Universal Nonlinear Infection Kernel from Heterogeneous Exposure on Higher-Order Networks

Abstract

The collocation of individuals in different environments is an important prerequisite for exposure to infectious diseases on a social network. Standard epidemic models fail to capture the potential complexity of this scenario by (1) neglecting the higher-order structure of contacts that typically occur through environments like workplaces, restaurants, and households, and (2) assuming a linear relationship between the exposure to infected contacts and the risk of infection. Here, we leverage a hypergraph model to embrace the heterogeneity of environments and the heterogeneity of individual participation in these environments. We find that combining heterogeneous exposure with the concept of minimal infective dose induces a universal nonlinear relationship between infected contacts and infection risk. Under nonlinear infection kernels, conventional epidemic wisdom breaks down with the emergence of discontinuous transitions, superexponential spread, and hysteresis.

Mathematical models of epidemics play an increasingly important role in public health efforts and pandemic preparedness [1]. By providing insights on the interplay of the biological and sociological aspects of epidemics, models can test potential interventions in silico and suggest potential outcomes [2]. However, large-scale forecasting comparisons show that statistical models often outperform mechanistic models that make assumptions about spreading dynamics [3].

In this Letter, we look at the interplay of two commonly used assumptions in disease models: Random mixing and the linearity between infection risk and exposures to infected individuals. In almost all disease models, doubling the number of contacts between susceptible and infectious individuals doubles the risk of infection for the susceptible individuals. Some past work in mathematical biology has considered nonlinear infection rates [4,5], but these models are rarely used in practice. Other fields such as sociology do often consider generalized contagion models, often dubbed complex contagions [6,7]. In these complex contagions, having a nonlinear relationship between infection rate and sources of infection allows the model to consider mechanisms such as social reinforcement [8], where a set of multiple exposures can have more impact than the mere sum of unique exposures.

The mathematical convenience of assuming random mixing when modeling infectious diseases comes at the price that all contacts between susceptible and infectious individuals are effectively equivalent. This assumption has often been lifted using heterogeneous mathematical models where individuals are distinguished by some individual features such as their intrinsic susceptibility or reaction to the infection [9,10], relaxing the mass-action assumption directly [11,12], or by specifying an underlying contact network [13,14].

Moreover, the linearity assumption says that all increments in the total exposure to infectious individuals (measured for example as a viral inoculum) are equivalent. Evidence associated with the minimal infective dose of different infectious diseases shows that not all exposures are equal and that some minimal dose might be required for an infection to likely occur. More precisely, the ID50 value is a measure of the dose needed to cause an infection in 50% of individuals. These concepts are needed because our immune system is usually able to handle microscopic challenges from viruses and bacteria alike. While an infective dose of tuberculosis might only require between 1 and 5 bacteria [15], some enterics might require up to 10⁹ pathogenic particles [16], and others like common respiratory infections still require further study [17]. There are indeed multiple different physical mechanisms behind immune evasion—for example some airborne viruses need to find their receptors on lung epithelial cells, while some bacteria might instead require interaction with the immune system [18]. These mechanisms are reviewed in Refs. [18–22], and all of them combine to determine the ID50 of specific pathogens. Likewise, the decay or clearing rates of pathogens in noninfectious courses can also vary a lot, potentially requiring days for bacteria to hours for airborne viral infections. For example, mathematical models for the pathogenesis of SARS-CoV-2 or influenza A use decay rates of the order of 7–18 hours but empirical estimates vary wildly (see Refs. [23,24] and references therein).

To study the effect of simultaneously relaxing these two assumptions, we consider a social structure where individuals attend a certain number of environments such as work places, gyms, or supermarkets. This division of contact structure in environments is motivated by the known role of superspreading events, which are, for example, critical to the ongoing spread of COVID-19 [25–32]. While variations at the individual level are often used to explain superspreading [33], we focus here on the variability of environments and of temporal patterns [34–40] at the group level, which undoubtedly affects epidemics [27], especially when a certain exposure within a certain time window is needed to confidently spark an infection. Interestingly, available case data highlight how there is no expected size or duration for such events. Transmission is highly context dependent on the settings (e.g., ventilation) and activity (e.g., singing or shouting) such that the resulting superspreading events are heterogeneous in size, duration, and attack rate, as shown in Fig. 1(a). Higher-order contact structures and heterogeneous temporal patterns are therefore key ingredients for more realistic models of spreading dynamics.

