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    Behavioural movement strategies in cyclic models

    In this work, we performed stochastic simulations of a cyclic nonhierarchical system composed of 5 species. To this purpose, we implemented a standard numerical algorithm largely used to study spatial biological systems11,13,41. We considered a generalisation of the rock-paper-scissors game for 5 species, whose rules are illustrated in Fig. 1a. The arrows indicate a cyclic dominance among the species. Accordingly, individuals of species i beat individuals of species (i+1), with (i=1,2,3,4,5).The dynamics of individuals’ spatial organisation occurs in a square lattice with periodic boundary conditions, following the rules: selection, reproduction, and mobility. We assumed the May-Leonard implementation so that the total number of individuals is not conserved43. Each grid point contains at most one individual, which means that the maximum number of individuals is ({mathcal {N}}), the total number of grid points.Initially, the number of individuals is the same for all species, i.e., (I_i,=,{mathcal {N}}/5), with (i=1,2,3,4,5) (there are no empty spaces in the initial state). We prepared the initial conditions by distributing each individual at a random grid point. At each timestep, one interaction occurs, changing the spatial configuration of individuals. The possible interactions are:

    Selection: (i j rightarrow i otimes ,), with (j = i+1), where (otimes) means an empty space; every time one selection interaction occurs, the grid point occupied by the individual of species (i+1) vanishes.

    Reproduction: (i otimes rightarrow i i,); when one reproduction is realised an individual of species i fills the empty space.

    Mobility: (i odot rightarrow odot i,), where (odot) means either an individual of any species or an empty site; an individual of species i switches positions with another individual of any species or with an empty space.

    In our stochastic simulations, selection, reproduction, and mobilities interactions occur with the following probabilities: s, r and m, respectively. We assumed that individuals of all species have the same probabilities of selecting, reproducing and moving. The interactions were implemented by assuming the von Neumann neighbourhood, i.e., individuals may interact with one of their four nearest neighbours. The simulation algorithm follows three steps: i) sorting an active individual; ii) raffling one interaction to be executed; iii) drawing one of the four nearest neighbours to suffer the sorted interaction (the only exception is the directional mobility, where the neighbour is chosen according to the movement tactic). If the interaction is executed, one timestep is counted. Otherwise, the three steps are redone. Our time unit is called generation, defined as the necessary time to ({mathcal {N}}) timesteps to occur.In our model, individuals of one out of the species can move into the direction with more individuals of a target species. The choice is based on the strategy assumed by species. We assumed three sorts of directional movement tactics:

    Attack tactic: an individual of species i moves into the direction with more individuals of species (i+1);

    Anticipation tactic: an individual of species i goes towards the direction with a larger number of individuals of species (i+2);

    Safeguard tactic: an individual of species i walk into the direction with a larger concentration of individuals of species (i-2).

