The network model

In our model, we examine a system with N agents that dynamically form and dissolve contacts with each other over time, leading to a heterogeneous network. Each agent is represented as a node in a graph, and an edge is drawn between two nodes when the corresponding agents are in contact. The time-dependent adjacency matrix A(t) has an entry a_ij(t) = 1 if agent i is in contact with agent j at time t, and a_ij(t) = 0 otherwise. The network is undirected and unweighted, meaning thatA(t) is a symmetric matrix with entries consisting of 0 and 1. The status of each edge a_ij is governed by a two-state continuous-time Markov process, where edges are created at rate and removed at rate . These rates are expressed in inverse time units. We fix , ensuring that each connection persists for an average duration of one time unit. Note that can, in principle, vary heterogeneously, but we fix them here to establish a time scale for the system. Each agent possesses its own pair of contact parameters, and , which are used to determine the edge rates [57]. For example, creates a neutral network, where nodes connect randomly, while generates an assortative network, where nodes with similar degree tend to connect. In summary, we obtain the reactions

The rates can be calculated from a distribution that represents the expected number of contacts over the simulation time frame [0,T]. The probability of creating a link between agent i and agent j during the simulation is given by 1 − , hence the rates can be determined by solving

This equation ensures that, throughout the simulation, the expected number of contacts for each agent aligns with the specified distribution. The expected number of contacts per agent should remain constant, , rather than scaling with the size of the system. This implies that the creation rates scale inversely with the number of agents, (since ).

Contact-driven agent-based contagion models

In the simplest form, the spreading is modeled by dividing each agent into one of two compartments, either susceptible S or infected I. Upon contact, a susceptible agent i may become infected by an infected agent at a rate . This reaction is dependent on the relationship of two agents and hence we call it contact dependent event. While this reaction is fundamental to all spreading models, it can be expanded by incorporating additional spontaneous transitions. These transitions do not rely on contacts and affect only a single agent at a time. Adaptivity is introduced by considering the diagnosis of infected agents, thereby adding a diagnosed compartment D. The infected agent i transitions to the D compartment with rate . Once diagnosed, agent i loses all existing contacts and forms new contacts at a reduced rate, , where . This system will be referred to as SID. We also consider models in which the infectious period eventually ends and infected or diagnosed agents become recovered. Agents in the recovered compartment R are not eligible for transmitting infection or becoming infectious. Recovery occurs with rate and we refer to this model as SIDR. Lastly, we consider the case in which recovered agents could become susceptible again (SIDRS). The spontaneous transition from R to S occurs at rate . In summary, we consider the following reactions

The SI model contains reaction , the SID model contains , the SIDR model contains and the SIDRS model contains all listed reactions.

Transitions between compartments are primarily spontaneous; that is, they depend solely on the individual agent’s state. However, transitions from S to I are contact-driven and depend on both the agent states and the network structure.

Moreover, agents can make transient contact adjustments based on the epidemic’s progression, governed by the rate . These changes depend on the epidemic’s state, as agents may be aware of the number of diagnosed individuals, causing to scale with D(t). As more agents become diagnosed, the remaining susceptible agents reduce their contact behavior more quickly. This behavior was observed, for example, during the 2022 mpox outbreak in communities of men who have sex with men [58]. We define the rates and to represent the reduction of contacts and the return to normal behavior, respectively:

(1)

with some arbitrary scaling factor c>0. If an agent i adjusts its behavior, we set its contact creation rate to , so that it behaves like a diagnosed agent. This behavior is reversed at a rate , which scales inversely with the diagnosed compartment’s size.

Exact stochastic simulation of the system

At each time t the set of possible transition events is denoted by . The set of transitions can be partitioned into ‘network events’, ‘contact dependent events’ and ‘spontaneous transition events’. In our model the contact dependent events are infection events. The network events consist of one event per edge which is either a creation event if a_ij(t) = 0 and a deletion event if a_ij(t) = 1. The infection events contain the infection of the susceptible agents. All other transitions between the compartments are collected in the transition events, which are diagnosis or recovery. The waiting times for all events are exponentially distributed. The parameter is the corresponding rate. This implies that the time to the next event in the system is exponentially distributed with parameter

where , and are the sums of all rates in the set of network events, infection events, and transition events, respectively. The kth reaction is chosen proportionally to its rate, i.e. . Executing the transition and updating the contact network and the epidemic state accordingly gives the state of the system at a new time step. This procedure can be repeated until a time T is reached or a specific criterion such as (pandemic over) is met, see Fig 1A.

