Age-differentiated incentives for adaptive behavior during epidemics produce oscillatory and chaotic dynamics

Ronan F. Arthur; May Levin; Alexandre Labrogere; Marcus W. Feldman

doi:10.1371/journal.pcbi.1011217

Abstract

Heterogeneity in contact patterns, mortality rates, and transmissibility among and between different age classes can have significant effects on epidemic outcomes. Adaptive behavior in response to the spread of an infectious pathogen may give rise to complex epidemiological dynamics. Here we model an infectious disease in which adaptive behavior incentives, and mortality rates, can vary between two and three age classes. The model indicates that age-dependent variability in infection aversion can produce more complex epidemic dynamics at lower levels of pathogen transmissibility and that those at less risk of infection can still drive complexity in the dynamics of those at higher risk of infection. Policymakers should consider the interdependence of such heterogeneous groups when making decisions.

Author summary

Behavior during epidemics is driven by incentives, which may vary among different age classes. Here we include age-differentiated incentives in an adaptive behavioral epidemic model with two and three age classes. Our results indicate that this heterogeneity can drive the system into oscillations, chaos, and collapse. Those with less incentives to adopt protective behavior for themselves (i.e., the youth) can still drive epidemic dynamics and associated complexity in those who have more incentives to avoid contacts during epidemics (i.e., the elderly). This heterogeneity and interdependence should be taken into account when making public health policy that affects individual contact rates during an epidemic.

Citation: Arthur RF, Levin M, Labrogere A, Feldman MW (2023) Age-differentiated incentives for adaptive behavior during epidemics produce oscillatory and chaotic dynamics. PLoS Comput Biol 19(9): e1011217. https://doi.org/10.1371/journal.pcbi.1011217

Editor: Feng Fu, Dartmouth College, UNITED STATES

Received: May 26, 2023; Accepted: August 11, 2023; Published: September 5, 2023

Copyright: © 2023 Arthur et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript. Code for the modeling can be found at the following website: https://github.com/levinmay/adaptiveBehavior.

Funding: RA received support from the John Templeton Foundation, grant #61809. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors report no competing interests.

Introduction

One of the principal failings of the attempts to model and predict future trends and dynamics of infectious disease epidemics has been the lack of incorporation of human behavior into these models [1]. The social drivers of infectious disease dynamics have been relatively neglected in the academic literature compared with the vast resources and attention paid to the biomedical and, to a lesser extent, the ecological drivers of disease [2, 3]. COVID-19, however, has shone a spotlight on this discrepancy as socio-cultural factors have played an important role in the pandemic—from the political polarization of risk perception in the United States [4] to the social, cultural, and demographic factors associated with vaccine hesitancy [5]. Models should incorporate relevant social phenomena, as well as their interactions, as the study of each phenomenon in isolation may not be informative or useful. For example, adaptive age-specific preventative behaviors motivated by differentiated risk of COVID-19 mortality should be included. The resulting contact patterns and associated infection dynamics are expected to interact in their influence on epidemic trajectories.

The mortality rate of COVID-19 is highly age-specific [6] with a log-linear increase of infection fatality ratio by age among individuals over 30 [7]. This contributes to skewed risk perceptions across the population’s age structure and economic activity associated with COVID-19 risk [8, 9]. Heterogeneity in contact patterns, mortality rates, and transmissibility among and between different age classes is known to have significant effects on epidemic outcomes [10, 11]. In Germany, for example, younger adults and teenagers were likely the main drivers of COVID-19 transmission dynamics during the first three pandemic waves [12]. Goldstein et al. [13] found that vaccinating the elderly first against COVID-19 would save the most lives, although the model neglected key features of transmission dynamics [14]. Further, Acemoglu and colleagues showed that optimal lockdown policies during COVID-19 are age-specific, with strict lockdown policies on the oldest group and reduction of interactions between age classes being most effective [15]. Their model, however, assumed exogenous targeted policies and did not incorporate adaptive behavior, a cornerstone of COVID-19 dynamics.

As risk of transmission of a dangerous infection increases during an epidemic, individuals and governments tend to react by mitigating that risk with adaptive behavior. Adaptive behavior has enjoyed growing attention in the disease modeling literature [16, 17], with techniques using game theory [18], fear-infection parallel contagions [19, 20], and network- and agent-based approaches [21]. Early work by Capasso et al. [22] experimented with the introduction of a negative feedback mechanism in the traditional susceptible-infected-recovered (SIR) model [23] and represented reduced contacts as a function of the number of infected. Philipson formalized the economic theory of adaptive behavior showing that rational agents following dynamic incentives may lead to oscillations around an indifference point, or equilibrium [24]. Adaptive behavior can cause oscillatory dynamics because a system with a negative feedback can continuously overshoot adjustments [25, 26].

Adaptive behavior may also lead to complex dynamics that are characteristic of deterministic, chaotic systems, especially with delays in adaptive response [27]. Empirical evidence from the COVID-19 pandemic suggests that the behavior of the epidemic has been chaotic in a majority of countries [28]. An investigation of the second derivative of infections over time during COVID-19 found that the pandemic qualitatively met Henrí Poincaré’s criteria for chaos in deterministic dynamical systems: a large number of solutions, dynamic sensitivity, and numerical unpredictability [29]. Measles models show chaotic dynamics that may also characterize the observed dynamics [30, 31]. In the SEIR framework with a periodically varying contact rate that represented seasonal changes, sustained oscillations [32] and period-doubling bifurcations [33] were found, and one case with a particularly high degree of contact led to chaotic dynamics [34]. While age structure heterogeneity in susceptibility and seasonal variance in contact rates due to school attendance characterize measles modeling [35, 36], age structure heterogeneity in dynamic, adaptive contact rates have not been modeled for COVID-19.

Here, we model an infectious disease with adaptive behavior incentives of the form used in a previous model [27] and include different mortality rates for three age classes: the “young” (ages 20-49), the “middle-aged” (50-64), and the “old” (65+) (à la Acemoglu et al. [15]).