    In the standard model, individuals of all species move randomly.We considered that only individuals of species 1 perform the directional movement tactics, as illustrated in Fig. 1b. The solid, dashed, and dashed-dotted lines represent the Attack, Anticipation, and Safeguard tactics, respectively. The concentric circumference arcs show that individuals of species 2, 3, 4, and 5 always move randomly. For implementing a directional movement, the algorithm follows the steps: i) it is assumed a disc of radius R (the perception radius), in the active individual’s neighbourhood; ii) it is defined four circular sectors in the directions of the four nearest neighbours; iii) according to the movement tactic, the target species is defined: species 2, 3, and 4, for Attack, Anticipation, and Safeguard tactics, respectively; iv) it is counted the number of individuals of the target species within each circular sector. Individuals on the borders are assumed to be part of both circular sectors; v) the circular sector that contains more individuals of the target species is chosen. In the event of a tie, a draw between the tied directions is made; vi) the active individual switches positions with the immediate neighbour in the chosen direction. The swap is also executed in case of the neighbour grid point is empty.To observe the spatial patterns, we first performed a single simulation for the standard model, Attack, Anticipation, and Safeguard tactics. The realisations run in square lattices with (500^2) grid points, for a timespan of 5000 generations. We captured 500 snapshots of the lattice (in intervals of 10 generations), that were used to make the videos of the dynamics of the spatial patterns showed in https://youtu.be/Ndvk6Rg57m4 (standard), https://youtu.be/JGhkDAHSo74 (Attack), https://youtu.be/ZZp9QlOfv2Q (Anticipation), and https://youtu.be/eFxWdLhIOuQ (Safeguard). The final snapshots were depicted in Fig. 2a–d. Individuals of species 1, 2, 3, 4, and 5 are identified with the colours ruby, blue, pink, green, and yellow, respectively; while white dots represent empty spaces. The simulations were performed assuming selection, reproduction, and mobility probabilities: (s = r = m = 1/3). The perception radius was assumed to be (R=3).The population dynamics were studied by means of the spatial density (rho _i), defined as the fraction of the grid occupied by individuals of species i at time t, i.e., (rho _i = I_i/{mathcal {N}}), where (i=0) stands for empty spaces and (i=1,…,5) represent the species 1, 2, 3, 4, and 5. The temporal changes in spatial densities of the simulations showed in Fig. 2 were depicted in Fig. 3, where the grey, ruby, blue, pink, green, and yellow lines represent the densities of empty spaces and species 1, 2, 3, 4, and 5, respectively. We also computed how the selection risk of individuals of species i changes in time. To this purpose, the algorithm counts the total number of individuals of species i at the beginning of each generation. It is then counted the number of times that individuals of species i are killed during the generation. The ratio between the number of selected individuals and the initial amount is defined as the selection risk of species i, (zeta _i). The results were averaged for every 50 generations. Figure 4 shows (zeta _i,(%)) as a function of the time for the simulations presented in Fig. 2. The ruby, blue, pink, green, and yellow lines show the selection risks of individuals of species 1, 2, 3, 4, and 5, respectively.To quantify the spatial organisation of the species, we studied the spatial autocorrelation function. This quantity measures how individuals of a same species are spatially correlated, indicating spatial domain sizes. Following the procedure carried out in literature41,42,44,45,46, we first calculated the Fourier transform of the spectral density as (C({{vec{r^{prime}}}}) = {mathcal{F}}^{{ – 1}} { S({{vec{k}}})} /C(0)), where the spectral density (S({{vec{k}}})) is given by (S({{vec{k}}}) = sumlimits_{{k_{x} ,k_{y} }} {mkern 1mu} varphi ({{vec{kappa }}})), with (varphi ({{vec{kappa }}}) = {mathcal{F}}{mkern 1mu} { phi ({{vec{r}}}) – langle phi rangle }). The function (phi ({{vec{r}}})) represents the species in the position ({{vec{r}}}) in the lattice (we assumed 0, 1, 2, 3, 4, and 5, for empty sites, and individuals of species 1, 2, 3, 4, and 5, respectively). We then computed the spatial autocorrelation function as$$C(vec{r^{prime}}) = sumlimits_{{|{{vec{r^{prime}}}}| = x + y}} {frac{{C({{vec{r^{prime}}}})}}{{min (2N – (x + y + 1),(x + y + 1))}}}.