key: cord-0931396-d9ne9z8v
authors: Cascante-Vega, J.; Torres-Florez, S.; Cordovez, J.; Santos-Vega, M.
title: How disease risk awareness modulates transmission: coupling infectious disease models with behavioral dynamics
date: 2021-04-19
journal: nan
DOI: 10.1101/2021.04.13.21255395
sha: f90d649b2b765fad66a88d31921318aa9e7722e7
doc_id: 931396
cord_uid: d9ne9z8v

Epidemiological models often assume that individuals do not change their behavior or that those aspects are implicitly incorporated in parameters in the models. Typically these assumption is included in the contact rate between infectious and susceptible individuals. For example models incorporate time variable contact rates to account for the effect of behavior or other interventions than in general terms reduce transmission. However, adaptive behaviors are expected to emerge and to play an important role in the transmission dynamics across populations. Here, we propose a theoretical framework to couple transmission dynamics with behavioral dynamics due to infection awareness. We first model the dynamics of social behavior by using a game theory framework. Then we coupled the model with an epidemiological model that captures the disease dynamics by assuming that individuals are more aware of that epidemiological state (i.e. fraction of infected individuals) and reduces their contacts. Our results from a mechanistic modeling framework show that as individuals increase their awareness the steady-state value of the final fraction of infected individuals in a susceptible-infected-susceptible (SIS) model decreases. We also extend our results to a spatial framework, incorporating a spatially-defined theoretical contact network (social network) and we made the awareness parameter dependent on a global or local contact structure. Our results show that even when individuals increase their awareness of the disease, the spatial structure itself defines the steady state solution of the system, in which more connected networks (networks with random or constant degree distributions) results in a population with no change in their behavior. Our work then shows that explicitly incorporating dynamics about the behavioral response dynamics might significantly change the predicted course of the epidemic and therefore highlights the importance of accounting for this source of variation in the epidemiological models.

Traditional infectious disease transmission models allocate the population into 2 compartments that captures different disease states, and aim to parametrize the rates of 3 transition between those states in a manner that reflects the underlying biology of the 4 disease [1, 2] . Numerous factors influence transmission and are important to consider or 5 model directly in epidemiological models. On of those is the understanding of the effect 6 of individuals behavior on the disease population dynamics, which has recently been 7 highlighted as a response to reduce contact rates and therefore pathogen transmission 8 across populations [3] . Typically, as a disease spreads in a population, individual 9 behavior can change and therefore infection probability can be reduced or 10 amplified [4] [5] [6] [7] . Recently, changes in individual contact rates driven by changes in the 11 has been discussed and explicitly modeled [8, 9] . Although, these models do not 12 incorporate an explicit mechanism by which individuals could modify their behavior. 13 The have demonstrated how behavioral aspects play an essential role in disease 14 dynamics. Over time, behavior is expected to vary as population-level disease awareness 15 is modulated by increases in risk and the proportion of the population that has been 16 infected (e.g., risk awareness might be highest near the peak of the epidemic) [9] . 17 However the impact of behavior-time dynamics in controlling transmission is likely 18 intertwined with other variables that directly impact transmission, and can be 19 estimated in a time-variable contact rate. Recently, COVID-19 have highlighted the 20 importance of sustained social distancing to reduce infection risk within the population 21 so health systems capacity is not saturated [3] [4] [5] . As epidemics unfold, individuals 22 amass information provided by public health institutions, concerning the status of an 23 epidemic or are strongly influenced by beliefs of the disease in their population. This 24 heterogeneity in the levels of information gathered in certain communities, could 25 modulate the level of adherence to certain interventions. A classic example is the Ebola 26 outbreak in Sierra Leone in where risk communication played an essential role in 27 controlling transmission due to high infection probability given contact with and 28 infectious individual of this disease [CITE 29 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5782897/pdf/17-1028.pdf, ]. 30 Mechanistic models that aim to explain dynamics of human behavior often rely on a 31 game theory framework, where behavior is represented by the strategy that an 32 individual adopts in a decision-making process [10] [11] [12] [13] . However, most coupled 33 epidemiological-game theory models assume static games, and therefore do not capture 34 the dynamics of how individuals change their behavior over time, as an epidemic how different behavioral strategies generate a payoff (i.e., advantage in evolutionary 38 fitness), they model decisions across individuals and therefore how the adoption of a 39 given strategy changes their frequency in time [16] [17] [18] [19] . To date, models that accounts 40 for behavioral dynamics have not been coupled with epidemiological models to address 41 the evolution of behavioral strategies in connections with disease transmission.