Model specifications

We begin with a susceptible-infected-susceptible (SIS) compartmental disease model [37], which includes an adaptive contact behavior that maximizes a utility function, as in Arthur et al. [27]. With susceptible and infectious individuals, denoted by S_t and I_t, respectively, at time t, and N_t the population size, we have: (1) (2) (3) where b₀ represents transmissibility given contact, γ represents the removal rate, and represents a contact rate at time t chosen to maximize utility U(c) as a function of contacts: (4) Here, c represents contacts per unit time, represents the optimal contact rate when the disease risk is non-existent, α₁ represents the utility lost for deviating from contacts and α₂ represents the utility lost if infected. Here, Δ represents delayed information, such that an individual may base their perception of infection risk on prevalence during past time periods, rather than the current one. Maximizing Eq 4 with respect to c yields , the contact rate chosen at each time step to maximize utility.

To incorporate age structure, we stratify the population into k discrete age classes, represented by A₁, …, A_k. The number in sub-population A_i is given by N_i. We assume the size of each age class does not change: N_it = N_i and let . The utility function (as in Eq 4) for each age class i is given by (5) where α_1i represents the utility loss of reduced contacts for A_i, represents the target contact rate for A_i, and α_2i represents the utility loss (i.e., aversion) to infection based on a delayed perception of population-level prevalence for A_i. Here it is assumed that each age class perceives its risk according to the disease prevalence of the whole population (i.e., ), rather than just of their own group.

Interactions between and within age classes at time t are defined in terms of a dynamic contact matrix M_t (censu Ram & Schaposnik, 2021 [38]), (6) where represents the within-group contact of A_i, optimized at time step t to maximize utility in A_i (as in Eq 4), and c_ij represents the contact between A_i and A_j for i ≠ j. For simplicity, institutional behavior change is assumed to only affect the within-group contact (e.g., via school and workplace closures and nursing home quarantines), and thus, between-group contact rates are assumed fixed and not adaptive to changing infection risks. It is assumed that c_ij = c_ji.

Using the contact matrix (Eq 6), the transition between susceptible and infected disease states for age class A_i, as in Eqs 1–3, can be expressed as (7) (8) (9)

We assume the following constraints:

The aversion to infection is greatest in the elderly and least in the young, i.e., (10)
The target contact rate is greatest in the young and least in the elderly, i.e., (11)
The recovery rate from the infected category to the susceptible is the same for all age classes, namely, for all i, j, (12)
The number of infecteds in any age class A_i may never be greater than the population size of that class or less than zero, i.e., for all i and t, (13)

Analytical results

Equilibria

To understand the dynamic behavior of the number of infecteds in each age class, we first examine conditions for the existence of equilibria. If I_tb₀ is small relative to N, then on linearizing with respect to I in Eq 5, the optimal value of c_i at time t for A_i is found to be (14)

Define the parameter α_i as (15)

We begin with a 2-age-class model, where A₁ represents the youth and A₂ represents the middle-aged and elderly, in which case, Eqs 6–13, with k = 2, become (16) (17) (18) with S_1t + I_1t = N₁, S_2t + I_2t = N₂, and N = N₁ + N₂.

For simplicity, we assume the population sizes of all groups are equal (i.e., ). Then, substituting from Eq 14 and α₁ from Eq 15, with Δ = 0, we obtain (19) (20) (21)

By symmetry, (22) (23)

A fixed point (i.e., equilibrium) exists for A_i when I_i(t+1) = I_it for i = 1, 2. Thus, equilibria are the roots of the two simultaneous polynomial equations (24) and (25)

Stability

To analyze local stability, we calculate the Jacobian of the differential equations corresponding to Eqs 21 and 22, where I_1(t+1) = f₁(I_1t, I_2t) and I_2(t+1) = f₂(I_2t, I_2t) and evaluate the Jacobian at the equilibrium. Differentiating each function with respect to each variable I₁ and I₂, (26) (27) (28) (29)

Computational results

The 2-population model

We first examine the numerical iteration of the discrete time SIS recursions (Eqs 21 and 22) without time-delay, and set default values for all parameters. No empirical research has analyzed comparative utility lost or gained as a function of contacts. Thus, the parameter set has been chosen to demonstrate possible dynamic regimes, rather than to parameterize COVID-19 or other infectious diseases. These default parameters are: N₁ = 5000, N₂ = 5000, I₀ = 1, γ₁ = 0.1, γ₂ = 0.1, , , c₁₂ = 0.005, α₁ = 1, α₂₁ = 20, α₂₂ = 40, b₀ = 0.009.

By increasing the transmissibility parameter b₀ from the default value, we see a progression of dynamical regimes across critical thresholds from simple convergence to cyclic behavior in 2, 4, and 6-point cycles, chaos, and collapse (Figs 1, 2 and 3). By increasing the contact rate between the 2 populations, c₁₂, there is a progression from simple convergence to a 2-pt cycle, 6-pt cycle, chaos, back to a 6-pt cycle, and finally an asynchronous 8-pt and 2-pt cycle (Figs 4 and 5). Bifurcation diagrams in Figs 2 and 5 indicate sustained dynamic patterns for a continuous parameter change in either b₀ (Fig 2) or c₁₂ (Fig 5). When c₁₂ = 0 and is only based on prevalence in the same age category, we see what would happen in a homogeneous population without age structure with the same parameters (Fig 6). Without this interdependent age-structure, Youth dynamics are even more complex, while Elderly dynamics have less complexity.

Download:

Fig 1. Number of infected individuals over time for two-population age-structured model using default parameters and varying transmissibility b₀.

A) b₀ = 0.009, Convergence to stable equilibria; B) b₀ = 0.013, 2-point cycle; C) b₀ = 0.02, 4-point cycle; D) b₀ = 0.021, chaos; E) b₀ = 0.0235, 6-pt cycle; F) b₀ = 0.025, Collapse.

https://doi.org/10.1371/journal.pcbi.1011217.g001

Download:

Fig 2. Bifurcation diagram of equilibria, oscillatory dynamics, and chaotic behavior as a function of transmissibility b₀.

