Temporal and probabilistic comparisons of epidemic interventions

Our framework assumes that the spreading process of the disease being studied follows undirected percolation dynamics over a contact network and can therefore be analyzed as a branching process.Even though the underlying transmission dynamics occur in continuous time, we determine the probability of infection according to discretized generational time. This is represented in Fig.  2, where each solid-line arrow is a transmission event labeled with the probability of infection. We map these stochastic transmissions to a discretized epidemic generation. This discretization is shown in Fig.  2 with the vertical dotted lines representing time passing. At each generation, the branching process of transmissions from each infectious individual provides the new infectious individuals for the next generation. Even if transmission occurs in continuous time, the discrete-time mapping places individuals in the subsequent generation (sometimes underestimating time to transmission, sometimes overestimating it). The system is then updated and this process continues for the time-frame set. This assumption that transmission aligns with generational time makes analytical calculations and tracking of active cases easier even though it introduces small errors given that transmissions are pulled backward and forward in time, see Fig.  2 caption for an example case. This is an approximation of a spreading process Kenah \BBA Robins (\APACyear2007) but was recently shown to provide accurate temporal forecasts when compared to continuous-time simulations A.J. Allen \BOthers. (\APACyear2022).

In our case studies, we also assume that contacts in the population follows a geometric distribution. The aim of this assumption is to have a heterogeneous network of contacts. The geometric distribution is the discrete equivalent of the exponential distribution, which has been observed in real-world contact patterns [1-2]. With this distribution, we calculate the average number of secondary cases, $R_{0}$, to be 3. Though contact networks are inherently temporal, we here assume a static contact network except for the removal of connections due to network-based interventions. When applying an intervention, specifically a vaccination strategy, we assume that vaccination offers perfect protection. Likewise, the intricacies of vaccine efficacy (e.g. waning of immunity or the need for multiple doses) will not be covered in this work but could be incorporated in the framework. Our goal is instead to provide a general model of disease spread and showcase how a few specific types of interventions can be included in temporal, probabilistic, and analytical forecasts. The software associated with our model is available at Refs. A. Allen (\APACyear2021, \APACyear\bibnodate).

Forecasts, in our framework, are defined as the time evolution of our branching process approach and will not be directly validated with data. While the final states predicted by our general approach have been previously validated with empirical data [1], data to produce temporal forecasts of interventions are not available. Further validation would require contact distributions, epidemiological parameters, and incidence rates within communities before and after interventions. Instead, we rely on simulations for validation.

PGFs allow us to include inherent heterogeneity in epidemiological forecasting by calculating the probability distribution associated with specific network transmission trees. Generating functions offer elegant derivations of many statistical properties Wilf (\APACyear2005); Newman (\APACyear2002).

For epidemiological forecasting purposes, the focus is on the PGF of the network degree distribution, defined as

where the $k$th coefficient, $p_{k}$, is the probability of randomly choosing a node with degree $k$ from the network. The average degree of the network, $\langle k\rangle$, is found by differentiating Eq. (1) and evaluating at $x=1$,

This result is used to generate the distribution of potential transmissions, or the excess degree distribution,

The probability of reaching a node with degree $k$ from a randomly chosen edge is represented by the coefficients $q_{k}\propto(k+1)p_{k+1}$ due to the fact that a node of degree $k+1$ is $k+1$ times more likely to be connected to a random edge than a node of degree 1. The node of degree $k+1$ then has $k$ remaining edges to transmit through, which corresponds to the derivative and renormalization of the original PGF.

To incorporate the disease spread through the excess degree distribution, $q_{k}$, it is necessary to include, $p_{l|k}$, the probability of $\ell$ transmissions from a single infectious node, given that it has excess degree $k$,

where $T$ is the probability of transmission and is further explained in Sec. 2.3. Therefore, the number of infections caused by “patient zero” is equal to the probability of having degree $k$ and transmitting the disease to $\ell$ of those $k$ neighbors. This is defined as $G_{0}(x\mathchar 24635\relax\;T)$, given by

As $G_{1}(x)$ is derived from $G_{0}(x)$, so can $G_{1}(x\mathchar 24635\relax\;T)$ be derived from $G_{0}(x\mathchar 24635\relax\;T)$. In a static network, $G_{1}(x\mathchar 24635\relax\;T)$ represents the PGF for the probability distribution of the number of infections caused by a single node, i.e., the secondary case distribution.