Both epidemiological and behavioral dynamics have been studied under space-time 43 couplings resulting in complex dynamics depending on the contact or social network 44 where interactions between individuals take place [7, 20] . This highlights the importance 45 of understanding both coupled and dynamical models in defined social networks where 46 contacts represent interaction between individuals-nodes. Understand how networks 47 topology could modulate disease awareness and ultimately how epidemics unfold [21, 22] . 48 In this work, we introduce a novel framework for coupling both epidemiological and 49 behavioral models. Our framework consists of a traditional epidemiological model [2] 50 and a replicator dynamics model [10, 11] .By combining these two approaches we were 51 able to dynamically couple individuals behaviors with transmision intensity. Our work 52 shed lights on the interaction between social and behavioral dynamics affecting the 53 epidemiology of the disease. Specifically, our results show that even when individuals 54 increase the disease awareness, the contact network itself defines the steady state 55 solution of the system. In addition, more connected networks (networks with random or 56 constant degree distributions) results in a population with no change in their behavior. 57 Our work shows that explicitly incorporating responsive behavioral dynamics can 58 significantly change the predicted course of an epidemic and highlights the importance 59 of accounting for this source of variation.

Materials and methods 61 We study these coupled epidemiological and social dynamics using two dyfferent models: 62 (1) We couple two Ordinary Differential Equations systems (ODEs), one to describe the 63 state transitions in the epidemiological model and the other to describe the replicator (Here cooperation is understood as the strategy that reduce epidemiological risk but not 70 necessarily provides the highest payoff). Then, following the rationale that individuals 71 who do not cooperate might have a higher FOI, we discount the defector 72 (non-cooperator) payoff by the fraction of infected individuals I/N times a parameter 73 that we call the awareness σ. (2) Our other approach was couple the epidemiological 74 dynamics in an explicit contact network, G where nodes represent the individuals and 75 the edges represent the contacts [6] . We then sequentially update the replicator 76 equation within the network. Similar to the coupled ODE models, we assume the FOI is 77 inversely proportional to the number of cooperating individuals however as the network 78 model considers contacts of an individual we attempt to model individual access to the 79 state of infectious individuals in two settings that we call global and local information. 80 This follows the rationale that individuals can be informed of the disease at two 

We consider an SIS (susceptible -infected -susceptible) model (Eq. 4) which is a 88 variation of the Kermack & McKendrick model for diseases without or with short term 89 immunity, which forms the basis of almost all the communicable disease models studied. 90 In our SIS model, the population is divided into two classes, where Susceptible (S) can 91 be infected by those already Infected (I) and subsequently become susceptible again due 92 to lack of immunity [2] . We then model strategic interaction between individuals using 93 the replicator dynamics (RD) a concept from evolutionary game theory where 94 population's behavioral traits are described using biologically inspired operations such 95 as natural selection [10, 11, 23] . Under this dynamics, the percentage growth rate of 96 cooperatorsċ/c is equal to the excess of payoff respect to the populations payoff. We set 97 f c and f n as the fitness of cooperators and non-cooperators respectively. Therefore, the 98 average fitness can be obtained asf = c · f c + n · f n .

We then model the payoff using the Prisoners Dilemma (PD), an archetypical model 100 displaying the conflict between selfishness and public good [24] . It is a 2 × 2 game with 101 two strategies cooperate or non-cooperate. The payoff modeling pair meetings is defined 102 in Eq. 2. Here we set S = 0.5 and T = −0.5 as in the Prisoners Dilemma [24] . Note 103 that the row referring to the payoff that receives non-cooperator individuals is 104 discounted by the awareness of the disease g(σ, I).

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021. ;  Then the fitness of each strategy f c and f n can be written as shown in Eq. 3. Note 106 that the fitness is modeled as the expected payoff given current frequency of cooperators 107 c and non-cooperators n. Similarly the average fitnessf is then the expected given the 108 fitness of each one of the populations.