Points in the figure represent sustained long-term results in the model; thus any b₀ value with, for example, two corresponding y-axis points represents a sustained oscillation between two values for infected individuals.

https://doi.org/10.1371/journal.pcbi.1011217.g002

Download:

Fig 3. Phase plane of non-zero equilibrium and stability.

Heat maps of dynamic regimes as a function of the relationship between aversion to infection in the youth (α₂₁) and in the elderly (α₂₂). Dynamic regimes are indicated by color and A–F are increasing in transmissibility (b₀): A) b₀ = 0.009; B) b₀ = 0.013; C) b₀ = 0.02; D) b₀ = 0.021; E) b₀ = 0.0235; F) b₀ = 0.025.

https://doi.org/10.1371/journal.pcbi.1011217.g003

Download:

Fig 4. Number of infected individuals over time for two-population age-structured model using default parameters and varying between-group contact rate c₁₂.

A) c₁₂ = 0.005, Convergence to stable equilibria; B) c₁₂ = 0.025, 2-point cycle; C) c₁₂ = 0.04, 6-point cycle; D) c₁₂ = 0.05, chaos; E) c₁₂ = 0.08, 6-pt cycle; F) c₁₂ = 0.15, 8-pt cycle for Old and 2-pt cycle for Young.

https://doi.org/10.1371/journal.pcbi.1011217.g004

Download:

Fig 5. Bifurcation diagram of equilibria, oscillatory dynamics, and chaotic behavior as a function of inter-group contact rate c₁₂.

Points in the figure represent sustained long-term results in the model; thus any c₁₂ value with, for example, two corresponding y-axis points represents a sustained oscillation between two values for infected individuals.

https://doi.org/10.1371/journal.pcbi.1011217.g005

Download:

Fig 6. Number of infected individuals for Youth and Elderly populations over time with default parameters, each population size set to N, no between-group contact (i.e., c₁₂ = 0), and varying transmissibility b₀.

A) b₀ = 0.009; B) b₀ = 0.013 =,; C) b₀ = 0.02; D) b₀ = 0.021; E) b₀ = 0.0235; F) b₀ = 0.025.

https://doi.org/10.1371/journal.pcbi.1011217.g006

The aversion to infection for the elderly and the youth given by α₂₂ and α₂₁, respectively, may produce more complex dynamics at higher values, but the relationship between the two values is also important (Fig 3). Simpler dynamics (e.g., equilibrium and 2-point cycles) can be found closer to the line y = x (i.e., α₂₂ = α₂₁), while highly differentiated aversions to infection are more likely to produce chaos or oscillations of higher periodicity.

The 3-population model

For our 3-age-class model, A₁ represents the youth, A₂ represents the middle-aged, and A₃ represents the elderly. Then the analogous equations to Eqs 6–13, for k = 3, are (30) (31) (32) (33) with (34)

Equilibria for the system defined by Eqs 31–33 solve I_i(t+1) = I_it for i = [1, 2, 3] and are the roots of three simultaneous polynomial equations.

We set default values for parameters and initial conditions, such that N₁ = 5000, N₂ = 5000, N₃ = 5000, I₀ = 1, γ₁ = 0.1, γ₂ = 0.1, γ₃ = 0.1, , , , c₁₂ = 0.005, c₁₃ = 0.003, c₂₃ = 0.007, α₁ = 1, α₂₁ = 20, α₂₂ = 30, α₂₃ = 40, b₀ = 0.01.

By increasing the transmissibility b₀, the model goes from simple convergence to a 2-point cycle to chaos into a 2-pt cycle to collapse (Fig 7). By uniformly increasing the between-group contact rates c₁₂, c₁₃, and c₂₃, the model exhibits convergence, a 2-pt cycle, a 4-pt cycle, chaos, a 5-pt cycle, and a 6-pt cycle (Fig 8). The amplitude of oscillations and the variance of chaotic dynamics are greatest for the elderly population and least for the youth population. When a time-delay is introduced (Δ = 1), dynamic complexity appears at smaller levels of b₀ (see Fig 9) than when Δ = 0 (Fig 7).

Download:

Fig 7. Number of infected individuals over time in each of three populations using default parameters and varying transmissibility b₀.

A) b₀ = 0.01, Convergence to stable equilibria; B) b₀ = 0.013, 2-point cycle; C)b₀ = 0.022, Chaos into a 2-pt cycle cycle; D) b₀ = 0.05, Collapse.

https://doi.org/10.1371/journal.pcbi.1011217.g007

Download:

Fig 8. Number of infected individuals over time in each of three populations using default parameters and varying between-group contact rates c₁₂, c₁₃, and c₂₃.

A) c₁₂ = 0.005, c₁₃ = 0.003, c₂₃ = 0.007, Convergence to stable equilibria; B) c₁₂ = 0.008, c₁₃ = 0.01, c₂₃ = 0.01, 2-point cycle; C) c₁₂ = 0.022, c₁₃ = 0.02, c₂₃ = 0.024, 4-point cycle; D) c₁₂ = 0.032, c₁₃ = 0.03, c₂₃ = 0.034, Chaos; E) c₁₂ = 0.05, c₁₃ = 0.048, c₂₃ = 0.052, 5-pt cycle; F) c₁₂ = 0.06, c₁₃ = 0.058, c₂₃ = 0.062, 6-pt cycle.

https://doi.org/10.1371/journal.pcbi.1011217.g008

Download:

Fig 9. Disease dynamics for the 3-population model when delta = 1.

Number of infected individuals over time in each of three populations using default parameters, time-delay Δ = 1, and varying transmissibility b₀. A) b₀ = 0.01, Damped oscillator; B) b₀ = 0.013, Chaos; C) b₀ = 0.022, Collapse; D) b₀ = 0.05, Collapse.

https://doi.org/10.1371/journal.pcbi.1011217.g009

Discussion

Our model has generalized previous work on adaptive behavior during epidemics to include age structure with heterogeneous aversion to infection and target contact rates. Results from the 2- and 3-population adaptive behavior models indicate that, for certain parameter values, stable equilibria can be reached for each sub-population i, including 0 for all age groups and an equilibrium between 0 and N_i. In the 2-population case, the equilibrium for each population can be derived numerically. With increasing values of b₀ and c_ij (the transmissibility and between-group contact rate, respectively), the system may converge to a non-zero equilibrium, oscillate perpetually with 2-, 4-, 6-, 8-, and 10-period intervals, become chaotic, and collapse. Complexity of dynamics can be found at lower levels of transmissibility with greater differentiation between aversions to infection (Fig 3). For both the 2-population and 3-population models, the younger population has a higher non-zero equilibrium size than the older population, and the older population has greater amplitude of oscillations and variance of chaotic dynamics than the younger population.

