Controlling epidemic spread by social distancing: Do it well or not at all

Spatial model

For simplicity, we assume that individuals are located at nodes of a square lattice representing their geographical distribution. We assume no-contact boundary conditions, but we have found that boundary wrapping has little effect on the results. Each node of the lattice holds a single individual, which may be either susceptible, infected, or removed/recovered. On the lattice we define a neighbourhood of radius r centered at a node (i ₀,j ₀), determined by Euclidean distance, $\left\{(i,j)\mathrm{s.t.}\sqrt{{(i-i_0)}^2+{(j-j_0)}^2}\le r\right\}$. For each individual two types of neighbourhoods are considered, describing two network topologies, see Figure 1. The infection neighbourhood is characterised by radius r _i and describes spread of the infection, whereas the awareness neighbourhood defines the area of influence on individual decisions, and is determined by the radius r _a . Both radii can range from 0 (no neighbours) to infinity (whole population), although in practice infinity is represented by a finite value big enough to encompass the whole population. The awareness neighbourhood is uniform for all individuals and fixed for the duration of each epidemic, whereas the infection neighbourhood varies locally depending on the infection pressure (for details see below). The number of individuals in each neighbourhood is proportional to the radius squared and we denote by N _a the number of nodes in each awareness neighbourhood and by N _i the number of nodes in each infection neighbourhood. Initially all individuals are assigned the same baseline infection neighbourhood with radius $r_i^{\left(0\right)}$ leading to the number of individuals $N_i^{\left(0\right)}$ in each infection neighbourhood.

The epidemiological model is based upon an SIR (Susceptible-Infected-Recovered) model [38] (see Figure 2). All individuals are initially susceptible (S) and the epidemic is initiated by the introduction of few infected individuals (I) at random locations in the network. Each susceptible individual is affected by all infected individuals within its infection neighbourhood independently of their distance. Each contact between an infected and a susceptible individual may result in the susceptible becoming infected, with probability p per contact (and per time step). At each time step, each infected individual may become recovered, with probability q. Recovered individuals cannot be reinfected. The epidemic ends when all infected individuals recover. The simulation proceeds through discrete time steps (so that each probability can be interpreted as a hazard) up to a maximum number of steps, chosen to be sufficiently large to allow all epidemics to run to completion. Although our model is generic and so does not depend on a choice of a particular time step, we envisage our model to apply to relatively fast-spreading epidemics. Thus, for simplicity we assume the time step equal to one day with most epidemics finished in less than 120 days.

Social distancing is introduced by allowing susceptible individuals to temporarily reduce the number of contacts they make in response to the presence of nearby infection. Thus, each susceptible individual can detect the current infection pressure amongst its neighbours within the awareness radius, r _a . The local infection pressure, Θ, is the ratio of the number of infected individuals in radius r _a to the total number of neighbours within that radius. For any given susceptible individual, Θmay take values ranging from 0, meaning that there are no infected neighbours in radius r _a , to 1, meaning that all neighbours within radius r _a are infected.

see Figure 3. This formula ensures that if no infection is present within a susceptible’s awareness radius (Θ=0), it will resume full contact by reverting to the initial, maximum contact radius $r_i^{\left(0\right)}$. Increased local infection pressure causes the infection radius and hence the number of contacts to decrease.

Lower values of αrepresent more cautious (more risk-averse) attitudes to risk, and result in a greater reduction of r _i for a given Θ. For completeness, Figure 3 also illustrates ‘risk-neutral’ and ‘risk-seeking’ responses, represented by the upper two lines. The risk-neutral case is one in which susceptibles do not modify their behaviour, and is equivalent to no control being used: here $r_i^{\prime }=r_i^{\left(0\right)}$, regardless of the infection pressure. The risk-seeking case represents a response in which susceptibles seek to increase their contacts as the infection pressure increases, modelled by the formula $N_i=N_i^{\left(0\right)}\left(1+{\Theta }^{\alpha }\right)$. Risk-seeking response was not considered in this study, though we note that such seemingly perverse behaviour was observed [39] in a virtual ‘epidemic’ in an online computer game.

