Adaptive survival movement strategy to local epidemic outbreaks in cyclic models

We investigate a cyclic game system composed of five species whose selection interactions follow the generalised rock-paper-scissors model, illustrated in Fig. 1. Accordingly, individuals of species $i$ dominate over individuals of species $i+1$, with $i=1,2,3,4,5$; we assume the identification $i=i+5\,\alpha$, where $\alpha$ is an integer. We consider that an epidemic spreads through the system, with organisms of all species equally susceptible to infection and reinfection by a disease transmitted person to person. All infected organisms become viral vectors to contaminate their immediate neighbours; some sick individuals die while others are cured.

In our model, organisms of one out of the species perform behavioural movement strategies to protect themselves against nearby enemies and infection by approaching viral vectors. Individuals usually perform the Safeguard strategy, moving towards the direction with the highest density of enemies of their enemies [22]. Therefore, they profit from being close to guards, decreasing the risk of being killed in the cyclic game. However, if the density of sick individuals in the vicinity represents a danger of imminent contamination, the individual opts for the Social Distancing tactic [27]. The decision to move to the areas with lower population density is individual, being triggered every time a surge affects the organism’s neighbourhood making the local density of sick individuals higher than a given threshold.

We use square lattices with periodic boundary conditions to implement our stochastic simulations. As the spatial interactions may lead to organisms’ births and deaths, we use the May-Leonard implementation: the total number of individuals is not conserved [41]. We assume that each grid point contains at most one individual, which limits the maximum number of individuals to equal the total number of grid points, $\mathcal{N}$. The densities of individuals of species $i$ at time $t$ is defined as $\rho_{i}=I_{i}/\mathcal{N}$, where $I_{i}$ is number of individuals of species $i$, with $i=1,2,3,4,5$.

We identify the organisms using the notation $h_{i}$ and $s_{i}$ for healthy and sick individuals; the use of the identification $i$ stands for individuals irrespective of illness or health, with $i=1,2,3,4,5$. This notation allows the description of the spatial interactions as [27]:

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  Selection: $i\ j\to i\ \otimes\,$, with $j=i+1$, where $\otimes$ means an empty space; selection interactions happens between individuals, irrespective of the healthy conditions;

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  Reproduction: $i\ \otimes\to i\ h_{i}\,$; a reproduction interaction generates a new healthy individual.

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  Mobility: $i\ \odot\to\odot\ i\,$, where $\odot$ means either an individual of any species or an empty space; when moving, a healthy or a sick individual switches position with another organism of any species or with an empty site;

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  Infection: $s_{i}\ h_{j}\to s_{i}\ s_{j}\,$, with $i,j=1,2,3,4,5$: a sick individual passes the disease to another organism of any species;

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  Cure: $s_{i}\to h_{i}\,$: an ill individual is naturally cured of the disease without gaining immunity, thus being subject to reinfection;

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  Death: $s_{i}\to\otimes\,$: when a sick organism dies because of the disease, its position is left empty.

Empty spaces are created when individuals are selected, or sick organisms die because of the disease severity.

The spatial interactions are executed by assuming the von Neumann neighbourhood: individuals may interact with one of their four immediate neighbours. At each time step, one of the possible implementations is chosen aleatory, according to the respective interaction probabilities: i) healthy individuals: $s_{h}$ (selection), $r_{h}$ (reproduction), $m_{h}$ (mobility); ii) sick organisms: $s_{s}$ (selection), $r_{s}$ (reproduction), $m_{s}$ (mobility), $\omega$ (infection), $c$ (cure), and $d$ (death by disease complications). The parameters are the same for all organisms of every species. Our algorithm follows three steps: i) randomly selecting an active individual; ii) using the model probabilities to choose one interaction to be executed; iii) drawing one of the four nearest neighbours to suffer the action, except for the directional movement strategy, where the organism chooses the direction to move according to the specific survival strategy. If the interaction is implemented, one timestep is counted. Otherwise, the three steps are redone. We assume the time unit called generation, defined as the necessary time to $\mathcal{N}$ timesteps to occur.

We investigate a locally adaptive self-preservation strategy of individuals that use environmental cues to move cleverly according to the current survival conditions. This means that each organism interprets the signals received from the neighbourhood to identify the existence of local disease surges, thus maximising the protection from disease infection and enemies’ attacks.

Each organism usually moves to minimise the chances of being caught and eliminated in the spatial competition game, performing the Safeguard strategy - this is independent of the organism’s geographic position [22, 27]. When performing the Safeguard strategy, an individual of species $i$ moves in the direction with the highest concentration of individuals of species $i-2$. This provides protection against individuals of species $i-1$, according to Fig. 1. However, the Safeguard tactic increases the chances of contamination during a local epidemic since the virus is transmitted from individual to individual. Because of this, organisms adapt to respond to the arrival of a local disease surge, temporarily switching from the Safeguard to the Social Distancing strategy, moving towards the direction with more empty spaces.

