Robustness of behaviorally induced oscillations in epidemic models under a low rate of imported cases

This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a nonoscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters.

Figure 1

Hopf-bifurcation curves in the $({\beta }_a,{\alpha }_i)$ parameter space for $\varepsilon =0$ (dashed line), $\varepsilon ={10}^{-5}$ (dotted line) and $\varepsilon ={10}^{-4}$ (solid line). Parameters: $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1$.

Figure 2

Stability modulus of the Jacobian matrix at the endemic equilibrium of Eq. (2) as a function of ${\alpha }_i$. Parameters: $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_a=0.1, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon ={10}^{-4}$.

Figure 3

Evolution of $i\left(t\right)$ for $t\in [1500,3000]$ predicted by Eq. (2) and outputs from Gillespie and FGA (adaptive mean over 50 experiments). Parameters of the simulations: $N=1000$, degree $k=50$ for the Gillespie algorithm, $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon ={10}^{-4}, {\beta }_a=0.1, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$. (a) ${\alpha }_i=0.04$, (b) ${\alpha }_i=0.14$.

Figure 4

Hopf diagram of SAUIS-$\varepsilon$ obtained according to the description in Sec. 4d, using (a) Gillespie algorithm and (b) FGA with $N=1000$ nodes and time $T=3000$. Parameters: degree $k=50$ (for the Gillespie algorithm), $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon ={10}^{-4}, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$, 50 experiments for each pair $({\beta }_a,{\alpha }_i)$. The gradient of colors evidences the amplitude of the signal corresponding to the fraction of infected nodes. The black line is the theoretical Hopf-bifurcation curve.

Figure 5

Fraction $p$ of stochastic simulations with positive prevalence up to time $T=3000$ for $\varepsilon =0$ and $N=10000$ using FGA algorithm. The Hop-bifurcation curve (white curve within the extinction region) is included for a better visualisation of the extinction range. Parameters: $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon =0, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$, 100 experiments for each pair $({\beta }_a,{\alpha }_i)$.

Figure 6

(a) Time evolution of a stochastic simulation of an epidemic on a random regular network of size $N=1000$ and degree $k=100$ showing the input of imported cases (infections from abroad) for $\varepsilon ={10}^{-4}$ (black dots on the time axis). (b) Fraction of infected nodes in the same simulation until the beginning of the fourth flare-up where $\varepsilon =0$ (lockdown). Dashed line: Fraction of infected nodes without lockdown. Parameters: $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_a=0.05, {\beta }_u=0.5, {\alpha }_i=0.15, {\alpha }_a=0.01, {\nu }_a=1, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$.

Figure 7

Periodograms of SAUIS-$\varepsilon$ obtained using FGA with $N=10000$ nodes and time $T=3000$ (adaptive mean over 50 experiments). The number of points used is $M=2^{11}$ for panels (a) and (c), and $M=2^7$ for panels (b) and (d), according to the procedure in Sec. 4d. Parameters: $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_a=0.1, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon ={10}^{-4}, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$. (a, b) ${\alpha }_i=0.04$. (c, d) ${\alpha }_i=0.14$.

Figure 8

Comparison of the algebraic Hopf-bifurcation curve for regular random networks with two approximations obtained integrating Eq. (1) up to two different times and using for $CV=0.1\%$ to discriminate between periodic and nonperiodic solutions.

Figure 9

Hopf diagram obtained according to the description in Sec. 4d of the FGA with $N=10000$ nodes and time $T=3000$. The gradient of colors evidences the amplitude of the averaged signal of the fraction of infected nodes. The black line is the algebraic Hopf-bifurcation curve for $\varepsilon ={10}^{-4}$.

Figure 10

Hopf diagram of SAUIS-$\varepsilon$ obtained according to the description in Sec. 4d with $N=1000$ nodes and time $T=3000$, in two different network architectures of mean degree 50: (a) Poisson and (b) Exponential. Parameters: $\delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon ={10}^{-4}, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$, 50 experiments for each pair $({\beta }_a,{\alpha }_i)$. The gradient of colors evidences the amplitude of the signal corresponding to the fraction of infected nodes. The black line is the theoretical Hopf-bifurcation curve.

Figure 11

Evolution of $i\left(t\right)$ for $t\in [1500,3000]$ predicted by Eq. (2) (solid line), adaptive mean of outputs from FGA (over 50 experiments, dashed line) and a single experiment from FGA (dotted line). Parameters of the simulations: $N=10000, \delta =1, {\delta }_a=0.01, {\delta }_u=0.05, \beta =3, {\beta }_a=0.1, {\beta }_u=0.5, {\alpha }_a=0.01, {\nu }_a=1, \varepsilon ={10}^{-4}, i\left(0\right)=0.1, a\left(0\right)=u\left(0\right)=0.2$. (a) ${\alpha }_i=0.04$, (b) ${\alpha }_i=0.14$, (c) ${\alpha }_i=0.16$, (d) ${\alpha }_i=0.26$.