Understanding contagion dynamics through microscopic processes in active Brownian particles

Together with the universally recognized SIR model, several approaches have been employed to understand the contagion dynamics of interacting particles. Here, Active Brownian particles (ABP) are introduced to model the contagion dynamics of living agents that perform a horizontal transmission of an infectious disease in space and time. By performing an ensemble average description of the ABP simulations, we statistically describe susceptible, infected, and recovered groups in terms of particle densities, activity, contagious rates, and random recovery times. Our results show that ABP reproduces the time dependence observed in traditional compartmental models such as the Susceptible-Infected-Recovery (SIR) models and allows us to explore the critical densities and the contagious radius that facilitates the virus spread. Furthermore, we derive a first-principles analytical expression for the contagion rate in terms of microscopic parameters, without considering free parameters as the classical SIR-based models. This approach offers a novel alternative to incorporate microscopic processes into analyzing SIR-based models with applications in a wide range of biological systems.

Scientific Reports | (2020) 10:20845 | https://doi.org/10.1038/s41598-020-77860-y www.nature.com/scientificreports/ modeled using self-propelled models with a gravitational term that describes the ambient information 30,31 . Other approaches to the dynamics of infectious diseases in humans 32 have also been explored using a non-linear wave approach by means of reaction-diffusion equations to model the effect of random motion in the SIR dynamics. In our case, those conditions can be reproduced in the limit where the activity is zero, and the particles perform only Brownian motion. In this, aspect AM offers the possibility to simulate the way that the walkers interact, to obtain the typically S(t), I(t), and R(t) curves. Besides, AM introduce an empirical potential modeling the repulsion between particles and an additional rotational diffusion term that controls the medium's exploration. This is crucially different from the non-linear wave equations, leading to AM simulations to model complex effects such as clusterization formation or bimodal phase separation, which usually are not captured by the typical non-linear wave approaches.
Here, we explore infection propagation through active vectors that carry an internal state using an AM based model with underlying microscopic processes. The infection occurs through horizontal transmission when a susceptible agent comes into contact with an infected agent such as viruses propagating in salmon hatcheries, honeycombs 33,34 , or in mesoscale organisms such as cats with influenza or humans carrying flu 2,3,35 .