Now, PGFs traditionally do not keep track of time as the branching process unfolds; however, Noël et al. developed a piece-wise generating function that tracks the branching process via generations Noël \BOthers. (\APACyear2009). Mathematically, for a static network, this is given by

where $G_{0}(x\mathchar 24635\relax\;T)$ defines the distribution for the first generation and $G_{1}(x\mathchar 24635\relax\;T)$ defines all future generations. In this work we expand on the framework laid out above to demonstrate the effect that temporal behaviors can have on the branching process.

Following Noël et al. Noël \BOthers. (\APACyear2009), we calculate the cumulative case distribution. To do so, we use a simple generation scheme illustrated in Fig. 1: Any transmission from a node infected in generation $g$ is considered to be in epidemic generation $g+1$ regardless of the exact timing of the transmission event. From this, let $s_{g}$ be the number of cumulative cases at generation $g$ and let $m_{g}$ be the number of infectious nodes strictly belonging to generation $g$. Note that in this way, $s_{g}=\sum_{g^{\prime}=0}^{g}m_{g^{\prime}}$. We denote $\psi_{sm}^{g}$ the probability of having $s$ total infections by the end of the $g$-th generation with $m$ becoming infected (and thus being infectious) during that generation. We also denote

the associated PGF. As demonstrated in Ref. Noël \BOthers. (\APACyear2009), $\Psi_{0}^{g}(x,y)$ is derived via a recursive function for the probability of $s_{g-1}$ total infections in generation $g-1$. Each new infection, $m_{g-1}$, in $g-1$ has its own possible transmission connections, $G_{g-1}(xy\mathchar 24635\relax\;T)$, incorporating all possible transmission events leading up to generation $g$. Mathematically, this is given by

Successive iterations of Eq. (2.2) from an initial condition (e.g., $\Psi_{0}^{0}(x,y)=xy$ for a single patient zero) then allows to compute $\psi_{sm}^{g}$ at the desired generation $g$.

Given a time point, intervention strategies can be implemented, altering the future dynamics of the disease spread. From Eq. (6), we generalize the piece-wise generating function to adhere to the intervention strategy being utilized. Given the type of intervention strategy, there can be multiple generations with an intervention implemented. So, to capture interacting temporal features of the disease spread and intervention, each epidemic generation is defined by its own PGF,

as contact patterns change along with a new transmissibility expression, $T_{g}$, which will be derived in the following section. We represent this model in Fig. 1, where the branching process is dynamically slowed by an intervention rollout.

PGFs model a stochastic process which encapsulates the random nature of disease spread. The probability of a current infectious person causing a new infection, or the probability of transmission, is captured in $T$. We will follow Susceptible-Infectious-Recovered (SIR) dynamics which could depend on the time since infection $t’$, the time-dependent transmission rate $\beta(t’)$, and time-dependent recovery rate $\gamma(t’)$. One could then calculate a general probability for transmitting before recovery, but the exact calculation is often model-dependent. We will follow most models and consider that transmission and recovery as simple Poisson processes occurring at fixed rate $\beta$ and $\gamma$ respectively. However, transmission occurs only if the contact is not immune, which is true with probability $(1-V_{g})$, where $V_{g}$ is proportion of the population that has been vaccinated by generation $g$. In other words, $V_{g}$ is the cumulative proportion of the population vaccinated. Assuming infectiousness of a node in generation $g$ lasts for some random time $\ignorespaces\tau$ then the probability of the individual transmitting infection to another individual is

When evaluating the probability of a particular $\ignorespaces\tau$ , the cumulative distribution function over $\tau$ is evaluated, shown by

The above derivation uses the average rate of recovery, $\gamma$. Taking the derivative of Eq. (2.3) gives the probability mass function over $\ignorespaces\tau$ ,

