Constant contacts model
We assume the affected population is composed of two risk-groups, a fraction p of the population is composed of risk-takers (RT) and the remaining fraction (1-p) are risk-evaders (RE). We differentiate the RT and RE subpopulations by assuming the RE population face a reduced likelihood of infection due to adopting precautionary behaviors. On the other hand, we assume RT do not follow public health recommendations, thus facing a higher risk of infection, relative to the RE population. Political or ideological reasons, economic stress, the lack of reasonable alternatives, epidemic politicization or the lack of trust in public health authorities are some of the documented factors that potentially lead the population to risk the dangers of COVID-19 infection44, 45.
Previous mathematical models consider complex within-host disease dynamics46 or the impact of exogenous factors on the COVID-19 transmission dynamics47. In this study, we focus on incorporating individual heterogeneous adaptive behavioral responses, based on group-specific infection risk perceptions. Our model of disease progression assumes that individuals in each behavioral group may show the following health status: Susceptible (S), infectious Exposed (E), Infectious symptomatic (I), infectious Asymptomatic (A), and Recovered (R). We consider a pre-symptomatic infectious health status (E), following evidence suggesting that exposed individuals exhibit a period of viral shedding38, 48,49,50,51. RT susceptible individuals ((S_1)) can get infected by making contacts with either: symptomatic ones (I) with a baseline per-contact likelihood of disease transmission (beta), exposed individuals ((E_1) and (E_2)) with reduced per-contact likelihood of infection (rho beta) , or asymptomatic individuals ((A_1) and (A_2)) with reduced per-contact likelihood of infection (alpha beta). Similarly RE susceptible individuals ((S_2)) may get infected by making contacts with symptomatic, exposed or asymptomatic individuals at respective likelihoods, (epsilon beta), (rho epsilon beta), and (alpha epsilon beta), where (0<epsilon <1) corresponds to the reduction of the infection risk given by adopting precautionary behaviors. We assume (C^*) is the optimal contact rate in the absence of disease transmission, which remains constant in the absence of behavioral adaptations. Due to the absence of adequate data on the specific infectiousness of exposed and asymptomatic COVID-19 infected individuals49, we assume these subpopulations are less infectious than symptomatic ones, with (rho =0.25), and (alpha =0.4). Our model assumes that on average, (1/kappa) days after infection, a proportion (sigma) of exposed individuals remain asymptomatic, while the rest develop symptoms. Finally, we assume a similar infectious period of (1/gamma) days for symptomatic and asymptomatic individuals. Our model for disease progression is sketched in Fig. 1A, and mathematically formalized by the system of ordinary differential equations in Fig. 1B, where ({dot{f}}) stands for the time derivative of f.
Constant contacts disease model. The affected population is divided into two behavioral groups: risk-takers (RT) and risk evaders (RE), denoted with the subscripts 1 and 2, respectively. Thus, (S_1, E_1,) and (A_1) are the susceptible, exposed, and asymptomatic subpopulations of risk-takers, respectively; and similarly (S_2, E_2,) and (A_2) for risk evaders. We consider a single symptomatic subpopulation since we assume homogeneous behavior of individuals in this health class.
Heterogeneous adaptive behavior
We study the impact of interactions between infected individuals and susceptible individuals dynamically adapting their behavior in response to the perceived risk of infection. Our model focuses on infected individuals with mild or no symptoms and excludes individuals with severe symptoms, since these have minimal interaction with the general population. The economic-epidemiological model we use incorporates the interdependence between the epidemic burden and individual behavioral responses. Figure 2 shows a schematic of the coupling between the mean field epidemiological model and the Markov Decision Processes we use to model heterogeneous adaptive behavioral responses. We formulate a mean-field epidemiological model incorporating explicit contact rates to evaluate the progression of the epidemic, while simultaneously using a Markov Decision Framework to model individual adaptive behavioral responses.
Coupling disease dynamics and forward looking Markov Decision Processes. At each time step the sequential decision process sets a feedback loop between the epidemic state and individual behavioral responses: (i) the current disease prevalence defines potential future health state transitions, (ii) a projection of the system’s future state over the planning horizon sets the optimization problem, (iii) we find the population-specific optimal contact rates that maximize the expected utility of the susceptible population, over the planning horizon and simulate the epidemic model one step forward.
Notice that the optimization processes for RT and RE susceptible individuals are intrinsically coupled. The contact rates selected by a given population affect the overall population’s activity, which in turn impact the population’s mixing. At each time step, we decouple these processes by computing the optimal contact rate at time (t+1) for a given risk group, assuming the contact rate of the other risk-group to be the same as the one observed at time t, the latest sample available. At each time step, the group-specific optimization process incorporates a projection of the system’s future state, by assuming the current prevalence remains constant over the planning horizon. Variations on the future system’s state projection, and on the projection period length, deeply impact the solutions of the optimization problems, consequently impacting the outcomes of the behavioral responses and the epidemic dynamics. By solving the group-specific optimization process at each time-step, we get the privately optimal contact rate for each risk-group. This allows us to simulate the epidemic one step forward and to find the next step disease prevalence level. We iterate this process over the epidemic period to simulate the coevolution of the epidemic process and behavioral adaptations.