Our model is built on a number of simplifying assumptions. First, the model is of the SIS type and does not consider an Exposed class (E) or a Recovered class (R). The inclusion of E or R (in, e.g., SEIRS models) may decrease the number of susceptible individuals and thus slow some of the oscillatory or chaotic dynamics found here. The inclusion of R for SIR models depletes susceptibles and prevents sustained dynamic patterns. Risk tolerance and reactions to shifting prevalence are stratified by age class, but assumed homogeneous within each class. In reality, individuals would have heterogeneous risk tolerance according to their age, political affiliation, and other characteristics. We modeled adaptive behavior with dynamic within-group contact rates, but assumed between-group contact rates were fixed. While this was justified by COVID-19 public health policies that controlled within-group contact more than between-group contact, between-group contact is also responsive to prevalence. Expanding this model to finer age categories than the ones used in this paper should include empirically-derived contact rates and differentiated optimization for between-group contacts. Further, we constrained the model with four mathematical assumptions (Eqs 10–13), some of which may not always hold in real-world scenarios. The relaxation of the above assumptions may yield different model outcomes. We note the model was not trained on real-world data to produce parameters as empirical data on adaptive behavior and incentives is lacking. While the dynamical regimes found here exist in theory, we do not know where in practice real-world disease systems may be. However, COVID-19 exhibited oscillatory patterns and chaotic dynamics [28], suggesting that this theoretical work may be directly relevant to diseases with strong adaptive behavior components.

Without contact with other groups, the Elderly age class exhibits simpler dynamics (e.g., monotonic convergence) (Fig 6). While the older population has a higher risk associated with infection, the younger population’s lower aversion and higher baseline contact rates affect the epidemic dynamics in the older population. Thus, transmission in the young may not only lead to transmission in the elderly, but also increase the variance of the elderly dynamics and destabilize them. Indeed, even at lower levels of infection aversion for the elderly, the more differentiated the aversions to infection between the two age classes, the more complex the dynamics (Fig 3). Children are known to be the primary drivers of Influenza transmission, although severe morbidity and mortality are mostly seen in older age groups [39]. The importance of the youth in the dynamics of our model provides theoretical support for COVID-19 lockdown policies that reduce between-group interaction (i.e., c_ij), as found by Acemoglu et al. [15]. Transmission within the household is a key point of risk for the elderly and middle-aged, and high levels of transmission in the youth are likely to significantly affect disease outcomes in other age categories.

Our model may be usefully compared to other adaptive behavior models that look at sub-populations with differentiated characteristics or reactions to epidemic dynamics. It is worth noting that the justification for the bifurcated structure need not be restricted to age-based differentiation, but can also include political party affiliation, income levels, or other demographic, social, behavioral, physical, or geographic differences. Some studies use structured 2-population models with varying homophily and outgroup aversion or varying awareness separation and mixing separation [40, 41]. Results from these models indicate that heterogeneous populations, even when simply structured compartmentally as two populations, can produce greater complexity in epidemic dynamics, including large second waves and interconnected dual epidemics. Our results broadly agree with this theme: a 2-population model with varying infection aversion can produce more complexity at lower levels of transmissibility, and the difference between aversions can also drive complexity (e.g., Fig 3).

In our previous work [27], we compared the complex dynamics associated with endogenous behavior change during epidemics to similar theoretical behavior in ecological systems. For example, the logistic map is an extensively-studied simple model of population growth with limited resources that exhibits similar dynamics to our adaptive behavior models: convergence, oscillations, chaos, and collapse [42]. The driver of these dynamics in the logistic map is r, the growth rate. In our model, two of the principal drivers are b₀, the transmissibility, and c_ij, the between-group contact rate, as the epidemic’s reproduction number R₀ is directly proportional to transmissiblity and contact rate. Often, when a single population is modeled in ecology (e.g., in a fishery [43]), the carrying capacity operates as an attractor, above which the population is attracted downwards and below which the population is attracted upwards. With endogenous behavior, the equilibrium, or indifference point, is also an attractor, below which the population is motivated to relax protective behaviors and above which the population is motivated to adopt protective behaviors. It may be productive to extend this parallel further in the case of multiple populations. For example, the predator-prey model considers two interdependent populations. When the prey population is high, the predator population grows, but when the prey are low in number, the predators decrease. While our epidemiological model does not include such direct competition or predation, the behavior is somewhat similar: when the young population has high prevalence of disease, the prevalence in the older population increases until the indifference point is crossed. When the young population has low prevalence, the prevalence in the older population follows suit. If a comparison between ecology and epidemiology is appropriate, it follows that careful study of the literature in theoretical ecology may provide insights relevant to epidemiology, a field with as yet comparatively little exploration of such complex system dynamics.

We recommend further theoretical work in both adaptive behavior modeling and complexity in epidemiology. A systems perspective may better represent the inherent complexities and heterogeneities of real-world epidemics. Social drivers of disease have been relatively neglected in the literature [2], but played an important role in COVID-19 outcomes. For example, the divergence of risk assessment by age class that characterizes our model was a social phenomenon, though biologically motivated. Other complex phenomena important to COVID-19 include simultaneous asynchronous epidemics, political bifurcation of attitudes and practices, and the co-evolution of the human immune system and the virus. As public health policies depend on our ability to forecast different scenarios under a high degree of uncertainty and complexity, such modeling will play an important role in improving policy and health outcomes in future epidemics.