We can then compute the total probability of transmission by calculating the average probability of an individual transmitting before its recovery, given that the individual recovers at time $\tau$. The average transmissibility for a generation, $T_{g}$, is therefore

with this derivation following Refs. Hébert-Dufresne \BOthers. (\APACyear2013); Hébert-Dufresne \BBA Althouse (\APACyear2015). The expression for $T_{g}$ allows us to interpret the probability of transmission as the probability a transmission occurs first in a superposition of Poisson processes, transmission and recovery with rates $\ignorespaces\beta$ and $\ignorespaces\gamma$ respectively.

In our model, the passage of time to the next generation must also be included, which is determined by the product of the average excess degree, $q=G^{\prime}_{1}(1)$, and the transmission rate yielding $q\beta$ A.J. Allen \BOthers. (\APACyear2022). Allen et al. confirms similarities in the mapping between continuous time and discretized generational time given this rate for passage of time. Section 3.3 discusses the continuous-time simulations that exhibit the validation of our generational approach. Treating this as another Poisson process, and allowing for interventions, we find the probability of a single person causing an infection leading to the next generation to be

where again $V_{g}$ is the cumulative proportion of the population vaccinated by at $g$, and where $q_{g}$ is the generation-dependent average excess degree and is defined at Eq. (21). Similarly, the probability that the next generation occurs before a given person either transmits the disease or recovers is

To encapsulate the probability transmission given Eqs. (14) and (15) for each generation, we combine the probability of a single person causing an infection leading to the next generation with the sum of probabilities that the next generation occurs before a particular transmission or recovery event,

This last expression closes our mapping of continuous-time SIR dynamics to a discrete-time branching process. The same recipe can be used to map other compartmental models to branching processes by including additional mechanisms. Alternatively, a SEIR with a fixed latent period of one epidemic generation can be implemented by setting $t_{g}=0$ and $w_{g}=1$ to delay transmission. Regardless, once transmissions dynamics are mapped to a discrete-time (or generational) branching process, our general framework is agnostic to the details of the transmission mechanisms.

In this work, we consider the uniform intervention as a random vaccination strategy. This intervention strategy is implemented by randomly vaccinating susceptible nodes in the population with uniform probability Pastor-Satorras \BBA Vespignani (\APACyear2002). The $V_{g}$ term of Eq. (14) and Eq. (15) represents the probability of a node being vaccinated, along with the proportion of the population to be vaccinated at generation $g$. This quantity is therefore always a fraction between 0 and 1.

When a vaccination intervention is implemented at only one generation, meaning in a single intervention, the vaccinated population $V_{g}$ changes as a simple step-function. Realistically, vaccination interventions are implemented over time and over multiple generations, which we can incorporate into our modeling framework by defining a rollout strategy. Under a vaccination rollout, the cumulative percentage of the population to be vaccinated is spread over multiple generations, slowly affecting the growth of the epidemic spread along each of its active generational branches. For multiple generations, we can state that the total proportion of the population vaccinated over all generations, or cumulative percentage vaccinated, is given by

where each generational proportion vaccinated is defined as

When implementing random vaccination, the entire network is uniformly vaccinated, resulting in

for all $k$. It is important to note that random vaccination does not alter the general structure of the degree distribution or the average excess degree due to the condition set in Eq. (19).

Figure 3(A) showcases the difference in the probability distributions of cumulative infections on a system that has no intervention implemented (dotted lines), and one with a random rollout of 0.5% occurring at generations 4, 6, 8, and 10 (solid lines). Given that the intervention does not occur until generation 4, the distributions for generation 2 are exactly the same. For generations occurring after generation 4, the distributions begin to deviate from one another.

Figure 3(B) validates the extended PGF formalism for multiple interventions, with the theoretical distributions shown alongside numerical simulations following an event-driven, continuous time framework A.J. Allen \BOthers. (\APACyear2022). The analytical distributions show a bit of an overestimation at generation 10, which will be discussed in Sec. 3.3.