The paradigm in our model of human adaptive behavior is given by the trade-off between increasing contacts and the differential risk of infection that these carry for the RT and the RE populations. Individuals in each health class seek to maximize the expected utility due to social interactions, while trying to minimize infection risk, according to the transmission dynamics determined by the constant contact model.
Aside from the conditions that motivate heterogeneous behaviors, in our model individual behavior differs across health classes and risk groups, but individuals with similar health status and within the same risk group are assumed to behave similarly. We may expect some individuals to be satisfied with lower social activity than others however, for simplicity, we assume individual contact preferences have homogeneous and time-invariant structure. That is, individuals similarly value contacts over time, and independently of the epidemic state. Moreover, individual decentralized decisions are assumed to be taken from privately optimal perspectives. The cost-benefit trade-off perceived by individuals does not consider the aggregate effects that changing the pool of contacts has on others’ decisions.
We incorporate heterogeneous behavior in the constant contacts model by weighting the population in each health-class with the corresponding risk-group specific contact rates. Under the adaptive behavior model, the mixing is proportional to the population distribution among health-classes, and conditional on the behaviors determining the dynamic contact rates. Taking the constant contacts model as a baseline, we derive the incidence terms for the RT and RE susceptible individuals by considering the proportion of contacts that a typical individual in the (S_1) and (S_2) compartments makes with other infectious individuals,
$$begin{aligned}&beta C_t^{S_1} S_1 frac{rho (C_t^{E_1}E_1+C_t^{E_2}E_2) + alpha (C_t^{A_1} A_1+ C_t^{A_2} A_2) + C_t^{I} I}{sum _h C_t^h h}, text {and}nonumber &quad epsilon beta C_t^{S_2} S_2 frac{rho (C_t^{E_1}E_1+C_t^{E_2}E_2) + alpha (C_t^{A_1} A_1+ C_t^{A_2} A_2) + C_t^{I} I}{sum _h C_t^h h}, end{aligned}$$
(1)
where (sum _h C_t^h h) is the total population activity, for individuals in health classes (hin {S_1,S_2,E_1,E_2,I,A_1,A_2,R}) selecting contact rates ({C_t^{S_1},C_t^{S_2},C_t^{E_1},C_t^{E_2},C_t^{I},C_t^{A_1},C_t^{A_2},C_t^R}), at time t.
Since we assume economic productivity depends exclusively on social interactions, individuals determine the daily optimal contact choices at each time step by maximizing their expected utility (V_t(h)), depending on their current health status (hin {S_1,S_2,E_1,E_2,I,A_1,A_2,R}). The health-specific expected utilities (V_t(h)) comprise the potential benefits obtained by making the optimal contact choice at each future time step during the group-specific planning horizon (tau _i). The expected utilities account for potential future transitions to other health states, weighted by the respective transition probabilities, which are given by the system’s current state (the population distribution among health states and their respective contact choices). Individuals evaluate the future benefits/costs assuming the population distribution remains constant during the planning periods. Preferences are assumed single-peaked, so that individuals have a unique optimal contact rate in the absence of disease dynamics. Following the work by Morin et al.24, we assume a utility function of the particular form (u(C_t^h)=left( b C_{t}^{h}-(C_{t}^{h})^ 2right) ^{nu }) , where b is the per-day maximum number of contacts possible, (nu) is the utility function shape parameter, and (C_{t}^{h}) is the contact rate of a typical individual with health status h at time t. We assume individuals within the same risk-group obtain benefits based on a time-invariant utility function shape, regardless of their health status, except symptomatic individuals who get no utility during the infectious period. We use variations of the (nu) parameter to modify the marginal benefits of increasing contacts across groups. In other words, our model of heterogeneous behavior assumes individuals across groups show differential disposition to reduce their daily number of contacts. Moreover, we assume risk assessment remains constant over time, therefore the risk-group-specific utility function remains invariant over the epidemic period.
Finally, we incorporate the role of uncertain information on the decision-making process. We let the perceived health status represent a source of information uncertainty, where non-symptomatic individuals (exposed and asymptomatic), unaware of the infection risk they pose to others, may perceive themselves—and be perceived by others—as not presenting a risk of infection25.