FIG. 1.

Modeling contagions and superspreading events through higher-order networks. (a) Scatter plot of superspreading events of COVID-19 where the number of people involved (size), the duration of the event, and the resulting proportion of infected individuals (attack rate) are all available (extracted from Refs. [41,42]; see the Supplemental Material [43]). (b),(c) Framework for contagions on hypergraphs [52], where the size m of the hyperedges (environment), the hyperdegree k of the nodes (individuals), and the participation time to the environment τ are all heterogeneous, distributed according to $\widehat{P}(m)$, $\tilde{P}(k)$, and P(τ), respectively. For the sake of simplicity, we assume the same distribution P(τ) for all environments. (b) At each time step t, an individual participates for a time τ (drawn independently) to each environment. (c) An individual gets infected with probability θ_m(ρ) in the environment at time step t, which depends on the size m and the fraction infected ρ.

Mathematically, we represent the contact structure as a hypergraph [53–55] where each environment is described by a hyperedge connecting m nodes (individuals) and where each node is incident to k hyperedges. All hyperedges of a same size m are considered equivalent, although this assumption is relaxed in the Supplemental Material [43] to consider additional sources of heterogeneity. To model heterogeneous temporal patterns, we consider a discrete-time process, where at each time step t = 1, 2, …, we draw for each individual a participation time τ ∈ [1, τ_max] for each environment to which they are connected [Fig 1(b)]. The time steps correspond to fixed temporal windows of size τ_max, during which susceptible individuals can get infected through their participation to environments.

We first study the impact of the spatiotemporal collocation patterns on the infection kernel θ_m(ρ), the probability of getting infected in an environment of size m when a fraction ρ of the other participants are infectious [Fig. 1(c)]. We then analyze the properties of the resulting contagion process.

Universal infection kernel from heterogeneous exposure.—

Let us consider a susceptible individual participating to an environment of size m for a duration τ, where a fraction ρ of the other participants are infectious. During this exposure period, some of the other m − 1 individuals might participate to the environment as well. We assume that the considered individual receives an infective dose κ ∈ [0, ∞) from the infectious individuals, distributed according to π(κ; λ), where λ ≡ ⟨κ⟩. A reasonable assumption is that the mean dose received is proportional to the time spent in the environment and to the proportion of infectious people, λ = βf(m)τρ, where β is a rate of dose accumulation and f(m), unitless, modulates the typical number of contacts in environments frequented by m individuals.

While this is not a strict requirement for our results to hold (see the Supplemental Material [43]), we further assume that the random variable for the dose can be written as κ = λu, where u is a random variable that is independent of λ. In this case, u can be seen as an intrinsic property of the contagion process—determined by rates of viral shedding, diffusion in the environment, variability of human interactions, etc.—while λ acts as a scale parameter, i.e., π(κ; λ) = π(κ/λ; 1)/λ ≡ π(κ/λ)/λ.

To incorporate the concept of minimal infective dose, we assume that an individual develops the disease if κ > K, a perspective analogous to standard threshold models [56–58] and related to the assumption that successful host invasion necessitates multiple attempts by the pathogen [59]. The probability of getting infected in the environment is then

$$\overline{\mathrm{\Pi }}(K/\lambda )={\int }_{K/\lambda }^{\infty }\pi (\kappa )d\kappa .$$ (1)

The infection kernel θ_m(ρ) is calculated by averaging $\overline{\mathrm{\Pi }}(K/\lambda )$ over P(τ). We focus here on the case of heterogeneous exposure periods modeled with a Pareto distribution P(τ) = C_ατ^−α−1, where C_α is a normalization constant, α > 0, and τ ∈ [1, τ_max]. However, for our dose mechanism to be well defined, we can only average over participation times $\tau \in [1,\mathcal{T}]$, where $\mathcal{T}\le {\tau }_{\max }$ is the “clearing window,” i.e., the characteristic time for the immune system to get rid of any dose κ < K. If we assume that this clearing window is sufficiently large compared to τ_c ≡ K/βf(m)ρ, the characteristic time to get infected in the environment, we can neglect events where $\tau \ge \mathcal{T}$ as they do not contribute significantly to the infection kernel [43]. We therefore redefine our support as $\tau \in [1,\mathcal{T}]$ and the infection kernel is

$$\begin{array}{l}{\theta }_m(\rho )={\int }_1^{\mathcal{T}}\overline{\mathrm{\Pi }}\left({\tau }_c/\tau \right)P(\tau )d\tau \\ =\frac{C_{\alpha }}{\alpha }\left[\overline{\mathrm{\Pi }}\left({\tau }_c\right)-\overline{\mathrm{\Pi }}\left({\tau }_c/\mathcal{T}\right){\mathcal{T}}^{-\alpha }+{\tau }_c^{-\alpha }{\int }_{{\tau }_c/\mathcal{T}}^{{\tau }_c}\pi (y)y^{\alpha }dy\right].\end{array}$$ (2)