Every time an organism decides to perform the Social Distancing instead of Safeguard strategy, the vulnerability to being killed in the spatial game increases. Therefore, each organism scans the vicinity to conclude how to move. This means that the choice of the survival strategy is local, based on each organism’s reality. We define a real parameter $\beta$, with $0\,\leq\,\beta\,\leq\,1$, to define the social distancing trigger parameter: a threshold assumed by the organisms as the minimum local density of sick individuals to perform the Social Distancing strategy. Only if the local density of sick individuals is higher than $\beta$, the Social Distancing is performed. Otherwise, the Safeguard tactic is executed.

Adapting the survival movement strategy also depends on the organisms’ conditioning, meaning their cognitive ability to learn how to react to the presence of sick individuals. We then introduce the adaptation factor, $\lambda$, a real parameter, $0\leq\lambda\leq 1$, which indicates the organism’s conditioning to respond to a local surge appropriately. For $\lambda=0$, the organism cannot identify the presence of sick individuals, performing only the Safeguard strategy. In the limit $\lambda=1$, the organism is fully conditioned to react to the local surges correctly.

Every time an individual of species $1$ is stochastically chosen to move, the adaptation factor defines the probability of the survival movement strategy being implemented. Namely, $\lambda$ is defined as the probability of the organism being able to execute the behavioural strategy; $1-\lambda$ is the probability of moving randomly.

The numerical implementation of the strategic movement follows the steps [33, 27]:

1. 1.

   We define the perception radius $R$, measured in lattice spacing, as the maximum distance an individual can scan the environmental cues to use the information to decide what strategic movement is suitable at the moment.

2. 2.

   We implement a circular area of radius $\mathcal{R}$ centred in the active individual, defining the total grid points the organism is able to analyse.

3. 3.

   We calculate the density of sick organisms within the disc of radius $R$ surrounding the active individual. In case of the local density is below the threshold $\beta$, the organism decides to perform the Safeguard strategy; otherwise, the Social Distancing tactic is chosen.

4. 4.

   We separate the organism’s perception area into four circular sectors in the directions of the nearest neighbour. Thus, we count the number of empty spaces and organisms within each circular sector; individuals on the borders are assumed to be in both circular sectors.

5. 5.

   We define the movement direction according to the organism’s strategy choice: the direction with the larger number of organisms of species $i-2$ ($h_{i-2}+s_{i-2}$) for Safeguard or the direction with more empty spaces for Social Distancing. If more than one direction is equally attractive, a draw is done.

To study the effects of the adaptive survival strategy to local epidemic outbreaks in the spatial patterns, we calculate the characteristic length scale of the typical spatial domains occupied by each species. For this purpose, we compute the spatial autocorrelation function $C_{i}(r)$, with $i=1,2,3,4,5$, in terms of radial coordinate $r$. Defining the function $\phi_{i}(\vec{r})$ to describe the position $\vec{r}$ in the lattice occupied by individuals of species $i$, we use the mean value $\langle\phi_{i}\rangle$ to find the Fourier transform

and the spectral densities

The normalised inverse Fourier transform allows us to find the autocorrelation function for species $i$ as

which we write as a function of $r$ as

Defining the threshold $C_{i}(l_{i})=0.15$, we define the characteristic length scale for spatial domains of species $i$ as $l_{i}$, with $i=1,2,3,4,5$.

The impact of the locally adaptive organisms’ response to their survival is computed by employing two death risks: a) Selection Risk, $\zeta_{i}(t)$: the probability of an organism of species $i$ being killed by an individual of species $i-1$ at time $t$; b) Infection Risk, $\chi_{i}(t)$: the probability of a healthy organism of species $i$ being infected by an ill individual of any species at time $t$.

The numerical implementation of $\zeta_{i}(t)$ and $\chi_{i}(t)$ follows the algorithm:

1. 1.

   We count the total number of healthy and sick individuals of species $i$ when each generation commences;

2. 2.

   We calculate how many individuals of species $i$ are selected (killed by individuals of species $i-1$) during the generation;

3. 3.

   We compute the number of healthy organisms infected during the generation;

4. 4.

   The selection risk is the ratio between the number of selected individuals and the total number of organisms;

5. 5.

   The infection risk is the ratio between the number of infected individuals and the total number of healthy organisms.

Throughout this paper, all outcomes presented were obtained in simulations with $\mathcal{N}=500^{2}$ grid points, starting from random initial conditions and running for a timespan of $5000$ generations. The initial conditions are elaborated by allocating an individual of a random species at each point. The number of individuals of each species is the same at the beginning of the simulation, $I_{i}=\mathcal{N}/5$, with $1\%$ being sick - there are no empty spaces in the initial state. All simulations ran for the set of probabilities: $s_{h}=r_{h}=m_{h}=1/3$, $s_{s}=r_{s}=5/31$, $m_{s}=9/31$, $\omega=10/31$, and $c=d=1/31$. However, having repeated the simulations for other parameters, we verified that our conclusions are valid for other sets of interaction probabilities.