In the typical PD the non-cooperator strategy is the dominating strategy, but the 

The matrix payoff governing the dynamics of the game are then given by Eq. 2. We 125 assume non-cooperators' payoff is discounted by the awareness of the disease given the 126 current epidemic state g(σ, I). The level of consciousness that a population exhibits 127 named σ, and the awareness population given the current epidemic state is defined by 128 Eq. 5. Finally, the transmission probability is assumed to be inversely proportional to 129 the fraction of cooperators in the population, one might think in a family of functions 130 for modeling this but by simplicity we only assume contact rate decays exponentially 131 with the cooperator fraction c as is shown in the Eq. 6. Additionally this functional 132 response only needs to be parametrized with the nominal contact rate named β max in a 133 population where behavior is assumed to not impact transmission at all (i.e. σ = 0 an 134 unconscious population). . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021. ; https://doi.org/10.1101/2021.04.13.21255395 doi: medRxiv preprint regardless of their state of the disease, susceptible or infected. However, considering 139 explicit social networks give additional insight about the role of contacts, usually 140 studied within theoretical constructed networks [8, 25] . We implemented a SIS model on 141 a network assuming now susceptible individuals get infected by their infected neighbors 142 (individuals with common edges) by a infection probability β i and recover from the 143 disease with probability P r = 1/γ where γ is the recovery rate as used in the ODEs 144 model Eq. 4. Then each individual i has a state in the disease named Susceptible S or 145 Infected I as described by Eq. 7.

To extend the behavioral dynamics on a network we again use the imitation 147 dynamics update rule given a pay-off matrix as the one described by Eq. 2. We assume 148 that individuals tend to adopt the strategy of more successful neighbors, where success 149 is measured with the payoff as described before, more successful individuals correspond 150 to the ones with greatest payoff in a time step of the simulation. Then successful 151 individuals are pushed to both balance the social dilemma between cooperate or non 152 given the current state of the epidemic. Similar to the replicator dynamics, mean field 153 approximation assumes a two strategy game whose update rule is described by Eq. 8

where the probability of changing from the strategy used by the player j to the strategy 155 used by the player i, is given by the differences of payoffs of each player u j and u i 156 respectively and player irrationality K, which we set to 0.5.

Therefore the payoff of player i comes from playing the same symmetric two-player 158 game defined by the used 2 × 2 payoff matrix. In this case her total income can be 159 expressed as Eq. 9.

Where s i is the unit vector describing the the pure strategy played by the individual 161 i and the summation runs on the individual neighbors j ∈ N i defined by the network 162 structure [17, 20] . The infection rule is given by the infection probability of each node 163 β i , which similarly as the ODEs model this node infection probability will be inversely 164 proportional to the number of cooperating individuals in her neighborhood β i = β(c Ni ). 165 Again for simplicity we only assume that infection probability decays exponentially with 166 the fraction of cooperators in her neigh-boors as is shown in the Eq. 10. This functional 167 response again can only be parametrized using the nominal infection probability named 168 β max .

The social dynamics will consequently depend on the current state of the disease. 170 We discount the payoff of non-cooperator individuals by the awareness parameter σ and 171 the current state of the disease g(σ, I). However for modeling the effect on local vs 172 global spread of the information we assume this discount factor will depend on the state 173 of the disease in neighboring nodes (local information) or in the whole population 174 (global information) as described by Eq. 11. This approximation follows the rationality 175 that disease dynamics are governed by local outbreaks (infection among nodes and their 176 neighbors) as is shown in Fig. 1B . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021. ;  g(σ, I) =σ (11) Simulation and steady state analysis 178 For both models we assume individuals in average are infectious for 7 days and therefore 179 the recovery rate is γ = 1/7. We first study the steady state of both ODEs and network 180 models and characterize the dynamics of the system in the steady state. We calculated 181 the fraction of infected individualsĪ = I/N and the fraction of cooperators c. Given x(T − i)