References

1. Bedson J, Skrip A, Pedi D, Abramowitz S, Carter S, Jalloh MF, et al. A review and agenda for integrated disease models including social and behavioural factors. Nat Hum Behav. 2021; 5(7): 834–846. pmid:34183799
- View Article
- PubMed/NCBI
- Google Scholar
2. Arthur RF, Gurley ES, Salje H, Bloomfield LSP, Jones JH. Contact structure, mobility, environmental impact and behavior: the importance of social forces to infectious disease dynamics and disease ecology. Phil Trans Roy Soc B. 2017; 372: 20160454. pmid:28289265
- View Article
- PubMed/NCBI
- Google Scholar
3. Ferguson N. Capturing human behaviour. Nature. 2007 Apr; 446: 733. pmid:17429381
- View Article
- PubMed/NCBI
- Google Scholar
4. Kerr J, Panagopoulos C, van der Linden S. Political polarization on COVID-19 pandemic response in the United States. Pers Indiv Difer. 2021; 179: 110892. pmid:34866723
- View Article
- PubMed/NCBI
- Google Scholar
5. Latkin C, Dayton LA, Yi G, Konstantopoulos A, Park J, Maulsby C, et al. COVID-19 vaccine intentions in the United States, a social-ecological framework. Vaccine 2021; 39(16): 2288–2294. pmid:33771392
- View Article
- PubMed/NCBI
- Google Scholar
6. Bonanad C, Garcia-Blas S, Tarazona-Santabalbina F, Sanchis J, Bertomeu-Gonzalez , Facila L, et al. The effect of age on mortality in patients with COVID-19: a meta-analysis with 611,583 subjects. J Am Med Dir Assoc. 2020; 21(7): 915–918. pmid:32674819
- View Article
- PubMed/NCBI
- Google Scholar
7. O’Driscoll M, Ribeiro Dos Santos G, Wang L, Cummings DAT, Azman AS, Paireau J, et al. Age-specific mortality and immunity patterns of SARS-CoV-2. Nature. 2021; 590(7844): 140–145. pmid:33137809
- View Article
- PubMed/NCBI
- Google Scholar
8. Bundorf MK, DeMatteis J, Miller G, Polyakova M, Streeter JL, Wivagg J. Risk perceptions and protective behaviors: evidence from COVID-19 pandemic. Working paper. NBER. 2021 Apr.
9. Korn L, Siegers R, Eitze S, Sprengholz P, Taubert F, Boehm R, et al. Age differences in COVID-19 preventive behavior: a psychological perspective. Eur Psychol. 2021; 26(4): 359–372.
- View Article
- Google Scholar
10. Busenberg S, Cooke K, Iannelli M. Endemic thresholds and stability in a class of age-structured epidemics. SIAM J App Math. 1988; (48): 1379–1395.
- View Article
- Google Scholar
11. Inaba H. Threshold and stability results for an age-structured epidemic model. J Math Biol. 1990; 28: 411–434. pmid:2384720
- View Article
- PubMed/NCBI
- Google Scholar
12. Rodiah I, Vanella P, Kuhlmann A, Jaeger V, Harries M, Krause G, et al. Age-specific contribution of contacts to transmission of SARS-CoV-2 in Germany. Eur J Epidemiol. 2023 Jan; 38: 39–58. pmid:36593336
- View Article
- PubMed/NCBI
- Google Scholar
13. Goldstein JR, Cassidy T, Wachter KW. Vaccinating the oldest against COVID-19 saves both the most lives and most years of life. Proc Natl Acad Sci USA. 2021; 118(11):e2026322118. pmid:33632802
- View Article
- PubMed/NCBI
- Google Scholar
14. Dushoff J, Colijn C, Earn DJD, Bolker BM. Transmission dynamics are crucial to COVID-19 vaccination policy. Proc Natl Acad Sci USA. 2021; 118(29): e2105878118. pmid:34230099
- View Article
- PubMed/NCBI
- Google Scholar
15. Acemoglu D, Chernozhukov V, Werning I, Whinston MD. Optimal targeted lockdowns in a multigroup SIR model. Am Econ Rev: Insights. 2021 Dec; 3(4): 487–502.
- View Article
- Google Scholar
16. Fenichel EP, Castillo-Chavez C, Ceddia MG, Chowell G, Parra PAG, Hickling GJ, et al. Adaptive human behavior in epidemiological models. Proc Natl Acad Sci USA. 2011; 108(15): 6306–6311. pmid:21444809
- View Article
- PubMed/NCBI
- Google Scholar
17. Funk S, Salathé M, Jansen VAA. Modelling the influence of human behaviour on the spread of infectious diseases: a review. J Roy Soc Interface. 2010; 7(50): 1247–1256. pmid:20504800
- View Article
- PubMed/NCBI
- Google Scholar
18. Reluga TC. Game theory of social distancing in response to an epidemic. PLoS Comp Biol. 2010; 6(5): 1–9.
- View Article
- Google Scholar
19. Epstein JM, Parker J, Cummings D, Hammond R. Coupled contagion dynamics of fear and disease: Mathematical and computational explorations. PLoS ONE 2008; 3(12): e3955. pmid:19079607
- View Article
- PubMed/NCBI
- Google Scholar
20. Epstein JM., Hatna E, Crodelle J. Triple contagion: A two-fears epidemic model. J Roy Soc Interface. 2021 Aug; 18(181): 20210186. pmid:34343457
- View Article
- PubMed/NCBI
- Google Scholar
21. Bobashev GV, Goedecke DM, Yu F, Epstein J. A hybrid epidemic model: combining the advantages of agent-based and equation-based approaches. In: Henderson SG, Biller B, Hsieh MH, Shortle J, Tew JD, Barton RR, eds. Proceedings of the 2007 Winter Simulation Conference. 2007; pp.1532–1537.
22. Capasso V, Serio G. A generalization of the Kermack-McKendrick deterministic epidemic model. Math Biosci. 1978; 42(1-2): 43–61.
- View Article
- Google Scholar
23. McKendrick WKA. A contribution to the mathematical theory of epidemics. Proc Roy Soc London A. 1927; 115(772): 700–721.
- View Article
- Google Scholar
24. Philipson T. Economic epidemiology and infectious diseases. In: Culyer AJ, Newhouse JP, eds, Handbook of Health Economics, 2000; Volume 1B, Chapter 33, pp 1761–1799.
- View Article
- Google Scholar
25. D’Onofrio A, Manfredi P. Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases. J Theor Biol. 2009; 256(3): 473–478. pmid:18992258
- View Article
- PubMed/NCBI
- Google Scholar
26. Glaubitz A, Fu F. Oscillatory dynamics in the dilemma of social distancing. Proc Proc Roy Society A. 2020; 476(2243): 20200686.
- View Article
- Google Scholar
27. Arthur RF, Jones JH, Bonds MH, Ram Y, Feldman MF. Adaptive social contact rates induce complex dynamics during epidemics. PLoS Comp Biol. 2021; 17(2): 1–17. pmid:33566839
- View Article
- PubMed/NCBI
- Google Scholar
28. Sapkota N, Karwowski W, Davahli MR, Al-Juaid A, Taiar R, Murata A, et al. The chaotic behavior of the spread of infection during the COVID-19 pandemic in the United States and globally. IEEE Access. 2021; 9: 80692–80702. pmid:34786316
- View Article
- PubMed/NCBI
- Google Scholar
29. Jones A, Strigul N. Is spread of COVID-19 a chaotic epidemic? Chaos Solitons Fractals. 2021 Jan; 142: 110376. pmid:33100605
- View Article
- PubMed/NCBI
- Google Scholar
30. Bolker B, Grenfell BT. Space, persistence and dynamics of measles epidemics. Phil Trans Roy Soc London B. 1995; 348(1325): 309–320. pmid:8577828
- View Article
- PubMed/NCBI
- Google Scholar
31. Grenfell BT. Chance and chaos in measles dynamics. J Roy Stat Soc B. 1992; 54(2): 383–398.
- View Article
- Google Scholar
32. Dietz K. The incidence of infectious diseases under the influence of seasonal fluctuations. In: Berger J, Bühler WJ, Repges R, Tautu P, editors, Mathematical Models In Medicine. Lecture Notes in Biomathematics, Volume 11. 1976; Springer, Berlin, Heidelberg.
33. Aron JL, Schwartz IB. Seasonality and period-doubling bifurcations in an epidemic model. J Theor Biol. 1984; 110(4): 665–679. pmid:6521486
- View Article
- PubMed/NCBI
- Google Scholar
34. Schaffer WM. Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology? IMA J Math Appl Med. 1985; 2(4): 221–252. pmid:3870986
- View Article
- PubMed/NCBI
- Google Scholar
35. Anderson RM, May RM. Vaccination against rubella and measles: quantitative investigations of different policies. Epidemiol Infect. 1983; 90(2): 259–325. pmid:6833747
- View Article
- PubMed/NCBI
- Google Scholar
36. Schenzle D. An age-structured model of pre-and post-vaccination measles transmission. Math Med Biol. 1984; 1(2): 169–191. pmid:6600102
- View Article
- PubMed/NCBI
- Google Scholar
37. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc Roy Soc London A. 1927; 115(772) 700–721.
- View Article
- Google Scholar
38. Ram V, Schaposnik LP. A modified age-structured SIR model for COVID-19 type viruses. Sci Rep. 2021; 11: 15194. pmid:34312473
- View Article
- PubMed/NCBI
- Google Scholar
39. Brownstein JS, Kleinman KP, Mandl KD. Identifying pediatric age groups for influenza vaccination using a real-time regional surveillance system. Am J Epidemiol. 2005; 162(7): 686–693. pmid:16107568
- View Article
- PubMed/NCBI
- Google Scholar
40. Smaldino PE, Jones JH. Coupled dynamics of behaviour and disease contagion among antagonistic groups. Evol Hum Sci. 2021 (online); 3: e28. pmid:37588564
- View Article
- PubMed/NCBI
- Google Scholar
41. Harris MJ, Cardenas KJ, Mordecai EA. Social divisions and risk perception drive divergent epidemics and large later waves. Evol Hum Sci. 2023 (online); 5: e8. pmid:37587926
- View Article
- PubMed/NCBI
- Google Scholar
42. Ausloos M, Dirickx M, editors. The logistic map and the route to chaos: From the beginnings to modern applications. 2006. Springer, Berlin, Heidelberg.
43. Ricker W. Stock and recruitment. J Fish Res Board Canada. 1954; 11(5): 559–623.
- View Article
- Google Scholar