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Fig 1. Methodological overview.

A. Updating steps of the infection network using SSA (top) and HAS (bottom). SSA updates all network changes, whereas HAS jumps directly to the first infection event. B. Edge existence probability after when the edge exists at time t₀ (left) and when it did not exist at time t₀ (right). C. All rejection steps for edge a_ij until the transition is accepted. The blue area depicts the edge existence probability and the green area depicts the rescaled edge existence probability, while the dashed horizontal line shows the upper bound of the edge existence probability c_ij. The red crosses depict a random uniform number , which determine acceptance or rejection.. The left shows the process with naive upper bound B₀ from Eq 3, and the right shows the process with tight upper bound B from Eq 4. On the right, all rejection steps are bypassed, and the time to the last rejection step must be sampled after moving to the infection event. Red vertical lines show rejected time points. The grey vertical lines show rejected time points that were skipped.

https://doi.org/10.1371/journal.pcsy.0000049.g001

This algorithm is exact, i.e. all distributions and moments obtained converge to their exact values. However, the computational complexity is dominated by the contact dependent transitions. The expected running time of SSA is given by the expected number of updating steps multiplied with the expected running time per updating step [59]. The expected number of updating steps is the inverse of the expected time steps, which again is the inverse of the expected sum of reaction propensities. The time steps are inversely proportional to the sum of the rates. The expected running time of SSA is given by

The first term corresponds to the total effort of simulating the network, the second term comes form the effort of infections and the last term accounts for transition events (I to D and {I, D} to R). The expected effort of an updating step of the network or an infection is , since both actions require two agents, that need to be found. The expected size of the sum of the rates are

Note that the time step for the network process consistently scales as . In contrast, the time step of the epidemic propagation process scales as − , which approaches O(1), as the number of recovered individuals increases. This means that the expected running time of SSA scales according to the network process

High acceptance sampling (HAS) for adaptive networks

In the following section, we derive an exact sampling method for modeling disease spread on adaptive networks. The running time of the method does not scale with the network process but rather with the infection process, as . To achieve this, we employ acceptance-rejection sampling [60], which reduces simulation effort while generating exact trajectories. In the acceptance-rejection method, an upper bound B on the sum of the transition rates is used to determine the next time step. The kth reaction is selected with rate . Since B is an upper bound on the sum of all rates , there is a non-zero probability that none of the transitions will be selected. In this case only the time is updated and no transition occurs. This method is particularly useful when the rates are time-dependent. We introduce a tight upper bound B to maximise the number of acceptance steps.

The state of the network influences the contact-dependent infection rates. Instead of realising the process governing the existence of the edge a_ij over the entire simulation, we sample the state on demand using the probability of edge existence P(a_ij(t) = 1). We have the following Markovian master equation

Assuming as initial condition, this ODE has the two solutions:

(2)

This allows on-demand sampling of the edge state at any time.

Naive upper bound. If we assume that an infection can spread from any infected or diagnosed agent to any susceptible agent, we obtain the following naive upper bound B₀ on the sum of the propensities of the transitions of the spreading process (‘contact-dependent-’ and ‘spontaneous transitions’)

(3)

The time step is exponentially distributed with parameter B₀, and transition k is selected proportional to its rate . One step of acceptance-rejection sampling with the naive upper bound B₀ is done in the following way:

1. Calculate the naive upper bound B₀ using Eq 3
2. Sample the time increment and set
3. Sample the next event with probability
4. If the next event is a ‘spontaneous transition’, i.e. diagnosis or recovery, execute the event and go to step 1.
5. Else edge (i, j) is selected for a possible ‘contact-dependent transition’, i.e. infection. If for , infect agent j, else set . Go to step 1.

Here, we assume that the network is fully connected and sample the state of the edge between (i, j) at time t to determine if the infection is possible. The matrix records the times when the edges were last observed. In step 5, we can set the last state of edge (i, j) to 0 for the following reason: the previous state of the edge can only be 1 if it was initialized as such. If it was initialized to 0, the observed state will only change to 1 if the edge was chosen for a potential infection and was confirmed to exist. In this case, it becomes irrelevant for further spreading, since (i, j) is no longer a susceptible-infected pair. For the remainder of this paper, we assume the previous state of all edges is 0 and initialize the observation time to . This assumption ensures that, as the simulation begins at t = 0, the edge existence probability is close to equilibrium. Although this algorithm does not require simulating all network processes, it does not enhance the performance of SSA, as the rejection probability at each step remains high. Note that  +  (Fig 1B) for large t and c_ij = O(1/N).