For contact to occur between two individuals, each one must be within the contact neighbourhood of the other. At each time step, any two individuals a and b will make contact, provided that each is within the other’s contact radius, i.e., if d is the Euclidean distance between a and b, then i and j will make contact provided that d < r _i (a) and d < r _i (b). We assume that all individuals are aware of their own status (susceptible, infected, or recovered) and take this into account when deciding how to respond. Susceptible individuals reduce their contact neighbourhood because they know they are at risk of becoming infected, whereas infected and recovered individuals no longer have this risk and so do not reduce their contact neighbourhood.

Economic benefit

The value of R _∞ (no control) is obtained by taking the mean of 20 simulation runs where control is not used. Similarly, R _∞ (control) is the mean of 20 runs with control.

Secondly, the control produces an economic loss by reducing the number of economically beneficial contacts taking place between pairs of individuals. To count the contacts taking place during an epidemic, we sum up the number of contacts that take place at each time step, over a fixed reference period. This reference period is chosen to be equal to 900 time steps which exceeds the duration of the longest epidemic in our sample (most epidemics were significantly shorter than 900 steps). We identify the economic benefits with the infection neighbourhood. This is equivalent to the assumptions that each economically significant contact is associated with an infection risk.

For any given $r_i^{\left(0\right)}$, the value of contacts (no control) is fixed: each individual will make contact with all of its neighbours within radius $r_i^{\left(0\right)}$, so a fixed number of contacts occurs at each time step, and contacts (no control) can be calculated by multiplying this number by the length of the reference period. The value of contacts (control) is obtained by taking the mean of 20 simulation runs with control.

Small-world model

In order to construct the small-world model [36, 37], we start with the local-spread model as described above. A fixed number of local links (representing a proportion of all local links) is selected and those connections are ‘rewired’ to a random location outside the interaction neighbourhood determined by $r_i^{\left(0\right)}$, thus keeping the total number of links (local and long-range) constant. For simplicity we assume that the probability of passing an infection along any of the long-range links is the same as for local links.

As described above, in response to infection located in their awareness neighbourhood susceptible individuals reduce their number of local links. We assume that a similar behaviour governs small-world links. Thus, at each time step the number of active long-range links originating from a susceptible individual is proportional to (1−Θ ^α ), where Θrepresents the infection pressure within the awareness neighbourhood of this individual (as for local links). Infection can only pass along the long-range link if it is active and joins a susceptible and an infected individual. Finally, we assume that each active long-range link contributes to the calculation of the number of contacts in the same way as local links. Non-active long-range links do not contribute to the overall number of contacts.

Simulations

The model was implemented in NetLogo 4.0.4 [40], a simulation tool for agent-based stochastic simulation. Figure 4 shows a snapshot of a typical simulation run in NetLogo. Each run was replicated 20 times, so that in total over 2 million simulation runs were performed. Multiple, concurrent simulation runs were carried out on a grid of about 175 PCs with the help of the Condor distributed computing tool [41]. The statistical package R [42] was used to analyse the resulting data.

Effect of social distancing

Social distancing (characterised here by a relatively risk-averse attitude, α=0.25, see Figure 3) shifts the critical infectiousness p towards higher values but also generally increases the duration of the epidemic, Figure 5. For a given p, social distancing always decreases the final size of the epidemic (Figure 5a), but the effect on the duration depends on the value of p (Figure 5b). The different outcomes are indicated roughly by the four regions marked on the graphs, labelled A, B, C, and D, and with the approximate boundaries between them shown as vertical dotted lines in Figure 5 and in other figures below. For small values of p the disease is non-invasive in both cases, while both the final size and the duration of the epidemic are lower in presence of social distancing than without it (region A). In regions B and C, the epidemic without social distancing is invasive, but the behavioural changes render it non-invasive. However, in region B the duration of the epidemic is shortened by the distancing, whereas in region C it is longer. Finally, in region D the disease is invasive regardless of the social distancing, with slightly lower final size, but significantly longer duration. The approximate boundaries between regions are given by the transition between invasive and non-invasive disease for the case without control (boundary between A and B) and with control (boundary between C and D), respectively. In addition, the boundary between B and C is placed at the value of p where controlled and uncontrolled epidemics last approximately the same time, see Figure 5.