[ref1] 1. Bedson J, Skrip A, Pedi D, Abramowitz S, Carter S, Jalloh MF, et al. A review and agenda for integrated disease models including social and behavioural factors. Nat Hum Behav. 2021; 5(7): 834–846. pmid:34183799
View Article
PubMed/NCBI
Google Scholar

[2] View Article

[3] PubMed/NCBI

[4] Google Scholar

[ref2] 2. Arthur RF, Gurley ES, Salje H, Bloomfield LSP, Jones JH. Contact structure, mobility, environmental impact and behavior: the importance of social forces to infectious disease dynamics and disease ecology. Phil Trans Roy Soc B. 2017; 372: 20160454. pmid:28289265
View Article
PubMed/NCBI
Google Scholar

[6] View Article

[7] PubMed/NCBI

[8] Google Scholar

[ref3] 3. Ferguson N. Capturing human behaviour. Nature. 2007 Apr; 446: 733. pmid:17429381
View Article
PubMed/NCBI
Google Scholar

[10] View Article

[11] PubMed/NCBI

[12] Google Scholar

[ref4] 4. Kerr J, Panagopoulos C, van der Linden S. Political polarization on COVID-19 pandemic response in the United States. Pers Indiv Difer. 2021; 179: 110892. pmid:34866723
View Article
PubMed/NCBI
Google Scholar

[14] View Article

[15] PubMed/NCBI

[16] Google Scholar

[ref5] 5. Latkin C, Dayton LA, Yi G, Konstantopoulos A, Park J, Maulsby C, et al. COVID-19 vaccine intentions in the United States, a social-ecological framework. Vaccine 2021; 39(16): 2288–2294. pmid:33771392
View Article
PubMed/NCBI
Google Scholar