$$Subsequently, we found the scale of the spatial domains of species i, defined for (C(l_i)=0.15), where (l_i) is the characteristic length for species i.We calculated the autocorrelation function by running 100 simulations using lattices with (500^2) grid points, assuming (s = r = m = 1/3) and (R=3). Each simulation started from different random initial conditions. We then captured each species spatial configuration after 5000 generations to calculate the autocorrelation functions. Finally, we averaged the autocorrelation function in terms of the radial coordinate r and calculated the characteristic length for each species. We also calculated the standard deviation for the autocorrelation functions and the characteristic lengths. Figure 4 shows the comparison of the results for Attack, Anticipation, and Safeguard strategies with the standard model. The ruby, blue, pink, green, and yellow circles indicate the mean values for species 1, 2, 3, 4, and 5, respectively. In the case of standard model, the mean values are represented by grey circles, which are the same for all species. The error bars show that standard deviation. The horizontal black line represents (C(l_i), =, 0.15).To further explore the numerical results, we studied how the perception radius R influences species spatial densities and selection risks. We calculated the mean value of the spatial species densities, (langle , rho _i,rangle) and the mean value of selection risks, (langle , zeta _i,rangle) from a set of 100 simulations in lattices with (500^2) grid points, starting from different initial conditions for (R=1,2,3,4,5). We used (s,=r,=,m,=1/3) and a timespan of (t=5000) generations. The mean values and standard deviation were calculated using the second half of the simulations, thus eliminating the density fluctuations inherent in the pattern formation process. The results were shown in Fig. 6, where the circles represent the mean values and error bars indicate the standard deviation. The colours are the same as in Figs. 3 and 4. Furthermore, to verify the precision of the statistical results, we calculated the variation coefficient – the ratio between the standard deviation and the mean value. Supplementary Tables S1 and S2 show statistical outcomes for species densities and selections risks, respectively.We studied a more realistic scenario where not all individuals of species 1 can perform the directional movement tactics. For this reason, we defined the conditioning factor (alpha), with (0,le ,alpha ,le ,1), representing the proportion of individuals of species 1 that moves directionally. For (alpha =0) all individuals move randomly while for (alpha =1) all individuals move directionally. This means that every time an individual of species 1 is sorted to move, there is a probability (alpha) of the algorithm implementing the directional movement tactic, instead of randomly choosing one of its four immediate neighbours to switch positions. To understand the effects of the conditioning factor, we observed how the density of species 1 changes for the entire range of (alpha), with intervals of (Delta alpha = 0.1). The simulations were implemented for (R=3) and (s,=r,=,m,=1/3). It was computed the mean value of the spatial density of species 1, (langle , rho _1,rangle), and its standard deviation from a set of 100 different random initial conditions. The results were depicted in Fig. 7, where the green, red, and blue dashed lines show (langle , rho _1,rangle) as a function of (alpha). The error bars indicate the standard deviation.Finally, we aimed to investigate how the directional movement tactics jeopardise species coexistence for a wide mobility probability range. Because of this, we run 2000 simulations in lattices with (100^2) grid points for (0.05, More