The overall number of contacts taking place during an epidemic is affected by two factors: epidemic size and duration. Firstly, the size of the epidemic determines how many individuals are still susceptible, and therefore exercising the control (infected and recovered individuals do not change their behaviour in our model). A larger epidemic also creates a higher infection pressure, leading to increased reaction. Secondly, the longer the duration of the epidemic the more pronounced the cumulative effect of the reduction of contacts. Both factors combine to reduce the overall number of contacts differentially over p, see Figure 6a. The effect is relatively minor in regions A and B, but becomes very strong in region C, where the long duration of the epidemic combines with relatively large number of infections. Finally, in region D, although the epidemic leads to many cases, it lasts for a relatively short period of time. This results in a smaller drop in contact numbers.

The economic impact of the reduction in the number of contacts can be offset by a reduction in disease cases caused by the social distancing. The two factors are weighed by c, the relative cost of social interactions versus cost of infection. A typical dependence of the net economic benefit on p for medium values of c is shown in Figure 6b; we will explore the dependence on c later in the paper.

Overall, we see that the effect of the control may be neutral, beneficial, or detrimental, with the outcome depending on the infectiousness of the disease, p. The effect is different in different regions A-D. In region A the control makes little difference as the disease is not invasive even in the absence of social distancing. R _∞ is small and the epidemic is of short duration. Using the control causes a very slight reduction in the number of contacts, and a correspondingly very small reduction in the net benefit.

As p increases past the uncontrolled epidemic threshold value, indicated by the boundary between regions A and B in the graphs, the social distancing leads to a positive net benefit. The control reduces R _∞ almost to zero, and greatly shortens the duration of the epidemic. Although the number of contacts is also somewhat reduced, this is more than compensated for by the large reduction in R _∞ , so that the net benefit is strongly positive.

As p increases further, region C, the control becomes ineffective, R _∞ rises towards the levels seen without control, and the duration of the epidemic increases beyond no-control levels. The picture is of a control that is too weak to suppress the epidemic and is merely slowing down the speed of its spread without reducing its final impact. Figure 6a shows a further downside: by prolonging the epidemic, the control prolongs the period during which social distancing is practised, thus greatly reducing the number of contacts. This in turn results in an overall negative benefit. Eventually, as p enters region D, we reach the worst case scenario for the control, where using the control gives a much worse result than doing nothing. As p increases further beyond this point, we return to a situation where, as in region A, use of the control makes little difference to the severity of the epidemic or its duration. However, unlike region A, in this case the epidemic is invasive despite the control. The number of contacts and the overall benefit both improve slightly from the worst case scenario, though remaining low.

An interesting observation from these graphs is the close juxtaposition of the best case scenario with the worst case, as shown by the steep transition from positive to negative benefit taking place within region C. This implies that if social distancing is to be used to control disease, it is very important to get the parameters right. If the infectiousness of the disease is underestimated even slightly, the actual outcome of using the control can be substantially worse than anticipated. We shall see later on, when we consider different attitudes to risk and different values of the awareness radius, that the importance of getting the parameters exactly right is a recurring theme in this work.