Tight upper bound. To reduce the computational overhead of many rejection steps, we need to minimize the propensity bound B₀. When the pair (i, j) is selected, the infection spreads with probability . This means that the edge a_ij is equipped with two events, spreading and rejection, which have the rate and , respectively. The function is bounded above by c_ij, see Fig 1C, which means that the rate of acceptance of the spread over a_ij is bounded by . This introduces the tight constant bound for the pandemic transition rates

(4)

We use this upper bound to generate larger time steps and reduce the rejection probability compared to the process with the naive upper bound B₀. With the tight upper bound, many rejection steps are leaped over. Focusing on a single edge a_ij, we need the time between the last rejection of an infection along a_ij and the current time t. This time is exponentially distributed with parameter , but the actual sampled period t − is exponentially distributed with parameter . Therefore, we sample an additional time step − and obtain

The edge existence probability is then evaluated using a time span of length (). This ensures that the edge existence probability still accounts for leaped rejection steps, see Fig 1C.

Algorithmically, a step of HAS is performed in the following way:

1. Calculate the tight upper bound B using Eq 4
2. Sample the time increment and set
3. Sample the next event with probability
4. If the next event is a ‘spontaneous transition’, i.e. diagnosis or recovery, execute the event and go to step 1.
5. Else edge (i, j) is selected for a possible ‘contact-dependent transition’, i.e. infection. Sample − and set − . If for , infect agent j, else set . Go to step 1.

The calculation of B requires steps, but since only a single rate changes per time step, a subsequent update of B can be done with constant effort O(1).

HAS is exact, a proof can be found in S1 Text. For each infection event, the acceptance probability for a step of size is given by

The step size is exponentially distributed with parameter and hence the expected rejection probability is given by

This is dominated by the infection probability (infection before edge dissolving).

Numerical evaluation of exactness

We compare the outcomes of SID processes on adaptive temporal networks (model details in Methods section, simulated using both the SSA and the HAS methods. As can be seen in Fig 2, the first and second moments, as well as the distribution of agents across compartments at the end time, are indistinguishable between the two methods, confirming the exactness of the HAS method (see Fig 2). Sampling 100,000 trajectories took 1,768 seconds with SSA, whereas it required only 59 seconds with HAS.

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Fig 2. Comparison of numerical results for an SID model simulated with SSA vs. HAS.

The network system consists of N = 100 agents with an initial infection size of . Infection rates are distributed , diagnosis rates . The expected number of contacts per time step is sampled from an exponential distribution with parameter 1, and each value is shifted by 1 to eliminate agents with 0 contacts. Each contact is expected to last for one time step (). Diagnosed agents cut their contacts in half (). A total of 100,000 trajectories were sampled for each method. A. Mean number of agents per compartment (SID) per time step. B. Variance of the number of agents per compartment per time step. C. Distribution of agents per compartment at the end of the simulation T = 24.

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Numerical comparison of sampling speed

In order to quantify the speed difference between SSA and HAS, we examined the runtimes of both methods in scenarios with varying parameters. The varied parameters are system size (Fig 3A), edge existence probability (Fig 3B) and infection probability (Fig 3C). Simulations with SSA become infeasible for larger systems or systems with many contacts. Although HAS also grows polynomially with N, its exponent is lower than for SSA. In the analyzed simulation setup, sampling 1,000 trajectories of a system with 100,000 agents using HAS is faster than sampling a single trajectory of the same system using SSA. As the infection probability approaches 1, the runtimes of both methods converge. In this limit, the network dynamics completely determine the spreading, since every contact lead to infection. But the runtime of HAS remains constant for a wide range of infection probabilities.

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Fig 3. Comparison of computational speed between HAS and SSA.