When $1\ll {\tau }_c\ll \mathcal{T}$ and π(y) decreases faster than y^−α−1, then the integral on the right converges to a constant, the term in ${\mathcal{T}}^{-\alpha }$ can be neglected, and $\overline{\mathrm{\Pi }}\left({\tau }_c\right)\ll {\tau }_c^{-\alpha }$, which implies

$${\theta }_m(\rho )~D_{\alpha }{\tau }_c^{-\alpha }\propto {\rho }^{\alpha },$$ (3)

where D_α is some constant. This form of infection kernel is universal, driven by temporal patterns, and does not depend on the value of K (given K > 0) or on the particular form of π. We illustrate it in Fig. 2(a) using an exponential for the dose distribution—other cases such as the Weibull and the Fréchet distribution are presented in the Supplemental Material [43].

FIG. 2.

Properties of contagions with nonlinear infection kernels induced by heterogenous exposure. We use an exponential dose distribution π(κ; λ) ∝ e^−κ=λ with a power-law distribution of participation time P(τ) ∝ τ^−α−1, a clearing window $\mathcal{T}\to \infty$, and f(m) = 1. (a) Effective infection kernel using β = 0.1. The infection probability has a power law scaling θ_m(ρ) ∝ ρ^α. (b),(c) We use Poisson distributions for both $\tilde{P}(k)$ and $\widehat{P}(m)$, with ⟨k⟩ = 5 and ⟨m⟩ = 10, and set μ = 0.05. We use Eqs. (4)–(6) to evolve the system. (b) Supralinear kernels ν > 1 lead to a superexponential growth for the global prevalence I(t). We use β = 5 × 10⁻⁴, β = 0.025, and β = 0.077 for ν = 0.5, ν = 1, and ν = 1.5, respectively. $\overline{\tau }$ is the median exposure period. (c) The phase diagram for stable solutions in the stationary state (t → ∞) can be continuous or discontinuous with a bistable regime. Sublinear and linear kernels ν ≤ 1 lead to a continuous phase transition, and the invasion threshold β_c vanishes for ν → 0. Supralinear kernels ν > 1 can lead to a discontinuous phase transition with a bistable regime.

Let us stress that the condition $1\ll {\tau }_c\ll \mathcal{T}$ is not restrictive. On the timescale on which we measure the exposure periods, τ_c ≫ 1 implies that the contagion does not spread too fast because otherwise the whole population would be rapidly infected, while $\mathcal{T}\gg {\tau }_c$ suggests that it is transmissible before the immune system is able to clear the dose received.

More broadly, we do not need to assume that π(y) decreases faster than y^−α−1 or that λ is a scale parameter for κ. In fact, π(κ; λ) does not even need to be a continuous distribution and P(τ) could be asymptotically power law for large τ. In this more general context, we still recover a universal infection kernel θ_m(ρ) ∝ ρ^ν in most cases, but ν is not always directly equal to α [43]. Linear infection kernels (ν = 1) are recovered only in some specific cases—for instance, when α = 1 or when we use a Poisson distribution for κ and K = 1. From now on, we make abstractions of the underlying distributions π(κ; λ) and P(τ) and focus on the resulting effective model parameterized by ν.

Epidemic spreading with nonlinear infection kernel.—

We now illustrate the consequences of nonlinear infection kernels for contagion on hypergraphs. To simplify the mathematical analysis, we consider a susceptible-infectious-susceptible (SIS) model. At each time step, an infected node becomes susceptible with probability μ, while a susceptible node gets infected through a hyperedge of size m with probability θ_m. If a node is part of multiple hyperedges, we assume that the probability of infection through each hyperedge is independent. This is reasonable if $k_{\max }\mathcal{T}\le {\tau }_{\max }$ for a maximal hyperdegree k_max. For instance, if $\mathcal{T}$ is of the order of a few hours, τ_max a week, and an individual participates to an environment once a day, the night allows the immune system to clear any dose κ < K accumulated the day before.