[18] View Article

[19] PubMed/NCBI

[20] Google Scholar

[ref6] 6. Bonanad C, Garcia-Blas S, Tarazona-Santabalbina F, Sanchis J, Bertomeu-Gonzalez , Facila L, et al. The effect of age on mortality in patients with COVID-19: a meta-analysis with 611,583 subjects. J Am Med Dir Assoc. 2020; 21(7): 915–918. pmid:32674819
View Article
PubMed/NCBI
Google Scholar

[22] View Article

[23] PubMed/NCBI

[24] Google Scholar

[ref7] 7. O’Driscoll M, Ribeiro Dos Santos G, Wang L, Cummings DAT, Azman AS, Paireau J, et al. Age-specific mortality and immunity patterns of SARS-CoV-2. Nature. 2021; 590(7844): 140–145. pmid:33137809
View Article
PubMed/NCBI
Google Scholar

[26] View Article

[27] PubMed/NCBI

[28] Google Scholar

[ref8] 8. Bundorf MK, DeMatteis J, Miller G, Polyakova M, Streeter JL, Wivagg J. Risk perceptions and protective behaviors: evidence from COVID-19 pandemic. Working paper. NBER. 2021 Apr.

[ref9] 9. Korn L, Siegers R, Eitze S, Sprengholz P, Taubert F, Boehm R, et al. Age differences in COVID-19 preventive behavior: a psychological perspective. Eur Psychol. 2021; 26(4): 359–372.
View Article
Google Scholar

[31] View Article

[32] Google Scholar

[ref10] 10. Busenberg S, Cooke K, Iannelli M. Endemic thresholds and stability in a class of age-structured epidemics. SIAM J App Math. 1988; (48): 1379–1395.
View Article
Google Scholar

[34] View Article

[35] Google Scholar

[ref11] 11. Inaba H. Threshold and stability results for an age-structured epidemic model. J Math Biol. 1990; 28: 411–434. pmid:2384720
View Article
PubMed/NCBI
Google Scholar

[37] View Article

[38] PubMed/NCBI

[39] Google Scholar

[ref12] 12. Rodiah I, Vanella P, Kuhlmann A, Jaeger V, Harries M, Krause G, et al. Age-specific contribution of contacts to transmission of SARS-CoV-2 in Germany. Eur J Epidemiol. 2023 Jan; 38: 39–58. pmid:36593336
View Article
PubMed/NCBI
Google Scholar

[41] View Article

[42] PubMed/NCBI

[43] Google Scholar

[ref13] 13. Goldstein JR, Cassidy T, Wachter KW. Vaccinating the oldest against COVID-19 saves both the most lives and most years of life. Proc Natl Acad Sci USA. 2021; 118(11):e2026322118. pmid:33632802
View Article
PubMed/NCBI
Google Scholar

[45] View Article

[46] PubMed/NCBI

[47] Google Scholar

[ref14] 14. Dushoff J, Colijn C, Earn DJD, Bolker BM. Transmission dynamics are crucial to COVID-19 vaccination policy. Proc Natl Acad Sci USA. 2021; 118(29): e2105878118. pmid:34230099
View Article
PubMed/NCBI
Google Scholar

[49] View Article

[50] PubMed/NCBI

[51] Google Scholar

[ref15] 15. Acemoglu D, Chernozhukov V, Werning I, Whinston MD. Optimal targeted lockdowns in a multigroup SIR model. Am Econ Rev: Insights. 2021 Dec; 3(4): 487–502.
View Article
Google Scholar

[53] View Article

[54] Google Scholar

[ref16] 16. Fenichel EP, Castillo-Chavez C, Ceddia MG, Chowell G, Parra PAG, Hickling GJ, et al. Adaptive human behavior in epidemiological models. Proc Natl Acad Sci USA. 2011; 108(15): 6306–6311. pmid:21444809
View Article
PubMed/NCBI
Google Scholar

[56] View Article

[57] PubMed/NCBI

[58] Google Scholar

[ref17] 17. Funk S, Salathé M, Jansen VAA. Modelling the influence of human behaviour on the spread of infectious diseases: a review. J Roy Soc Interface. 2010; 7(50): 1247–1256. pmid:20504800
View Article
PubMed/NCBI
Google Scholar

[60] View Article

[61] PubMed/NCBI

[62] Google Scholar

[ref18] 18. Reluga TC. Game theory of social distancing in response to an epidemic. PLoS Comp Biol. 2010; 6(5): 1–9.
View Article
Google Scholar

[64] View Article

[65] Google Scholar

[ref19] 19. Epstein JM, Parker J, Cummings D, Hammond R. Coupled contagion dynamics of fear and disease: Mathematical and computational explorations. PLoS ONE 2008; 3(12): e3955. pmid:19079607
View Article
PubMed/NCBI
Google Scholar

[67] View Article

[68] PubMed/NCBI

[69] Google Scholar

[ref20] 20. Epstein JM., Hatna E, Crodelle J. Triple contagion: A two-fears epidemic model. J Roy Soc Interface. 2021 Aug; 18(181): 20210186. pmid:34343457
View Article
PubMed/NCBI
Google Scholar

[71] View Article

[72] PubMed/NCBI

[73] Google Scholar

[ref21] 21. Bobashev GV, Goedecke DM, Yu F, Epstein J. A hybrid epidemic model: combining the advantages of agent-based and equation-based approaches. In: Henderson SG, Biller B, Hsieh MH, Shortle J, Tew JD, Barton RR, eds. Proceedings of the 2007 Winter Simulation Conference. 2007; pp.1532–1537.