The outcomes presented in Fig. 2 and 3 were obtained for $R=3$, $\lambda=1.0$, and $\beta=0.6$. The statistical analyses whose results appear in Figs. 4 to  7 were realised by performing a series of $100$ simulations, starting from different initial conditions; in each case, the standard deviation is represented by error bars. The parameters were assumed as follows: i) Figs. 4 and  5: $0\leq\beta\leq 1$, with intervals of $\Delta\beta=0.1$ (for fixed $R=3$ and $\lambda=1.0$); ii) Fig. 6: $0\leq R\leq 5$, with intervals of $\Delta R=1$ (for fixed $\beta=0.6$ and $\lambda=1.0$); iii) Fig. 7: $0\leq\lambda\leq 1$, with intervals of $\Delta\lambda=0.1$ (for fixed $R=3$ and $\beta=0.6$).

In Figs. 2(a) to 2(d), organisms of species $i$ are depicted by the respective species colours, following the scheme in Fig. 1, where red, blue, pink, green, and yellow stand for species $1$, $2$, $3$, $4$, and $5$, respectively; white dots show the empty sites. The organisms of species $1$ present in Figs. 2(a) to 2(d) are shown in Figd. 2(e) to 2(h), with purple and orange dots indicating the individuals performing Safeguard and Social Distancing, respectively; grey areas are empty or occupied by organisms of species $2$, $3$, $4$, $5$. The colours in Figs. 3 to  7 also follow the scheme in Fig. 1; the grey lines represents empty spaces.

The spatial patterns observed in Fig. 2 show that the asymmetry in the species segregation results from the local interactions where each organism of species $1$ moves strategically, optimising the results of the behavioural survival strategy according to its own reality. To quantify the characteristic length scale of the typical spatial areas occupied by each species, we ran a set of $100$ simulations for various values of $\beta$. The mean value of characteristic length scale $l_{i}$, with $i=1,2,3,4,5$ is depicted in Fig. 4; the error bars represent the standard deviation.

If organisms of species $1$ exclusively execute the Social Distancing tactic, independent of the perception of the local density of sick individuals ($\beta=0.0$), the average size of the areas occupied by them is larger than the others. In this limit case, individuals of species $5$ form the second-largest groups; the characteristic length scale of regions dominated by species $3$ is the shortest. The results are approximately the same for $0.0\,\leq\,\beta\,\leq\,0.3$. However, as the social distancing trigger grows (individuals live with less concern about the disease contamination), the difference in the average sizes of the regions of each species decreases. For $\beta\geq 0.8$, organisms of species $4$ are limited to the smallest areas; species $1$ occupies spatial domains with the second shortest characteristic length scale.

Performing a series of simulations, starting from different initial conditions, we computed the social distancing trigger parameter interferes with the organisms’ infection, selection risks, and densities of species. Figures 5(a), 5(b), and 5(c) depict $\zeta_{i}$, $\chi_{i}$, and $\rho_{i}$ as functions of $\beta$ - the simulations parameters are the same as in the previous sections.

Overall, the results show a crossover between scenarios where organisms perform exclusively one out of two survival movement strategies: the Social Distancing ($\beta\leq 0.3$) and Safeguard ($\beta\geq 0.9$) tactics.

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  If the social distancing trigger is low, the organisms are more careful against disease contamination. For $\beta\,\leq\,0.3$, the fear of being sick is too high that all organisms forget the risk of being killed by enemies for executing exclusively social distancing, as shown in Fig. 5(a). The consequence is that the infection risk reaches the minimum value while the selection risk is maximum for $\beta\,\leq\,0.3$: the more careful the organisms are against viral infection, the more exposed to being killed by individuals of species $5$.

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  Conversely, as $\beta$ grows, the organisms are more tolerant of the presence of viral vectors, concentrating more on approaching the enemies of their enemies. Thus, the selection risk decreases while infection risk grows for larger $\beta$. According to Fig. 5(b), for $\beta\geq 0.9$, the individuals’ negligence in avoiding disease contamination leads to the maximum infection risk. This happens because although the protection received from organisms of species $4$ against individuals of species $5$ is maximum, there is a chance of part of the guards being sick, thus, infecting the individuals of species $1$.

The results depicted in Fig. 5(c) show that the benefits of the adaptive survival strategy for species $1$ are maximised for $\beta=0.6$. In this case, species $1$ predominates in the cyclic game, with the density of organisms of species $1$ reaching the maximum.