (black solid curves) for $\text{Ibar}>{\text{Ibar}}_c$ as computed numerically by solving the full system of Eqs. (1). $\text{Ibar}=0.0252$, and the rest of the parameters are the same as in Fig. 1.

Figure 7

Inverse pulse decay time as a function of Ibar for $\text{Ibar}>{\text{Ibar}}_c$ as computed numerically by solving the full system of Eqs. (1) (the blue circles). At the threshold (the red square), the front still propagates, making $t_f$ infinite, so the inverse decay time is zero. The fit shown by the black dotted line corresponds to $1/t_f=0.52{(\text{Ibar}-{\text{Ibar}}_c)}^{1/2}$. The rest of the parameters are the same as in Fig. 1.