The results from this agent model highlight the importance of flight as a topic for research—even a small amount of flight can have a dramatic impact on epidemic dynamics.
First, to establish a baseline, we consider a simple form of the model in which fear propagates, but no one adapts their behavior in response. All agents are “ignorers”. Parameterized in this way, the model produces quintessential SIR curves, such as those shown in .
We consider a simple form of the model in which fear propagates, but no one adapts their behavior in response.
In a 30-run analysis, the average epidemic duration with all agents set to “ignorers” is 742.1 periods (SE 9.5), and average total incidence is 99.9%. (See the leftmost bars of , respectively)
(A&B): Epidemic duration and total incidence under three different parameter settings.
In the next version of the model we replace the population of “ignorers” with a population of “hiders.” Replacing the population with “hiders” drastically reduces incidence to an average of 27.8% and stops the epidemic earlier (in an average of 647 rounds). A population who systematically hides from an incoming epidemic will suffer many fewer cases of disease. See the middle bars of below.
Next, we introduce flight, but only a small amount—90% of agents still respond to fear by hiding (removing themselves from circulation); the remaining 10% flee. How does this small proportion of flight affect incidence and duration of the epidemic?
As shows, even this small amount of flight dramatically increases the size and speed of the epidemic (comparing the rightmost bars of 5a and b to the middle ones). Average incidence in the population is 64% and the average epidemic duration is 595 time periods. This small proportion of fleeing agents causes the population to suffer more than 2.3 times as many disease cases as in the all-hiders configuration.
Of course, the 10% of agents who are fleeing are also not hiding. By remaining in circulation, even ignoring (neither fleeing nor hiding) agents should have an inflammatory effect on the epidemic. So, is it the flight or simply the increased circulation from non-hiding which is driving the previous result? To answer this question, we ran the simulation with 90% “hiders” and 10% “ignorers.” As illustrates, the results from these runs differ noticeably from the runs with actual flight: we observed an average incidence of 32% and an average epidemic duration of 640 time periods. Flight has a substantial impact above and beyond increasing the number of “non-hiders”.
(A&B): A comparison of epidemic duration and total incidence with 10% “fleers” versus 10% “ignorers.”
Not only does flight increase incidence dramatically, but it also increases the rapidity and geographic scope of the epidemic. One way to measure the geographic spread of the bug is to begin the epidemic with an index case in one corner of the 2D lattice, and observe if and when the bug reaches the far diagonal corner. shows that epidemics rarely spread fully across the lattice with no flight—but almost always spread fully across the lattice with even a small amount of flight. Specifically, when merely 10% flee, the epidemic reaches the far corner an average of 92% of the time. However, when no agents flee, the epidemic only reaches the far corner 17% (all hiders) or 27% (90% hide / 10% ignore) of the time.
The percentage of runs (out of 30) for each parameter setting in which the epidemic spreads fully across the landscape, from an index case in one corner of the lattice all the way to the opposite corner.
Furthermore, in rare cases where the epidemic spreads fully across the lattice without flight, it takes much longer to do so than in cases with flight, as shown in . Without flight the epidemic takes roughly 600 time periods to cross the lattice. When fight is allowed, the average run takes 340 time periods to traverse the lattice, almost twice as fast.
In rare cases where the epidemic spreads fully across the lattice without flight, it takes much longer to do so than in cases with flight.
For an illustration of how flight spreads the epidemic quickly across the lattice, increasing both incidence and speed, see . Blue dots represent susceptibles (infected with neither fear nor pathogen); yellow dots, infected with fear alone; orange dots, acting on fear; red dots, infected with pathogen; white dots, recovered.
(A&B): Screenshots from the agent-based simulation model without and with flight.
In the first screen shot (9a), with no flight, yellow agents (infected with fear) form a moving buffer zone between the epidemic of pathogen and the susceptible agents. The latter thus have an opportunity to remove themselves from circulation because they are likely to contract fear before they are exposed to the pathogen (as per our earlier discussion of the fear R0), The second screenshot shows how a small amount of flight enables a few infected fleeing agents to pierce this buffer zone, introducing the pathogen quickly into the susceptible pool.
These specific quantitative results are summarized in . They are, of course, dependent on the specific parameters used above. But the larger qualitative point is robust. Behavioral adaptation need not damp the force of an epidemic. If flight is admitted, this form of “social distancing” can increase both the speed and size of an epidemic.
This exposition invites a great deal of further work, including development of the multi-patch (meta-population) ODEs with flight, full sensitivity analysis of the agent-based model, further “dialogue” between the two approaches, and calibration to historical cases.
However, the present effort clearly enforces the overarching point that infectious disease models must incorporate behavioral adaptation. In the development above, the adaptive repertoire is quite narrow, including only self-isolation and flight. But it can obviously be broadened substantially. Moreover, the model—while explored for contagious disease here—can be applied to a wide range of cases where momentous contagions of fear eventuate from events that are not themselves contagious, such as toxic chemical plumes or floods, fires, and earthquakes.