To seed the initial conditions we set the initial condition as 50% of the population as 188 cooperators. Note that as the non-cooperator strategy is the dominant strategy in the 189 payoff matrix this model is not sensible to the initial conditions. This seeding follows [20] suggesting that in steady state hubs of nodes might favor cooperation. 206 We therefore analyze how was the disease dynamics in those clusters. To determine the 207 effect of transmissibility of the disease we consider five values of 208 R 0 ∈ {6.3, 4.2, 3.1, 2.1, 1.5}. We do not consider R 0 values less than 1. In SI we include 209 further information of the clustering algorithm used. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021. ;  parameters R 0 and σ respectively. Temporal dynamics of both infected and cooperators 234 exhibit interesting changing behavior as transmissibility increased, leading the 235 dynamical system to their steady state faster [2, 26] . Then suggesting behavior 236 dynamics, i.e. cooperation will also be faster as a response of reducing the transmission 237 of the disease. We then study time dynamics under three levels of consciousness named 238 full, partial and medium consciousness as we indicated in the methods section and three 239 levels of transmission that relates pathogens from normal transmission ∼2 or highly 240 transmitted ∼6. Specifically we use R 0 ∈ {2.1, 4.2, 6.3} this is shown in Fig. 3 . Each . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021. ; https://doi.org/10.1101/2021.04.13.21255395 doi: medRxiv preprint cooperation (sustain it in time) in highly connected nodes [20] . Therefore, we 245 hypothesize that if both infected fraction and cooperators fraction might change across 246 scale-free network clusters. In consequence we search for clusters using a node clustering 247 algorithm on the network and plot the fraction of infected individuals; cooperating. Fig. 4 shows the temporal dynamics for T = 150 days after the seeded 253 index case. The clusters computed are also shown at top left side of Fig 4; note that we 254 only consider the three biggest hubs (top three clusters detected with more nodes). We 255 show the temporal dynamics in Fig. 4B , C, here the row B correspond to a maximum 256 level of awareness and the row C to a medium level of awareness, we do not show attempt to study clustering structure in the small-world or grid networks as their degree 265 distribution is uniform and therefore the hub might correspond to the whole network. 266 We restrict the analysis of hub dynamics to the scale-free graph. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021 Social dynamics play an essential role in the evolution of epidemics and the unfolding of 269 a disease across a population [6, 8] . Specifically, awareness of the state of the epidemic 270 or disease can result in different shapes or patterns in the epidemic curve [9, 27] . The 271 latter implies that the structure of population also plays an important role in both the 272 evolution of disease spread and social dynamics [20] . Real heterogeneous networks such 273 as the scale-free networks are used to resemble real world ones and have been shown to 274 favor cooperation in a social dilemma [20] . We developed a theoretical framework that 275 couples two models typically used for modeling spread of a pathogen in a population the 276 SIS model and for modeling the evolution cooperation in social dilemmas [2, 24] . To our 277 knowledge it is the first infectious disease model to couple social behavior with disease 278 dynamics, using replicator dynamics to examine how awareness of the disease state We found that the steady state of the network model varies greatly depending on the 299 network structure. For the small-world and grid network 2C and 2D we found that the 300 steady state of infected individuals 2C.1 and 2D.1 does not change as a function of the 301 awareness level σ of the population. We believe that this behavior emerges as a result of 302 the symmetrical spread of the disease on the graph, also as the degree distribution is 303 uniform across nodes the pathogen spread easily to all the network no matter the initial 304 nodes infected.

Comparing the final fraction of cooperators in the small-world versus the grid graph 306 Fig 2C.2 and Fig 2D. 2 respectively, we observed that despite the transmission strength 307 R 0 or the consciousness σ nearly 25% of the population cooperates (yellow color in the 308 heat-map). This is confirmed visually by looking at the steady state in Fig. 3 and Fig 7. 309 We hypothesize that these dynamics emerge by the natural structure of the contact 310 network, which naturally pushes the system towards cooperation and sustaining it in 311 time, as has been shown for other real world networks [20] .

Our framework accounts dynamically changes in the behavior of an individual, 313 diminishing their own risk of infection as well as the risk of those with whom it 314 interacts [28] . To incorporate this in the models, we considered individual-level contact 315 patterns to capture behavior towards a disease outbreak. This allows an analysis of the 316 disease and behavioral dynamics in complex population structures, and permits explicit 317 demographic predictions that can be studied for public health interventions [29] . We . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 19, 2021. ; https://doi.org/10.1101/2021.04.13.21255395 doi: medRxiv preprint disease and behavioral assumptions set for the R0 and risk awareness.

In general, we have evaluated the effect that communities have over the spread of 321 cooperative strategies towards the mitigation of a disease. We highlight the importance 322 of dense population structures in maintaining cooperative regime, which in turn 323 decreases the epidemic size within the whole population. This crucial characteristic of 324 heterogeneous scale-free graphs is absent in homogeneous scatter and small-world 325 graphs, where this population architecture fail to achieve cooperation in order to 326 counter the spread of an infectious disease. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 19, 2021. ; where d(V ) is the total degree of the graph [32, 33] .

Awareness-driven behavior changes can shift the shape of epidemics away from peaks and toward plateaus, shoulders, and oscillations

Containing papers of a mathematical and physical character

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When individual behaviour matters: homogeneous and network models in epidemiology

Fast unfolding of communities in large networks

Community detection in social networks

Community detection in large-scale social networks: state-of-the-art and future directions. Social Network Analysis and Mining

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We would like to acknowledge Pallavi Kache, Alejandro Feged-Rivadeneria, Tomas Rodríguez Barraquer and Pablo Cárdenas for their thoughtful comments on the manuscript.