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    High turbidity levels alter coral reef fish movement in a foraging task

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    Model-based projections for COVID-19 outbreak size and student-days lost to closure in Ontario childcare centres and primary schools

    Population structureThere are N households in the population, and a single educational institution (either a school or a school, dependent on scenarios to be introduced later) with M rooms and a maximum capacity dependent on the scenario being tested. Effective contacts between individuals occur within each household, as well as rooms and common areas (entrances, bathrooms, hallways, etc.) of the institution. All groups of individuals (households and rooms) in the model are assumed to be well-mixed.Each individual (agent) in the model is assigned an age, household, room in the childcare facility and an epidemiological status. Age is categorical, so that every individual is either considered a child (C) or an adult (A). Epidemiological status is divided into stages in the progression of the disease; agents can either be susceptible (S), exposed to the disease (E), presymptomatic (an initial asymptomatic infections period P), symptomatically infected (I), asymptomatically infected (A) or removed/recovered (R), as shown in Fig. 1b.In the model, some children in the population are enrolled as students in the institution and assigned a classroom based on assumed scenarios of classroom occupancy while some adults are assigned educator/caretaker roles in these classroom (again dependent on the occupancy scenario being tested). Assignments are made such that there is only one educator per household and that children do not attend the same institution as a educator in the household (if there is one), and vice versa.Interaction and disease progressionThe basic unit of time of the model is a single day, over which each attendee (of the institution) spends time at both home and at the institution. The first interactions of each day are established within each household, where all members of the household interact with each other. An asymptomatically infectious individual of age i will transmit the disease to a susceptible housemate with the age j with probability (beta ^H_{i,j}), while symptomatically infectious members will self-isolate (not interact with housemates) for a period of 14 days.The second set of interpersonal interactions occur within the institution. Individuals (both students and educators) in each room interact with each other, where an infectious individual of age i transmits the disease to some susceptible individual of age j with probability (beta ^C_{i,j}). To signify common areas within the building (such as hallways, bathrooms and entrances), each individual will then interact with every other individual in the institution. There, an infectious individual of age j will infect a susceptible individual of age i with probability (beta ^O_{i,j}).To simulate community transmission (for example, public transport, coffee shops and other sources of infection not explicitly modelled here), each susceptible attendee is infected with probability (lambda _S). Susceptible individuals not attending the institution in some capacity are infected at rate (lambda _N), where (lambda _N >lambda _S) to compensate for those consistent effective interactions outside of the institution that are neglected by the model (such as workplace interactions among essential workers and members of the public).Figure 1b shows the progression of the illness experienced by each individual in the model. In each day, susceptible (S) individuals exposed to the disease via community spread or interaction with infectious individuals (those with disease statuses P, A and I) become exposed (E), while previously exposed agents become presymptomatic (P) with probability (delta). Presymptomatic agents develop an infection in each day with probability (delta), where they can either become symptomatically infected (I) with probability (eta) or asymptomatically infected (A) with probability (1-eta).The capacity of the sole educational institution in the model is divided evenly between 5 rooms, with class size and student-educator ratio governed by one of three basic scenarios: seven students and three educators per room (7 : 3), eight students and two educators per room (8 : 2), and fifteen students and two educators per room (15 : 2). Classroom assignments for children can be either randomized or grouped by household (siblings are put in the same class).Symptomatically infected agents (I) are removed from the simulation after 1 day (status R) with probability (gamma _I), upon which they self-isolate for 14 days, and therefore no longer pose a risk to susceptible individuals. Asymptomatically infected agents (A) remain infectious but are presumed able to maintain regular effective contact with other individuals in the population due to their lack of noticeable symptoms; they recover during this period (status R) with probability (gamma _A). Disease statuses are updated at the end of each day, after which the cycles of interaction and infection reoccur the next day.The actions of symptomatic (status I) agents depend on age and role. Individuals that become symptomatic maintain a regular schedule for 1 day following initial infection (including effective interaction within the institution, if attending), after which they serve a mandatory 14-day isolation period at home during which they interaction with no one (including other members of their household). On the second day after the individual’s development of symptoms, their infection is considered