Effect of varying c

The parameter c represents the relative weight of a single contact between a pair of individuals, compared to a single case of infection. Changing c causes a vertical shift in the benefit, that is most significant in regions C and D. However it does not change the overall shape of the graph, Figure 7. If c=0 then there is no cost attached to reducing contacts, so using the control always results in a non-negative benefit (top line). As c is increased, reducing contacts has a greater impact, causing a strong downward shift in regions C and D of the graph where large numbers of contacts are lost. For the remaining graphs, c=0.05 is chosen because it best illustrates the generic behaviour of the net economic benefit.

Importance of risk attitude

The parameter αrepresents the attitude to risk that determines how social distancing is applied. A very low value of α means highly risk-averse behaviour, causing a strong control response in which susceptible individuals effectively close down all social contacts if there is any hint of disease within their awareness radius, Figure 3. Higher values of α represent more relaxed (less risk-averse) attitudes and will cause a weaker response, particularly at low infection pressure, Θ. When all individuals are highly risk averse (α= 0.05), social distancing leads to a positive benefit for even the most highly infectious diseases, see Figure 8. The only exception is region A where disease does not spread even in absence of social distancing, but changes in behaviour around the initial foci lead to slightly negative benefit. However, if the risk attitude is made more relaxed (α = 0.25 and α=0.55 in Figure 8), the control becomes less effective for more infectious diseases, resulting in a worse outcome than doing nothing.

Another way of thinking about these results is in terms of the four regions, A, B, C, and D, identified earlier. With a highly risk-averse value of α, regions C and D vanish, and region B (corresponding to the diseases for which the distancing is beneficial) extends from the invasion threshold value of p all the way to p=1. Relaxing (increasing) α causes region B to narrow and regions C and D to emerge. As αis increased further, the boundary between regions B and C shifts to the left, meaning that the control is beneficial for a narrower range of diseases.

In Figure 9, we take a different view of this data, choosing three different diseases (represented by different values of p) and analysing how the benefit changes as α is varied. For a non-invasive disease (p=0.01), using the control has little effect and varying α makes little difference, although a high value of αrepresents the optimal choice corresponding to a maximum value of the benefit. For a moderately infectious disease (represented by p=0.25), there is a positive benefit for lower (more risk averse) values of α, up until a sharp threshold value, after which the outcome is negative. As α is increased further, the outcome improves, though remaining negative. These graphs imply that it is very important to exercise sufficient caution in using social distancing. If the control is just slightly too relaxed and falls on the wrong side of the threshold, the result will be worse than doing nothing. For a more infectious disease (p=0.51), control is only beneficial if αis extremely risk-averse.

Small-world model

For the case described above when only short-range links are present in the network, the disease spreads locally and can therefore be controlled efficiently by individuals reducing their social activities. Addition of randomness increases the propensity of the disease agent to spread by allowing it to create new foci in regions where there is a high density of susceptible individuals. As a result, the switch from the invasive to the non-invasive disease occurs for smaller values of p[13, 36]. The long-range links can also make social distancing less effective, as individuals are responding to infection within their local neighbourhood and the disease can pass along the long-range links from outside this neighbourhood. However, even when 20% (3,000) of links are long-range, this does not render the control completely ineffective, Figure 10 (left column), although it requires a more cautious attitude (smaller α) for it to work. For networks dominated by long-range spread (as exemplified here by a small-world network with 30% long-range links and by a completely random network), the social distancing breaks down and the optimal solution is to avoid any reaction to the disease, Figure 10 (right column).

The process by which inclusion of long-range links breaks down the usefulness of the social distancing is, however, not trivial. Two contrasting cases need to be distinguished, corresponding to a very relaxed risk attitude (α∼1) and to a very strict response (α∼0), see Figure 10. Firstly, if individuals do not respond to the local infection load (large α), the epidemics are usually very quick (rightmost parts of Figure 10a,b) and infect most of the individuals in the population (Figure 10c,d). However, the number of contacts lost to social distancing is also small, resulting in the net benefit close to 0 (rightmost parts of Figure 10e,f).