Results
Active Brownian particles and SI model. Let us consider N moving particles in a rectangular box with area A and periodic boundary conditions, as shown in Fig. 1a. We model the ABP dynamics by considering both Weeks-Chandler-Andersen (WCA) potential (5) and rotational diffusion according to the following set of Langevin equations where � r i = (x i , y i ) is the instantaneous position, � F ij = −∇U ij is the force derived from the WCA potential, n i = (cos θ i , sin θ i ) is the director vector, and θ i = tan −1 (y i /x i ) is the orientation. Due to particle rotational diffusion, each angle θ i change randomly according to the Wiener process, where ξ θ i (t) = 0 and ξ θ i (t)ξ θ i (0) = 2D R δ(t) . Here, D R is assumed as a constant parameter that takes account particle's exploration of the medium. Then, active agents are allowed to move persistently in their given direction, with a persistence length l p = v 0 /2D R , until a random reorientation takes place 36 . Therefore when D R is small active agents can be considered as persistent explorers, which is true for several organisms in nature 12 .
First, we consider a simple SI model where infected I(t) and susceptible S(t) satisfy I(t) + S(t) = N . A contagious event occurs when a susceptible particle i is in contact with an infected particle j at a distant |� r i − � r j | ≤ R , where R is the contagion radius, as shown in Fig. 1b. For a set of parameters (N, L x , L y , R, v 0 ) we run N sim simulations to compute the ensemble average curves Fig. 1c, we Particle infection: pair interactions between particle i and j at a distant |� r i − � r j | ≤ R , where R is the contagion radius. Infected particle i is moving with velocity v 0ni and given position r i and interact through the contagion radius R with particle j which is moving with velocity v 0nj and position r j . For the SI model we only consider two states: susceptible S(t) (blue) and infected I(t) (red) such that S(t) + I(t) = N . In the SIR model, we introduced the recovered group R(t) (green) such that after a random recovery time τ i rec infected particle i becomes recovered. (c) Phase diagram for the SI model showing the number of infected particles as a function of the contagion radius R and the particle density ρ = N/A . The dashed black line represent the critical density ρ crit = 1/(πR 2 ) . For the simulation we consider N = 300 , v 0 = 1 , L x = L y , and I(0) = 1.
Scientific Reports | (2020) 10:20845 | https://doi.org/10.1038/s41598-020-77860-y www.nature.com/scientificreports/ show a phase diagram for the number of infected as a function of the contagion rate and the particle density for N = 300 . As expected, in the region of high density and large contagion radius, the infected group saturates reaching its maximum value. More importantly, we observe the existence of a critical density ρ c = 1/(πR 2 ) (black dashed line) above which the particles are immediately infected. It is worth to notice that in several situations, living organisms do not move randomly. This is especially true for human 13,37 , or even in animals 38 ; However, there are situations where the persistence of the movement is altered by ambient noise, for example, humans moving inside a supermarket or in a shopping mall, on those scenarios the movement is altered to avoid clashes between individuals as they also performed some random explorations of the medium. Although humans might not be considered as random particles, in a first approximation, humans and animals in all length scales can be considered as active random particles 12,14 such as fishes, mosquitos, bees, algae, and bacteria, among others 15 , in this case, active matter models can correctly describe agents interactions and collective motion 39 .
Microscopic expression contagion rate. Using a mean-free-path analysis (see Methods 4.2 for further details), we obtain the following analytical expression for the contagion rate r: In the low-density regime, ρ ≪ ρ crit , we obtain a linear scaling r ≈ √ 8ρRv 0 . Also, our model predicts a singularity at ρ = ρ crit for which r → ∞ . As a result, all particles are instantaneously infected. One critical observation is the dimensional-dependent nature of the contagion rate in our model. For instance, for N moving particles in a volume V, the mean-free-path analysis predicts a three-dimensional contagion rate and v rel is the average relative velocity between particles. Therefore, our active matter model predicts that distancing between infected particles is more critical in a three-dimensional system since r 3D ∝ R 2 . The latter can be crucial in biological systems where a 3D movement is present during the contagion dynamics 12,29 .
Our model has minimal but fundamental mechanisms to study clustering formation (CF) 39 or two-phase separation (TPS) 40 while varying the particle activity (velocity) or density 23,41 . In the scenario of virus propagation, a TPS could be relevant since CF might be included in the contagion dynamics by considering density gradients or hot spots in dilute or dense systems. The latter can be used to simulate quarantines in groups or on their city hall while some rangers continue moving in the space between clusters. In this case, we expect that the density-dependent contagion rates r and r 3D would be useful for novel mechanisms of infection that are not described by standard epidemic models 41 . Now, we shall establish the connection between our microscopic contagion rate given in Eq.
(2) and the characteristic epidemic curve for the SI model. At each discrete time t n = n t ( n ∈ N and �t > 0 ), the number of infected varies according to the Markovian model I n+1 = I n + p n S n , where p n = (r�t)(I n /N) and S n = N − I n are the contagion probability and number of susceptible at time t n , respectively. As a consequence, in the continuum limit, the curve I(t) evolves according to ( t → 0): The above equations can be written as Ṡ = −rIS/N and İ = rIS/N , which is the standard SI model. The logistic function I(t) = I(0)Ne rt /[(N − I(0)) + I(0)e rt ] gives the analytic solution of (3). To support our previous observations, in Fig. 2, we plot a comparison between the infected curve I(t) obtained from the ensemble average procedure and the logistic model given above. Here, we consider a system with N = 100 particles in a square box with lengths L x = L y = 100 , contagion radius R = 1 , and particle velocity v 0 = 1 . We observe a good agreement between the theory and simulations, revealing that one initial contagion grows logistically if the recovered group is neglected. However, a small asymmetry of the analytical logistic model is observed in Fig. 2. One suggestive approach is to fit the ensemble average with the generalized logistic model or Richard's model given by İ = rI p [1 − (I/N) q ] ( 0 ≤ p ≤ 1 ) which has been used in COVID-19 pandemic curves 42 . This could be useful for biological systems showing logistic-like behaviors with more complicated microscopic dynamics.
Furthermore, in the inset of Fig. 2, we compare the microscopic expression for the contagion rate defined in Eq. (2) and the predicted rate obtained in our simulations. We recover the predicted linear dependence of the contagion rate in terms of the particle density, which validates our microscopic model.