a disease outbreak centred in their assigned room, triggering the closure of that room for 14 days. All individuals assigned to that room are sent home, where they self-isolate for 14 days due to presumed exposure to the disease. Symptomatically infected children are not replaced, and simply return to their assigned classroom upon recovery. At the time of classroom reopening, any symptomatic educator is replaced by a substitute for the duration of their recovery, upon which they reprise their previous role in the institution; the selection of a substitute is made under previous constraints on educator selection (one educator per household. with no one chosen from households hosting any children currently enrolled in the institution).ParameterisationThe parameter values are given in Supplementary Table S4. The sizes of households in the simulation was determined from 2016 Statistics Canada census data on the distribution of family sizes42. We note that Statistics Canada data only report family sizes of 1, 2 or 3 children: the relative proportions for 3+ children were obtained by assuming that (65 %) of families of 3+ children had 3 children, (25%) had 4 children, (10%) had 5 children, and none had more than 5 children. Each educator was assumed to be a member of a household that did not have children attending the school. Again using census data, we assumed that (36%) of educators live in homes with no children, where an individual lives alone with probability 0.282, while households hosting 3, 4, 5, 6, and seven adults occur with probability 0.345, 0.152, 0.138, 0.055, 0.021 and 0.009 respectively. Others live with (ge 1) children in households following the size and composition distribution depending on the number of adults in the household. For single-parent households, a household with a single child occurs with probability 0.169, and households with 2, 3, 4 and 5 children occur with probabilities 0.079, 0.019, 0.007 and 0.003 respectively. With two-parent households, those probabilities become 0.284, 0.307, 0.086, 0.033 and 0.012.The age-specific transmission rates in households are given by the matrix:$$begin{aligned} begin{bmatrix} beta ^H_{1,1} &{} beta ^H_{1,2} \ beta ^H_{2,1} &{} beta ^H_{2,2} \ end{bmatrix} equiv beta ^H begin{bmatrix} c^H_{1,1} &{} c^H_{1,2} \ c^H_{2,1} &{} c^H_{2,2} \ end{bmatrix}, end{aligned}$$
    (1)
    where (c^H_{i,j}) gives the number of contacts per day reported between individuals of ages i and j estimated from data28 and the baseline transmission rate (beta ^H) is calibrated. To estimate (c^H_{i,j}) from the data in Ref.28, we used the non-physical contacts of age class 0–9 years and 25–44 years of age with themselves and one another in Canadian households. Based on a meta-analysis, the secondary attack rate of SARS-CoV-2 appears to be approximately (15 %) on average in both Asian and Western households43. Hence, we calibrated (beta ^H) such that a given susceptible person had a (15 %) chance of being infected by a single infected person in their own household over the duration of their infection averaged across all scenarios tested. As such, age specific transmission is given by the matrix$$begin{aligned} beta ^Hcdot begin{bmatrix} 0.5378 &{} 0.3916 \ 0.3632 &{} 0.3335 end{bmatrix}. end{aligned}$$
    (2)
    To determine (lambda _S) we used case notification data from Ontario during lockdown, when schools, workplaces, and schools were closed44. During this period, Ontario reported approximately 200 cases per day. The Ontario population size is 14.6 million, so this corresponds to a daily infection probability of (1.37 times 10^{-5}) per person. However, cases are under-ascertained by a significant factor in many countries. We assumed an under-ascertainment factor of 8.45 based on an empirical estimate of under-reporting45, meaning there are actually 8.45 times more cases than reported in Ontario, giving rise to (lambda _S = 1.16 times 10^{-4}) per day; (lambda _N) was set to (2cdot lambda _S). We emphasize that this number may fall later in the pandemic as testing capacity increases, although some individuals may still never get tested–especially schoolchildren, who are often asymptomatic.The age-specific transmission rates in the school rooms is given by the matrix$$begin{aligned} begin{bmatrix} beta ^C_{1,1} &{} beta ^C_{1,2} \ beta ^C_{2,1} &{} beta ^C_{2,2} \ end{bmatrix} equiv beta ^C begin{bmatrix} c^C_{1,1} &{} c^C_{1,2} \ c^C_{2,1} &{} c^C_{2,2} \ end{bmatrix} equiv beta ^C begin{bmatrix} 1.2356 &{} 0.0588 \ 0.1176 &{} 0.0451 end{bmatrix}, end{aligned}$$
    (3)
    where (c^C_{i,j}) is the number of contacts per day reported between age i and j estimated from data28. To estimate (c^C_{i,j}) from the data in Ref.28, we used the non-physical contacts of age class 0–9 years and 20–54 years of age, with themselves and one another, in Canadian schools. Epidemiological data on secondary attack rates in educational institutions are rare, since childcare centres and schools were closed early in the outbreak in most areas. We note that contacts in families are qualitatively similar in nature and duration to contacts in schools with small group sizes, although these contacts are generally more dispersed among the larger groups in rooms than among the smaller groups in households. On the other hand, rooms may represent equally favourable conditions for aerosol transmission, as opposed to close contact. Hence, we assumed that (beta ^C = alpha _C beta ^H), with a baseline value of (alpha _C = 0.75) based on more dispersed contacts expected in the larger room group, although we varied this assumption in sensitivity analysis.To determine (beta ^O) we assumed that (beta ^O = alpha _O beta ^C) where (alpha _O ll 1) to account for the fact that students spend less time in common areas than in their rooms. To estimate (alpha _O), we note that (beta ^O) is the probability that a given infected person transmits the infection to a given susceptible person. If students and staff have a probability p per hour of visiting a common area, then their chance of meeting a given other student/staff in the same area in that area is (p^2). We assumed that (p=0.05) and thus (alpha _O = 0.0025). The age-specific contact matrix for (beta ^O) was the same as that used for (beta ^C) (Eq. 3).Model initializationUpon population generation, each agent is initially susceptible (S). Individuals are assigned to households as described in the “Parameterisation” section, and children are assigned to rooms either randomly or by household. We assume that parents in households with more than one child will decide to enroll their children in the same institution for convenience with probability (xi =80%), so that each additional child in multi-child households will have probability (1-xi) of not being assigned to the institution being modelled.Households hosting educators are generated separately. As in the “Parameterisation” section, we assume that (36%) of educators live in adult-only houses, while the other educators live in houses with children, both household sizes following the distributions outlined in the “Parameterisation” section. The number of educator households is twice that required to fully supply the school due to the replacement process for symptomatic educators outlined in the “Disease Progression” section.Initially, a proportion of all susceptible agents (R_{init}) is marked as removed/recovered (R) to account for immunity caused by previous infection moving through the population. A single randomly chosen school attendee is chosen as a primary case and is made presymptomatic (P) to introduce a source of infection to the model. All simulations are run until there are no more potentially infectious (E, P, I, A) individuals left in the population and the institution is at full capacity. All results were averaged over 2000 trials.Estimating β
    H
    Agents in the simulation were divided into two classes: “children” (ages 0–9) and “adults” (ages 25–44). Available data on contact rates28 was stratified into age categories of width 5 years starting at age 0 (0–5, 5–9, 10–14, etc.). The mean number of contacts per day (c_{i,j}^H) for each class we considered (shown in Eq. 2) was estimated by taking the mean of the contact rates of all age classes fitting within our presumed age ranges for children and adults.For (beta ^H) calibration, we created populations by generating a sufficient number of households to fill the institution in each of the three tested scenarios; 15 : 2, 8 : 2 and 7 : 3. In each household, a single randomly chosen individual was infected (each member with equal probability) by assigning them a presymptomatic disease status P; all other members were marked as susceptible (disease status S). In each day of the simulation, each member of each household was allowed to interact with the infected member, becoming exposed to the disease with probability given in Eq. 2. Upon exposure, they were assigned disease status E. At the beginning of each subsequent day, presymptomatic individuals proceeded to infected statuses I and A, and infected agents were allowed to recover as dictated by Fig. 1b and Supplementary Table S4. This cycle of interaction and recovery within each household was allowed to continue until all infected individuals were recovered from illness.We did not allow exposed agents (status E) to progress to an infectious stage (I or A) since we were interested in finding out how many infections within the household would result from a single infected household member, as opposed to added secondary infections in later days. At the end of each trial, the specific probability of infection ((pi _n)) in each household (H_n) was calculated by dividing the number of exposed agents in the household ((E_n)) by the size of the household (|H_n|) less 1 (accounting for the member initially infected). Single occupant households ((|H_n|=1)) were excluded from the calculation. The total probability of infection (pi) was then taken as the mean of all (pi _n), so that$$begin{aligned} pi =frac{1}{D}sum _{n}pi _n=frac{1}{D}sum _{|H_n|ge 2}frac{E_n}{|H_n|-1}, end{aligned}$$
    (4)
    where D represents the total number of multiple occupancy households in the simulation. This modified disease simulation was run for 2000 trials each of different prospective values of (beta ^H) ranging from 0 to 0.21. The means of all corresponding final estimates of the infection rate were taken per value of (beta ^H), and the value corresponding to a infection rate of (15%) was interpolated.Simplifying assumptionsOur model makes simplifying assumptions that may influence its predictions. For instance, we assume that classrooms are homogeneously mixing and did not take social structure into account. Social structure might slow the spread of COVID-19 in classrooms. We also assumed that public health authorities will respond to a confirmed case by closing the classroom, although in practice, they may keep the class running if they think the case does not represent an infection risk to children or adults. This would reduce the number of student-days lost to closure. Similarly, we did not account for potential contacts between school children outside of classes, although students of a classroom that has been closed may still interact with their classmates outside of school. Other simplifying assumptions are mentioned in the “Discussion” section. More