To obtain some analytical insights, we introduce a degree-based mean-field theory [14], an approximation equivalent to consider an infinite size annealed hypergraph, where at each time step, the connections between nodes and hyperedges are shuffled. We validate our approach with Monte Carlo simulations on quenched hypergraphs in the Supplemental Material [43].

With the mean-field approximation, the marginal probability for a node to be infected at time t only depends on its hyperdegree, which we note ρ_k(t). The global prevalence is then simply $I(t)={\sum }_k{\rho }_k(t)\tilde{P}(k)$, and the evolution of the system is described by [14]

$${\rho }_k(t+1)=(1-\mu ){\rho }_k(t)+\left[1-{\rho }_k(t)\right]{\mathrm{\Theta }}_k(\overline{\rho }),$$ (4)

where ${\mathrm{\Theta }}_k(\overline{\rho })=1-{[1-\overline{\theta }(\overline{\rho })]}^k$ is the probability for a susceptible node of hyperdegree k to get infected. $\overline{\rho }(t)$ is the probability that a node belonging in any hyperedge is infected, and $\overline{\theta }(\overline{\rho })$ is the probability for a susceptible node to get infected in any hyperedge,

$$\overline{\rho }(t)=\sum _k{\rho }_k(t)\frac{k\tilde{P}(k)}{\langle k\rangle },\overline{\theta }(\overline{\rho })=\sum _m{\overline{\theta }}_m(\overline{\rho })\frac{m\widehat{P}(m)}{\langle m\rangle },$$ (5)

where ${\overline{\theta }}_m(\overline{\rho })$ is the probability for a node to get infected in a hyperedge of size m. Because of the annealed structure ${\overline{\theta }}_m(\overline{\rho })$ is just the average of θ_m(ρ) with ρ = i/(m − 1) over a binomial distribution,

$${\overline{\theta }}_m(\overline{\rho })=\sum _{i=0}^{m-1}\left(\begin{array}{c}m-1\\ i\end{array}\right){\overline{\rho }}^i{(1-\overline{\rho })}^{m-1-i}{\theta }_m\left(\frac{i}{m-1}\right),$$ (6)

with θ_m(ρ) defined at Eq. (2).

Figure 2(b) shows a first consequence of the nonlinear kernel: The appearance of superexponential growth for the global prevalence I(t) when ν > 1. Note that the growth is approximately exponential until a sufficiently high prevalence. For ν ≤ 1, we instead have a standard exponential growth until saturation is reached.

In the steady state of the epidemic dynamics, we obtain a self-consistent solution,

$${\rho }_k^{*}=\frac{{\mathrm{\Theta }}_k^{*}}{\mu +{\mathrm{\Theta }}_k^{*}},{\overline{\rho }}^{*}=\sum _k\tilde{P}(k)\frac{k}{\langle k\rangle }\frac{{\mathrm{\Theta }}_k^{*}}{\mu +{\mathrm{\Theta }}_k^{*}}\equiv G\left({\overline{\rho }}^{*}\right),$$ (7)

since Θ_k is a function of $\overline{\rho }$.

For contagions with a nonlinear infection kernel, the phase transition associated with the order parameter I* can be continuous or discontinuous with a bistable regime. Consequently, we define the invasion threshold β_c such that for all β > β_c, the absorbing state I* = 0 is unstable [dashed line in Fig. 2(c)]. We also define the persistence threshold β_p such that for all β < β_p, the absorbing state I = 0 is globally attractive [dotted line in Fig. 2(c)]. For continuous phase transitions, β_c and β_p coincide, and this is called the epidemic threshold; for a discontinuous phase transition, β_p < β_c, and for all β ∈ (β_p, β_c), there exists typically three solutions: $I_1^{*}=0$ and $I_2^{*}$, $I_3^{*}>0$ with $I_1^{*}$ and $I_3^{*}$ locally stable.

The invasion threshold β_c can be found by imposing G′(0) = 1. The persistence threshold β_p is obtained by imposing both ${\overline{\rho }}^{*}=G\left({\overline{\rho }}^{*}\right)$ and $G^{\prime }\left({\overline{\rho }}^{*}\right)=1$ for ${\overline{\rho }}^{*}>0$, and any tricritical point can be found by imposing G′(0) = 1 and G″(0) = 0.

