Effect of Human Behavior on the Evolution of Viral Strains During an Epidemic

Abstract

It is well known in the literature that human behavior can change as a reaction to disease observed in others, and that such behavioral changes can be an important factor in the spread of an epidemic. It has been noted that human behavioral traits in disease avoidance are under selection in the presence of infectious diseases. Here, we explore a complementary trend: the pathogen itself might experience a force of selection to become less "visible," or less "symptomatic," in the presence of such human behavioral trends. Using a stochastic SIR agent-based model, we investigated the co-evolution of two viral strains with cross-immunity, where the resident strain is symptomatic while the mutant strain is asymptomatic. We assumed that individuals exercised self-regulated social distancing (SD) behavior if one of their neighbors was infected with a symptomatic strain. We observed that the proportion of asymptomatic carriers increased over time with a stronger effect corresponding to higher levels of self-regulated SD. Adding mandated SD made the effect more significant, while the existence of a time-delay between the onset of infection and the change of behavior reduced the advantage of the asymptomatic strain. These results were consistent under random geometric networks, scale-free networks, and a synthetic network that represented the social behavior of the residents of New Orleans.

Keywords: Asymptomatic variant; Mandated social distancing; Network; Self-regulated social distancing; Symptomatic variant; Viral evolution.

Fig. 1

(Color figure online) The role of self-regulated SD in the spread of viruses. Time series are shown for four scenarios: No (${\sigma }_S=0$, black), low (${\sigma }_S=0.2$, blue), moderate (${\sigma }_S=0.4$, red), and high (${\sigma }_S=0.7$, green) self-regulated SD, and in the absence of mandated SD. Scale-free (left) and spatial (right) networks of 10, 000 individuals with average degree 10 are used. a, b The prevalence of $V_1$ (solid) and $V_2$ (dashed); c, d the proportion of $V_2$ ($V_2/(V_1+V_2)$). The remaining parameters are $\gamma +\delta =0.1$ per day, $\psi =0.0012$, ${\beta }_1={\beta }_2=0.028$ per day per contact for the scale-free and ${\beta }_1={\beta }_2=0.037$ per day per contact for the spatial network (corresponding to ${\mathcal{R}}_0=2.5$). Means and standard errors are shown for 5000 stochastic realizations

Fig. 2

(Color figure online) Selection for $V_2$ in the presence of a fitness cost. Time series of proportion of $V_2$ under moderate self-regulated SD, ${\sigma }_S=0.4$ (and with ${\sigma }_M=0$), are shown for $0\%$ fitness cost (${\beta }_2={\beta }_1$, black), $5\%$ fitness cost (${\beta }_2=0.95{\beta }_1$, blue), $10\%$ fitness cost (${\beta }_2=0.9{\beta }_1$, red), and $15\%$ fitness cost (${\beta }_2=0.85{\beta }_1$, green), for a scale-free and b spatial networks. All the other parameters are as in Fig. 1

Fig. 3

(Color figure online) The effect of mandated SD on the proportion of $V_2$. The proportion of the asymptomatic strain, $V_2$, is shown as a function time, for three different levels of mandated SD: a Scale-free network, ${\sigma }_M=0$ (black), ${\sigma }_M=0.2$ (blue), and ${\sigma }_M=0.4$, with ${\sigma }_S=0.4$; b spatial network, ${\sigma }_M=0$ (black), ${\sigma }_M=0.2$ (blue), and ${\sigma }_M=0.3$ (red), with ${\sigma }_S=0.2$. All the other parameters are as in Fig. 1. The levels for mandated and self-regulated SD are selected in such a way that ${\mathcal{R}}_0$ remains above one so an outbreak for $V_1$ is observed

Fig. 4

(Color figure online) The effect of delay of self-regulated SD on selection of $V_2$. The proportion of $V_2$ is shown as time series for the a scale-free and b spatial network, in the presence of time delay in the appearance of symptoms when infected by $V_1$. The different colors correspond to time delays of 0, 1, $\dots$, 5 days. Here, ${\sigma }_S=0.4$, ${\sigma }_M=0.0$, and the rest of the parameters are as in Fig.  1

Fig. 5

(Color figure online) Degree distribution of the New Orleans synthetic network. Red: the basic network; black: the network under school closure (see “Appendix 3”). The network includes 150, 000 nodes and has average degree 15.82 (with average degree 12.67 under school closure)