Susceptible, exposed, and asymptomatic individual behavior
We model the susceptible individuals’ daily optimal contact choice problem as a dynamic programming problem, the solution to which generates the privately optimal contact rate22,23,24,25. Note that, regardless of the behavioral group, individuals follows a SEIAR disease progression across health states. Therefore, a single set of Bellman’s equations formulates the optimization problem for both behavioral groups, where behavioral heterogeneity is captured by accordingly changing the health state transition probabilities. Formally, susceptible individuals’ daily optimal contact rate solves the Bellman’s equation,
$$begin{aligned} V_t(S_i)=max _{C_t^{S_i}}Big {ubig (C_t^{S_i}big )+delta big [(1-P^{S_iE_i})V_{t+1}(S_i)+P^{S_iE_i}( V_{t+1}(E_i)) big ] Big }, end{aligned}$$
(2)
where (V_t(S_i)) is the expected utility of risk group i susceptible individuals at time t, (V_{t+1}(S_i)) ((V_{t+1}(E_i))) is the expected utility of being susceptible (exposed) at time (t+1), and
$$begin{aligned} P^{S_1E_1}left( C_t^{S_1}right) =1-exp left( -beta C_t^{S_1} frac{rho (C_t^{E_1}E_1+C_t^{E_2}E_2)+alpha (C_t^{A_1} A_1+ C_t^{A_2} A_2) + C_t^{I} I}{sum _h C_t^h h}right) end{aligned}$$
(3)
is the probability of being infected at time t for RT individuals. Since RT and RE individuals have similar disease progressions, Eq. (2) also holds for RE, by adjusting the respective infection risk ((epsilon beta)) and the corresponding contact rate ((C_t^{S_2})), such that
$$begin{aligned} P^{S_2E_2}left( C_t^{S_2}right) = 1-exp left( -epsilon beta C_t^{S_2} frac{rho (C_t^{E_1}E_1+C_t^{E_2}E_2)+alpha (C_t^{A_1} A_1+ C_t^{A_2} A_2) + C_t^{I} I}{sum _h C_t^h h}right) . end{aligned}$$
(4)
The optimization problem formalized in Eq. (2) incorporates RT (RE) susceptible individuals’ immediate utility ( (u(C_t^{S_i})) ), plus the expected future utility discounted at a rate (delta). The susceptible individuals’ expected future utility accounts for the expected utility of remaining susceptible at the next time step, (V_{t+1}(S_i)), with probability (1-P^{S_iE_i}) and the expected utility of being infected (V_{t+1}(E_i)) (progressing to the (E_i) compartment), with probability (P^{S_iE_i}).
Notice that the solution of the optimization problem for susceptible individuals depends upon the expected utility of exposed individuals. Similarly to Eq. (2), we formulate the Bellman’s equation for exposed and asymptomatic individuals
$$begin{aligned} V_t(E_i) = u(C_t^{S_i})+delta big [(1-P^{E_i})V_{t+1}(E_i) + P^{E_i}big (sigma V_{t+1}(A_i)+(1-sigma )V_{t+1}(I)big ) big ], end{aligned}$$
(5)
where (P^{E_i}=1-e^{-kappa }) stands for the probability of progressing from the (E_i) health class to either (A_i) or I, with respective probabilities (sigma) and (1-sigma), and where the expected utility of asymptomatic individuals is given by,
$$begin{aligned} V_{t}(A_i) = u(C_t^{S_i})+delta big [(1-P^{A_iR}) V_{t+1}(A_i)+P^{A_iR}V_{t+1}(R)big ], end{aligned}$$
(6)
with (P^{A_iR}=1-e^{-gamma }) representing the probability of recovery.
We assume the absence of symptoms leads exposed and asymptomatic individuals to perceive themselves (and be perceived by others) as susceptible individuals, therefore becoming a source of uncertain information. This is incorporated in their corresponding immediate utilities on Eqs. (5) and (6), by using the term (u(C_t^{S_i})), where susceptible, exposed, and asymptomatic individuals within the same risk group choose their contact rates in the same way. In other words, we track individual risk-avoidance efforts over health-classes except while infected and recovered.
Symptomatic and recovered individual behavior
Since our model for disease progression does not consider potential reinfections, we assume there is no incentive for symptomatic and recovered individuals to adapt their behavior. It follows that symptomatic and recovered individuals make the daily number of contacts that maximizes their net benefits. The expected utility of symptomatic individuals, (V_t(I)), is given by the Bellman’s equation,
$$begin{aligned} V_{t}(I)=u(C_t^{*})+delta big [(1-P^{IR})V_{t+1}(I)+P^{IR}V_{t+1}(R) big ], end{aligned}$$
(7)
while the expected utility of recovered individuals, (V_t(R)), is formalized by,
$$begin{aligned} V_{t}(R)=u(C_t^{*})+delta V_{t+1}(R), end{aligned}$$
(8)
where (P^{IR}=1-exp ^{-gamma }) is the recovery probability.
Note that symptomatic and recovered individual utility expectations represent static problems, since they only depend upon the recovery rate and can be explicitly solved. Although we have not explicitly included a potential contact rate reduction of symptomatic individuals, for instance, due to altruism or sanctions, we can model it by limiting the maximum contacts available for this subpopulation.
Source: Ecology - nature.com