In the Supplemental Material [43], we obtain an exact self-consistent expression for the invasion threshold, and using again an asymptotic approximation, we find

$${\beta }_c\propto {\left(\frac{\mu \langle m\rangle \langle k\rangle }{\langle m{(m-1)}^{1-\nu }{[f(m)]}^{\nu }\rangle \langle k^2\rangle }\right)}^{1/\nu }.$$ (8)

This last expression depends in a intricate manner on both the moments of $\widehat{P}(m)$ and $\tilde{P}(k)$ and the nonlinearity of the infection kernel. As illustrated in Fig. 2(c), the invasion threshold can become very small for ν < 1 (vanishing for ν → 0), even for homogeneous $\tilde{P}(k)$ and $\widehat{P}(m)$. Note that a sublinear kernel (ν < 1) typically requires a distribution P(τ) ∝ τ^−α−1 with α < 1, which implies that the mean participation time ⟨τ⟩ diverges. In the Supplemental Material [43], we show that if instead we fix ⟨τ⟩ while varying α, there exists an optimal temporal heterogeneity α* that minimizes the invasion threshold β_c and maximizes early spread.

The minimal kernel exponent ν_c leading to a discontinuous phase transition is given by a tricritical point [43]. Although exact solutions require numerical evaluation, we get three insights from an asymptotic expansion: (1) ν > 1 is necessary in order to have a discontinuous phase transition, but it is not sufficient: ν_c depends on the first three moments of $\tilde{P}(k)$ and, in a more complicated manner, on the distribution $\widehat{P}(m)$. (2) It is necessary to have environments of size m > 2 to have a discontinuous phase transition. (3) A more heterogeneous $\tilde{P}(k)$ leads to a larger ν_c. Similar observations were made in Ref. [60] for m ≤ 3.

Conclusion.—

Our framework captures many properties usually overlooked for the sake of simplicity in epidemic models: the higher-order structure of contacts, the temporal heterogeneity of human activity, and the thresholding effects over the exposure due to the host immune system. In the Supplemental Material [43], we also demonstrate that our results are robust to variations in individual infectiousness or local transmission in different environments. In particular, we recover a universal nonlinear infection kernel that provides a connection between complex contagions based on nonlinear infection kernels [61] and threshold models [56–58].

Our results challenge a key assumption of most epidemic models and ask, “Why assume a linear relationship between the number of infectious contacts and the risk of infection?” This question is critical since three of the basic insights gathered from epidemic models break down under nonlinear infection kernels: They can lead to a discontinuous relationship between disease transmission and epidemic size, to a bistable regime where macroscopic outbreak and a disease-free state coexist, and to a superexponential growth. While the first two are difficult to assess for real contagions, superexponential spread has been observed for influenzalike illness [62].

Even though we considered the SIS model to simplify the analysis, the universal infection kernel θ_m(ρ) ∝ ρ^ν could be directly integrated in more realistic models such as susceptible-exposed-infectious-recovered or susceptible-infectious-recovered-susceptible where the same phenomenology typically carries over.

The phenomenology being drastically different from standard epidemiological models begs the following question: Why do linear models work? Even for a nonlinear kernel θ_m(ρ), the probability of infection ${\overline{\theta }}_m(\overline{\rho })$ (averaged over hyperedge configurations) is linear in $\overline{\rho }$ if $\overline{\rho }\ll 1$ [see Eq. (6)]. Therefore, linear models are a good approximation when the prevalence is sufficiently low but breaks down at higher prevalence, as clearly illustrated in Fig. 2(b) when ν = 1.5.

The mathematical framework we use to solve the SIS model hinges on a mean-field or annealed approximation, as in other studies [60,63–65], thereby suppressing dynamical correlations within hyperedges. As we show in the Supplemental Material [43], dynamical correlations can be captured using approximate master equations [44,45, 66–70], which are more complicated but provide similar results with better agreement to simulations. Future works could investigate more thoroughly the interplay between dynamical correlations, nonlinear kernels, and spatiotemporal heterogeneity.

Altogether, our conclusions stress the need to embrace heterogeneity in disease modeling—in the infection dynamics itself, in patterns of temporal activity, and in the higher-order structure of contact networks. Epidemics should be seen as the result of a collective process where higher-order structure and temporal patterns can drive complex dynamics.