[ref22] 22. Capasso V, Serio G. A generalization of the Kermack-McKendrick deterministic epidemic model. Math Biosci. 1978; 42(1-2): 43–61.
View Article
Google Scholar

[76] View Article

[77] Google Scholar

[ref23] 23. McKendrick WKA. A contribution to the mathematical theory of epidemics. Proc Roy Soc London A. 1927; 115(772): 700–721.
View Article
Google Scholar

[79] View Article

[80] Google Scholar

[ref24] 24. Philipson T. Economic epidemiology and infectious diseases. In: Culyer AJ, Newhouse JP, eds, Handbook of Health Economics, 2000; Volume 1B, Chapter 33, pp 1761–1799.
View Article
Google Scholar

[82] View Article

[83] Google Scholar

[ref25] 25. D’Onofrio A, Manfredi P. Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases. J Theor Biol. 2009; 256(3): 473–478. pmid:18992258
View Article
PubMed/NCBI
Google Scholar

[85] View Article

[86] PubMed/NCBI

[87] Google Scholar

[ref26] 26. Glaubitz A, Fu F. Oscillatory dynamics in the dilemma of social distancing. Proc Proc Roy Society A. 2020; 476(2243): 20200686.
View Article
Google Scholar

[89] View Article

[90] Google Scholar

[ref27] 27. Arthur RF, Jones JH, Bonds MH, Ram Y, Feldman MF. Adaptive social contact rates induce complex dynamics during epidemics. PLoS Comp Biol. 2021; 17(2): 1–17. pmid:33566839
View Article
PubMed/NCBI
Google Scholar

[92] View Article

[93] PubMed/NCBI

[94] Google Scholar

[ref28] 28. Sapkota N, Karwowski W, Davahli MR, Al-Juaid A, Taiar R, Murata A, et al. The chaotic behavior of the spread of infection during the COVID-19 pandemic in the United States and globally. IEEE Access. 2021; 9: 80692–80702. pmid:34786316
View Article
PubMed/NCBI
Google Scholar

[96] View Article

[97] PubMed/NCBI

[98] Google Scholar

[ref29] 29. Jones A, Strigul N. Is spread of COVID-19 a chaotic epidemic? Chaos Solitons Fractals. 2021 Jan; 142: 110376. pmid:33100605
View Article
PubMed/NCBI
Google Scholar

[100] View Article

[101] PubMed/NCBI

[102] Google Scholar

[ref30] 30. Bolker B, Grenfell BT. Space, persistence and dynamics of measles epidemics. Phil Trans Roy Soc London B. 1995; 348(1325): 309–320. pmid:8577828
View Article
PubMed/NCBI
Google Scholar

[104] View Article

[105] PubMed/NCBI

[106] Google Scholar

[ref31] 31. Grenfell BT. Chance and chaos in measles dynamics. J Roy Stat Soc B. 1992; 54(2): 383–398.
View Article
Google Scholar

[108] View Article

[109] Google Scholar

[ref32] 32. Dietz K. The incidence of infectious diseases under the influence of seasonal fluctuations. In: Berger J, Bühler WJ, Repges R, Tautu P, editors, Mathematical Models In Medicine. Lecture Notes in Biomathematics, Volume 11. 1976; Springer, Berlin, Heidelberg.

[ref33] 33. Aron JL, Schwartz IB. Seasonality and period-doubling bifurcations in an epidemic model. J Theor Biol. 1984; 110(4): 665–679. pmid:6521486
View Article
PubMed/NCBI
Google Scholar

[112] View Article

[113] PubMed/NCBI

[114] Google Scholar

[ref34] 34. Schaffer WM. Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology? IMA J Math Appl Med. 1985; 2(4): 221–252. pmid:3870986
View Article
PubMed/NCBI
Google Scholar

[116] View Article

[117] PubMed/NCBI

[118] Google Scholar

[ref35] 35. Anderson RM, May RM. Vaccination against rubella and measles: quantitative investigations of different policies. Epidemiol Infect. 1983; 90(2): 259–325. pmid:6833747
View Article
PubMed/NCBI
Google Scholar

[120] View Article

[121] PubMed/NCBI

[122] Google Scholar

[ref36] 36. Schenzle D. An age-structured model of pre-and post-vaccination measles transmission. Math Med Biol. 1984; 1(2): 169–191. pmid:6600102
View Article
PubMed/NCBI
Google Scholar

[124] View Article

[125] PubMed/NCBI

[126] Google Scholar

[ref37] 37. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc Roy Soc London A. 1927; 115(772) 700–721.
View Article
Google Scholar

[128] View Article

[129] Google Scholar

[ref38] 38. Ram V, Schaposnik LP. A modified age-structured SIR model for COVID-19 type viruses. Sci Rep. 2021; 11: 15194. pmid:34312473
View Article
PubMed/NCBI
Google Scholar

[131] View Article

[132] PubMed/NCBI

[133] Google Scholar

[ref39] 39. Brownstein JS, Kleinman KP, Mandl KD. Identifying pediatric age groups for influenza vaccination using a real-time regional surveillance system. Am J Epidemiol. 2005; 162(7): 686–693. pmid:16107568
View Article
PubMed/NCBI
Google Scholar

[135] View Article

[136] PubMed/NCBI

[137] Google Scholar

[ref40] 40. Smaldino PE, Jones JH. Coupled dynamics of behaviour and disease contagion among antagonistic groups. Evol Hum Sci. 2021 (online); 3: e28. pmid:37588564
View Article
PubMed/NCBI
Google Scholar

[139] View Article

[140] PubMed/NCBI

[141] Google Scholar

[ref41] 41. Harris MJ, Cardenas KJ, Mordecai EA. Social divisions and risk perception drive divergent epidemics and large later waves. Evol Hum Sci. 2023 (online); 5: e8. pmid:37587926
View Article
PubMed/NCBI
Google Scholar

[143] View Article

[144] PubMed/NCBI

[145] Google Scholar

[ref42] 42. Ausloos M, Dirickx M, editors. The logistic map and the route to chaos: From the beginnings to modern applications. 2006. Springer, Berlin, Heidelberg.

[ref43] 43. Ricker W. Stock and recruitment. J Fish Res Board Canada. 1954; 11(5): 559–623.
View Article
Google Scholar

[148] View Article

[149] Google Scholar

Figures

Abstract

Author summary

Introduction

Model specifications

Analytical results

Equilibria

Stability

Computational results

The 2-population model

The 3-population model

Discussion

References