To demonstrate a network intervention, in this work we show how a targeted vaccination strategy is implemented. The goal of targeted vaccination is to focus vaccination efforts on the group of nodes with the highest degrees in the network, or the individuals with the most contacts. This strategy results in reducing the impact of the individuals that have the most potential for creating a superspreading event, for an example see Rosenblatt \BOthers. (\APACyear2020).

Given a percentage of the population to vaccinate in $g$, as defined in Eq. (18), we start with degree classes $k^{\prime}=k_{max}$ and vaccinate a fraction $\delta_{k^{\prime}}$ of the degree class before moving to degree class $k^{\prime}-1$. To determine the fraction vaccinated for each degree class, we define

where this non-uniform intervention will alter the degree distribution and average excess degree.

By the independence assumption of the configuration model, each neighbor will be vaccinated in generation $g$ with probability equal to the probability that following a random edge leads to a vaccinated node, call this $H_{g}$. Thinking in terms of number of edges we thus compute

Therefore, the probability of a node being unvaccinated in $g$ is equal to $1-H_{g}$.

Now, the truncation of the degree distribution in $g$ alters the average excess degree $q_{g}$, along with the coefficients of Eqs. (1) and (3). To determine the new $q_{g}$, we must recompute $G^{\prime}_{g}(1)$. Multiplying this by the proportion of nodes that are unvaccinated gives an updated $q_{g}$,

which is then used to derive the new $T_{g}$ for a given $g$.

Similar to random vaccination described in Sec. 3.1, targeted vaccination can be implemented via one instance of vaccination, or multiple. Figure 3(C) shows the difference between a non-intervention strategy and a targeted rollout vaccination scheme of 0.5% at generations 4, 6, 8, and 10. A rollout strategy is conducted in the same manner for both random and targeted vaccination. Similar to random vaccination, the non-intervention leads a lower probability of seeing 100-400 cases than targeted (or random) vaccination. Does that mean the non-intervention is better? This question is answered in Fig. 4, which showcases how the weaker the intervention, the flatter the cumulative case distribution. These flatter distributions allow for there to be some chance of infecting more individuals over time. Figure 4 also shows that the stronger intervention, the more probability mass accumulates towards the smaller cumulative infection counts. This explains why interventions appear to do worse than no intervention at smaller values of cumulative case count. Even though this figure utilizes a targeted rollout vaccination strategy the same relationship holds for random vaccination strategies.

The simulations shown in Figs. 3(B) and (D) used to validate the theoretical framework were performed using an event-driven, continuous time approach on 150 distinct networks of 20,000 nodes with 500 simulations run on each network. This totals to 75,000 simulations per validation.

The analytical distributions of infection under different vaccination strategies capture the relationships between generations of infection, but tend to overestimate the number of cumulative infections compared to the continuous time simulations. This is due to a few factors; primarily, the finite-size effects of simulations on networks with 20,000 nodes, which support faster computational time but results in a sharper decrease in the size of epidemic generations than are captured by the branching process model, as discussed in Ref. A.J. Allen \BOthers. (\APACyear2022). Nonetheless, the important behavior of the distributions are captured in relation to one another, and across different vaccination strategies, allowing for comparison and ranking of the effects between strategies regardless of slight numerical precision errors.

Another source of discrepancy between the model and continuous time simulations is the inability of the model to account precisely for already-infected nodes by the time of an intervention. This quantity is estimated in Eq. (2.3), but may result in slight differences to the continuous-time simulations under which nodes who are already infected or recovered and are identified for targeted vaccination are not excluded, and are just ignored. This problem arises more for targeted vaccination efforts than random, since nodes in the targeted high degree classes are the same nodes that are likely to have been infected early in the spreading process. The results observed in simulation may experience a reduced disease burden on the population than the theoretical model which assumes an infinite supply of these high-degree nodes, because the simulation on a finite network has already burned through its supply of high-degree nodes, rendering them recovered before the vaccination intervention.

A simple comparison for differing interventions is a direct comparison of their cumulative probability distributions at a specific generation. Beyond this, there are metrics derived from the cumulative probability distributions that provide valuable information for decision makers.