Fig. 6

(Color figure online) New Orleans Network of size 150, 000 individuals: the role of self-regulated SD in the spread of viruses. Time series are shown for four scenarios of no (${\sigma }_S=0$, black), low (${\sigma }_S=0.2$, blue), moderate (${\sigma }_S=0.4$, red), and high (${\sigma }_S=0.7$, green) self-regulated SD, in the absence of mandated SD. a plot is the prevalence of $V_1$ (solid) and $V_2$ (dashed); b shows the proportion of $V_2$ ($V_2/(V_1+V_2)$). ${\beta }_1={\beta }_2=0.2$ and all the other parameters are as in Fig. 1 (corresponding to ${\mathcal{R}}_0=2.5$). Means and standard errors are shown for 1000 stochastic realizations

Fig. 7

(Color figure online) New Orleans Network of size 150, 000 individuals: the role of mandated SD in the spread of viruses. Time series are shown for four scenarios of no (${\sigma }_M=0$, black), low (${\sigma }_M=0.2$, blue), moderate (${\sigma }_M=0.4$, red), and high (${\sigma }_M=0.6$, green) mandated SD, in the presence of moderate self-regulated SD (${\sigma }_S=0.4$). a plot is the prevalence of $V_1$ (solid) and $V_2$ (dashed); b shows the proportion of $V_2$ ($V_2/(V_1+V_2)$). ${\beta }_1={\beta }_2=0.2$ and all the other parameters are as in Fig. 1 (corresponding to ${\mathcal{R}}_0=2.5$). Means and standard errors are shown for 1000 stochastic realizations

Fig. 8

Fraction of an advantageous virus, $V_2$. a The quantity z(t) obtained by solving Eqs. (1–1c) is plotted as a function of time for several values of $R_0$, obtained by changing the death rate, a. b The corresponding susceptible populations as functions of time. The rest of the parameters are ${\beta }_1=0.1,{\beta }_2=0.11,y_1(0)=0.001,y_2(0)=0.1y_1(0)$

Fig. 9

(Color figure online) Fraction of $V_2$ at the end of the epidemic. a Calculation of $t_{\mathit{end}}$, which represents the end of the epidemic, is illustrated. The blue line is the fraction of susceptible individuals, x(t), obtained as a solution of Eqs. (1–1c); $t_{\mathit{end}}=2t_1$, where $t_1$ corresponds to $x(t_1)=\frac{1}{2}(x(0)+x_{\infty })$. In other words, at time $t_1$ the population of susceptible individuals reaches halfway to its final value, $x_{\infty }$. b Quantity $y_2/(y_1+y_2)$ obtained by solving Eqs. (1–1c), is plotted at time $t_{\mathit{end}}$, as a function of the initial proportion of individuals infected with $V_1$, and ${\mathcal{R}}_0$. The rest of the parameters are ${\beta }_1=0.1,{\beta }_2=0.11,y_2(0)=0.1y_1(0)$

Fig. 10

Illustration of time-separated epidemics. a Numerical solution of system (1–1c) with initial conditions $y_{10}=0.01,y_{20}={10}^{-8},x_0=1-y_{10}-y_{20}$, where the mutant infection is characterized by an initial condition much lower than that of the wild type. The three populations are plotted as a function of time. The dashed lines show the system behavior in the absence of the mutant, and in particular, the quantity ${\overline{x}}_1$ is shown. b Numerical solutions of system (–) with $y_1(0)=0.01$ for $t\in [0,T]$ and system (8–10) with $y_2(T)=0.01$ for $t>T$. The other parameters are ${\beta }_1=0.13,{\beta }_2=2{\beta }_1,\gamma =0.1,T=200$

Fig. 11

(Color figure online) Time to reach infection peak for viruses $V_1$ (blue) and $V_2$ (red), as a function of ${\sigma }_S$ (the measure of $V_2$ advantage)

Fig. 12

(Color figure online) New Orleans Network under school closure: the role of self-regulated SD in the spread of viruses. Time series are shown for four scenarios of no (${\sigma }_S=0$, black), low (${\sigma }_S=0.2$, blue), moderate (${\sigma }_S=0.4$, red), and high (${\sigma }_S=0.7$, green) self-regulated SD, in the absence of mandated SD. a Plot is the prevalence of $V_1$ (solid) and $V_2$ (dashed); b shows the proportion of $V_2$ ($V_2/(V_1+V_2)$). ${\beta }_1={\beta }_2=0.29$ and all the other parameters are as in Fig.  1 in the main text (corresponding to ${\mathcal{R}}_0=2.5$). Means and standard errors are shown for 1000 stochastic realizations