Active Brownian particles and SIR model. Now, we include the recovered group R(t) into the dynamics.
In such a case, the total number of particles satisfy S(t) + I(t) + R(t) = N . First, we assume that the recovered group cannot be infected again, that is, particles gain immunity. Second, we neglect deaths since we are interested in the propagation itself. Third, we introduce a random recovery time τ i rec for each particle ( i = 1, ..., N ) such that τ i rec ∈ [τ min , τ max ] . Here, τ min and τ max are the minimum and maximum recovery in our simulations, respectively. In what follows, we use a uniformly distributed random number to generate the individual recovery times. This particular choice of the distribution impacts on the symmetry properties of the infected curve; thus, other probability distributions can be used to simulate a different scenario. Also, other relevant times, such as the incubation time, can be incorporated into the dynamics, which originates a delayed dynamics for the infected curve, as discussed in Ref. 43 . For simplicity, we only consider the effect of the recovery time on the dynamics.
We compare our simulations with the conventional SIR model, which is described by the set of differential equations Ṡ = −αIS , İ = αIS − βI , Ṙ = βI , where α and β are the infection and recovery rates, respectively 1 . (3) Scientific Reports | (2020) 10:20845 | https://doi.org/10.1038/s41598-020-77860-y www.nature.com/scientificreports/ We can find the optimal parameters α and β that improves the fit between the SIR model and our simulations. In Fig. 3, we observe a comparison between our simulations (ensemble average) and the SIR fit (dashed lines). In general, we numerically corroborate that our model cannot be fully explained in terms of the standard SIR model. In particular, the SIR model predicts an asymmetry curve for I(t), and the stationary states differ with our calculations. Our simulations show an asymmetric curve for the infected group, which has been previously observed in Ref. 44 . Using the relations α = r/N (r given in Eq. (2)) and β = 1/T prom ( T prom = (τ min + τ max )/2 ) into the equation İ = rIS/N − I/T prom we can find I(t). Noticing that relevant contributions to the product IS comes from the region where S(t) has a linear dependence, we use S(t) = S 0 − mt into the dynamics of I(t), and we found the following Gaussian curve: where t 0 = (rS 0 − N/T prom )/(rm) is the position of the peak and σ = [N/(rm)] 1/2 is the width of the Gaussian curve. In Fig. 4, we observe the good agreement between our simulations and the Gaussian model given in (4).
, and thus the ratio t 0 /σ is critical. In the low-density regime, we obtain I max ∝ exp [Rv 0 /A] illustrating that the contagion radius, available area, and velocity of particles strongly impact the maximum number of infected during the dynamics. On the other hand, the scaling σ ∝ [Rv 0 /A] −1/2 , tell us that any reduction of the maximum number of infected implies a flattened effect on the curve I(t), as expected in the standard SIR model. Further improvements or extensions of the current model can be performed by considering the incubation time, different particle velocities, time-dependent densities to model lock-down, or by including particle interactions modeled with microscopic pedestrian models 45 .

Conclusions
Active Matter (AM) simulations show that active Brownian particles that exchange an internal state can successfully reproduce the universally accepted SIR contagious curves, for horizontal disease transmission, by introducing the effects of contagious radii, particle velocity, and particles density. Theoretically, the SIR model assumes several empirical parameters in order to describe the contagious dynamics. Here, we introduce a first-principle analytical expression that successfully reproduces our simulations in terms of controllable microscopic parameters. Besides, our expression qualitatively recovers the SIR based models with good agreement with numerical simulations. Furthermore, we find an important dependence on the particle density and contagious radius in two and three dimensions, which opens a new forecast parameter in viruses propagation inside a specific population.
Although our study focuses mainly on particle density and contagious rates, we expect that our model can be improved by including complex interactions such as quorum-sensing to describe viruses propagation in birds or schools of fish or by adding external forces to describe human will 3,30,31 . Nevertheless, how these new interactions alter and couple the dynamics with compartmental models are exciting new questions that our work opens to the active matter community.

Methods
Brownian Dynamics Simulations in the overdamped limit. We performed Brownian dynamics simulations for N = 300 disk particles of radius a = 0.5 [m] bounded in a rectangular box with periodic boundary conditions. Particles are settled initially at random positions and orientation following a uniform distribution. Particles move according to Langevin equations (1) with self-propelled velocity v 0 = 1 [m/s] and rotational diffusion given by D R = 1 [rad 2 /s], where we set a new position and orientation for each particle using the Euler iteration method with a time step dt = 10 −3 . Since the particle dynamics is non-deterministic and particle encounters determine the contagious rate, we performed 100 different numerical simulations starting with a different random configuration. Particles perform pair-hard core interactions via the Weeks-Chandler-Andersen (WCA) potential, Here, ε is the interaction potential constant, r m locates the potential minimum, which is equal to the limit distance between particles r 0 = 2a . Although this interaction avoids particles overlapping its principal consequence, the particle trajectory deviations imitate living organisms' encounters. Particles also transmit the infection through an instantaneous pair-interaction, which sets a new length parameter on the problem, the contagious radii R. Then if the distance between a susceptible particle and an infected particle is less than R, the susceptible particle www.nature.com/scientificreports/ is labeled as infected. We vary the contagious radii from R = a, . . . , 6 , in steps of R = 0.5 , and the box length L = 100, . . . , 300 in increments of L = 10 8,46 .
Microscopic contagion rate. The microscopic contagion rate can be derived using the concept of mean free path , extensively used in the kinetic theory of gases and also used in Ref. 47 . In this context, represent the mean distance traveled by ABP between successive encounters with other particle at a distance d ij = R . In an active media with N moving particles = �|� v rel | 2 �τ c , with v rel and τ c being the relative velocity between particles and the mean contagion time, respectively. Here, ... denote the particle average. Thus, we estimate the contagion rate through the relation r = τ −1 c . Encounters between ABP's depends on the relative velocity First, we assume uncorrelated particle's velocities yielding �� v i · � v j � = 0 . Second, if the WCA potential does not drastically change the speed v 0 , we approximately obtain that �|� v ij rel | 2 � ≈ 2v 2 0 since �v 2 i � ≈ v 2 0 . By considering the total area swept for N particles in a time interval τ c as A sw = N(2R + πR 2 ) , we define the maximum contagion probability p c = A sw /A = 1 , and using the relation = √ 2v 0 τ c , we recover the analytical expression of the contagion rate given in (2).