As the individuals become more cautious in their response to the infection load, the duration of the epidemics goes up (Figure 10a) and the number of cases goes down (Figure 10c). In the second case, where the values of α are small, the behaviour depends crucially on the number of long-range links. For small number of such contacts, the epidemics are short (leftmost parts of Figure 10a) and stop after infecting a small number of individuals (leftmost parts of Figure 10c). As the number of long-range links increases to 30%, the duration of the epidemic increases (Figure 10b) and the number of cases goes up (Figure 10d) but remains short of infecting the whole population. In this case, epidemics progress slowly, infecting only few individuals at a time, but resulting in a massive reduction in the number of contacts over the whole period of their duration. Further addition of non-local links results in a faster epidemic affecting a large number of individuals (Figure 10b,d). For the particular choice of c in Figure 10 this results in a very similar value of the net benefit for 30% and 100% non-local links, although the balance will change when other values of c are selected (cf. Figure 7).

In this model we assume that the control is applied to both local and long-range links in the same way. If long-range links are excluded from the control, it becomes even less effective, but the results are qualitatively similar to those described above.

Importance of matching spatial scales: regular networks

The model introduces two spatial scales, the infection (and contact) neighbourhood and the awareness neighbourhood. The infection neighbourhood is associated with disease transmission and with economic benefits accruing from social contacts, whereas the role of the awareness neighbourhood is to provide an estimate of risks associated with infection. So far in the case of regular networks we have only considered cases in which the radius of the awareness neighbourhood r _a is equal to that of the maximum contact radius $r_i^{\left(0\right)}$. This corresponds to a situation when individuals base their decisions on the same social neighbourhood as the infection risk comes from. However, two other cases need to be considered. The individuals might be in contact potentially leading to infection with individuals whose status might be unknown to them. In our approach we simulate this situation by considering an awareness neighbourhood which is smaller than the infection neighbourhood. This results in underestimation of risks and leads to sub-optimal control of epidemic spread, as illustrated by $r_a=1<2=r_i^{\left(0\right)}$ in Figure 11a. The benefit is in this case a monotonically increasing function of α, leading to a maximum corresponding to a relaxed attitude (large α).

This behaviour is very different to the cases when $r_a\ge r_i^{\left(0\right)}$, i.e. when the individual bases the decisions on a sample equal or larger than the infection neighbourhood. The benefit is a non-monotonic function of α with a global positive maximum at low values of α, a rapid drop to highly negative values for intermediate αand improving for large α, Figure 11. However, extension of the awareness neighbourhood beyond $r_i^{\left(0\right)}$ reduces the range of αfor which the benefit is positive and significantly lowers the minimum value of the benefit (Figure 11). In particular, if the awareness neighbourhood includes the whole population, r _a =∞, the benefit is negative for all values of αconsidered here. Two mechanisms may contribute to produce this lack of efficiency. Firstly, it might be that individuals reduce contacts unnecessarily in response to awareness of disease cases that are too far away to pose a threat. Alternatively, it might be that individuals fail to protect themselves because the over-large awareness radius causes them to underestimate the level of infection amongst their local contacts. Examination of the simulation data suggests that a combination of these two mechanisms is responsible for this result, because making r _a larger than $r_i^{\left(0\right)}$ has the dual effect of causing an increase in R(∞) and a reduction in the number of contacts. Thus, the exact match of the awareness and infection neighbourhood appears to lead to an optimal case with the widest range of αfor which benefit is positive and the shallowest drop in the benefit for intermediate values of α.

Importance of matching spatial scales: small-world networks

Long-range links provide an alternative way in which disease can enter the local neighbourhood of the susceptible individuals. The question then arises whether global awareness of infection cases is able to restore the efficiency of social distancing when the network is dominated by random links? Figure 12 which shows the results for the random network suggests that this is not the case; the results for other numbers of non-local links are very similar. The loss of contacts resulting from over-reacting to infection far dominates the net benefit. However, for random networks with a very cautious response (very small α) global awareness is more efficient than local awareness, see Figure 12.