Fig. 13

(Color figure online) New Orleans Network under school closure: the role of mandated SD in the spread of viruses. Time series are shown for four scenarios of no (${\sigma }_M=0$, black), low (${\sigma }_M=0.2$, blue), moderate (${\sigma }_M=0.4$, red), and high (${\sigma }_M=0.6$, green) mandated SD, in the presence of moderate self-regulated SD (${\sigma }_S=0.4$). a plot is the prevalence of $V_1$ (solid) and $V_2$ (dashed); b shows the proportion of $V_2$ ($V_2/(V_1+V_2)$). ${\beta }_1={\beta }_2=0.29$ and all the other parameters are as in Fig.  1 in the main text (corresponding to ${\mathcal{R}}_0=2.5$). Means and standard errors are shown for 1000 stochastic realizations

Fig. 14

The effect of mandated SD on the proportion of $V_2$ for scenarios of low/high transmission rate and different initial conditions. The proportion of the asymptomatic strain, $V_2$, is shown as a function time, for 3 different levels of mandated SD (${\sigma }_M=0,0.2,0.4$) and fixed positive level of self-regulated SD (${\sigma }_S=0.4$) on scale-free network: a $\beta =0.0028$ per day and $\frac{V_2(0)}{V_1(0)}=0.1$; b $\beta =0.0028$ per day and $\frac{V_2(0)}{V_1(0)}=0.01$; c $\beta =0.0028$ per day and $\frac{V_2(0)}{V_1(0)}=0.002$; d $\beta =0.1$ per day and $\frac{V_2(0)}{V_1(0)}=0.1$; e $\beta =0.1$ per day and $\frac{V_2(0)}{V_1(0)}=0.01$; and f $\beta =0.1$ per day and $\frac{V_2(0)}{V_1(0)}=0.002$. All the other parameters are as in Fig. 1. The levels for mandated and self-regulated SD are selected in such a way that ${\mathcal{R}}_0$ remains above one so an outbreak for $V_1$ is observed

Fig. 15

The effect of mandated SD on cumulative infection of symptomatic strain for scenarios of low/high transmission rate and different initial conditions. The cumulative infection for symptomatic strain, $V_1+R_1$, is shown as a function time, for six different levels of mandated SD (${\sigma }_M=0,0.2,0.4$) and fixed positive level of self-regulated SD (${\sigma }_S=0.4$) on scale-free network. The parameters for a–f are as in Fig.  14

Fig. 16

The effect of mandated SD on cumulative infection of asymptomatic strain for scenarios of low/high transmission rate and different initial conditions. The cumulative infection for asymptomatic strain, $V_2+R_2$, is shown as a function time, for six different levels of mandated SD (${\sigma }_M=0,0.2,0.4$) and fixed positive level of self-regulated SD (${\sigma }_S=0.4$) on scale-free network. The parameters for a–f are as in Fig.  14

Fig. 17

The effect of self-regulated SD on cumulative infection of symptomatic/asymptomatic strains for various network structures. The cumulative infections for are shown as a function time, for four different levels of self-regulated SD (${\sigma }_S=0,0.2,0.4,0.7$) and no mandated SD (${\sigma }_M=0.0$), simulated on a, d scale-free network; b, e spatial network; c, f New Orleans network. All the other parameters are as in Fig.  1 for a–e and Fig. 6 for c, f

Fig. 18

Final sizes versus SD levels: the model in (1–1c) is used to quantify the amount of final sizes for $V_1$ (a), $V_2$ (b), and both (c), for different levels of SD strategies (${\sigma }_M$ and ${\sigma }_S$). The parameter values are $\beta =0.5$, $\gamma =0.1$, and for initial condition we have $y_{10}=0.05$ and $y_{20}=0.0001$, thus $\frac{y_{20}}{y_{10}}=0.002$

Fig. 19

Change in final sizes with respect to SD levels: The same model and parameter values as the one in Fig.  (18) are used to quantify the change of final sizes with respect change of SD strategies

Fig. 20

The region for which final epidemic sizes increase as a result of social distancing, for various transmission rates: The same model and recovery rate as those in Fig.  (18), initial condition $y_{10}=0.001$, $y_{20}=0.0001$ ($\frac{y_{20}}{y_{10}}=0.1$), and various transmission rates are used. The left column shows the region of SD parameter values, for which final epidemic size of $V_2$ increases with increasing mandated SD. The right column shows the region of SD parameter values, for which total final epidemic size increases with increasing self-regulated SD

Fig. 21

The region for which final epidemic sizes increase as a result of social distancing for various initial conditions: The same model and infection parameters as those in Fig.  (18), but various initial conditions are used. The left column shows the region of SD parameter values, for which final epidemic size of $V_2$ increases with increase in mandated SD. The right column shows the region of SD parameter values, for which total final epidemic size increases with increase in self-regulated SD