The base process involves N = 100 agents, starting with an initial infection size of . Infection and diagnosis rates are homogeneous, set at and , respectively. The expected number of contacts per time step is drawn from an exponential distribution with parameter 1, ensuring no agents have zero contacts by shifting each value by 1. Each contact persists on average for one time step. We sampled 1,000 trajectories for each parameter and method up to an end time of T = 24. A. System size varied from 100 to 100,000 agents. B. The probability of edge existence, defined as  +  , was adjusted between 0 and 1 while keeping the edge deletion rate constant and modifying . C. The infection probability, calculated as  +  , was varied by holding the edge deletion rate constant and altering the infection rate . The SSA algorithm reaches completion once every agent is infected, benefiting from a reduction in running time when the infection probability is close to 1. Computations were conducted on a Xeon Skylake 6130 with 3GB of RAM [82]

https://doi.org/10.1371/journal.pcsy.0000049.g003

Theoretical sampling speed of HAS

Similar to the complexity of SSA, the expected running time of HAS is given by

The first term is given by the infection events, which require an effort of (need to find two nodes in a list of size N) and the second term is given by the other spontaneous transitions that are independent of contacts. The expected size of the average transition rate is the same as with SSA, but the expected network infection rates are different

Identifying the infection event has size and hence the total effort of HAS scales with the infection process

If we assume that the compartments are on average of size O(N), the expected effort for HAS becomes , which is O(N) faster than SSA. However, the examples in Fig 3 suggest that the speed increase could be higher. Most of the time either the susceptible compartment S or the infectious compartment I  +  D is of size O(N) while the other is smaller, which can move the complexity of HAS closer to O(N). Consequently, the speed increase compared to SSA approaches .

Transient behaviour changes in response to infection incidence

We study the spread of an infection in an SIRD model on four types of contact network, each with the same mean node degree but differing in degree distribution: uniform, exponential, Poisson, and scale-free. The first three distributions are characterized by their mean, while the scale-free network is generated using a discrete one-parameter Weibull distribution (see S1 Fig). Each network consists of agents, with each agent having an average of 10 contacts per week. In Fig 4A we test different infection probabilities  +  , where i and j is a connected S-I-pair (in the absence of diagnosis), whereas in panels B-E, an S-I contact has a 55% probability of transmitting the disease. Half of the infected agents are getting diagnosed, at which point the diagnosed agents lose all contacts until recovery. Recovery occurs, on average, 5 days post-infection, corresponding to a recovery rate of . These parameters result in a mpox like outbreak, which is unfolding at a faster time scale in populations with varying contact distributions [6, 61].

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Fig 4. Transient behaviour changes on different network types.

We simulated an outbreak on an SIDR model across different network degree distributions and scenarios. In the first scenario, agents reduce their contacts after diagnosis, in the second scenario undiagnosed agents are aware and additionally changed their contact behaviour in response to infection incidence (number of diagnosed agents). The level of contact reduction is . The populations consist of agents, an S-I contact has a 55% probability of transmitting the disease, the diagnosis probability is 50% and recovery occurs on average 5 days post-infection. A. Final outbreak size for different values of infection probability for the three network types and scenarios as % of N. The colors correspond to the colors of the distributions above. Dotted lines represent the aware population, dash-dotted lines show a scenario without diagnosis as a control. B. Mean number of diagnoses per day for each network and population type. C. Timeline depicting the mean degree of infected agents for each network and population type over the simulation course D. Distribution of final outbreak sizes as percent of the total population after 100 simulations for each network and scenario. E. Mean node degree over time per network and population type.

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We study two different scenarios per network: (i) a ‘naive’ population and (ii) an ‘aware’ population. In the scenario with the naive population, agents only change their contact behavior upon diagnosis. In contrast, the aware population consists of susceptible agents who can spontaneously reduce their contacts and mimic a diagnosed agent. The aware population has access to information about the number of diagnosed agents, D(t). We use the rates and from Eq 1 and set c = 10. The contact reduction rate for diagnosed agents is set to . We simulate 100 trajectories per network and population type.

The final outbreak size is influenced by the variance in node degrees (Fig 4A). Networks characterized by short-tailed degree distributions lead to outbreaks of similar sizes. In contrast, the long-tailed distribution result in larger outbreaks at lower infection probabilities, but the maximum outbreak size at high infection probabilities is smaller compared to the other network types. agents with numerous contacts become infected during the early stages of the outbreak and, after recovery, leave behind a sparsely connected network where the infection cannot spread effectively [62].

The epidemic curve, defined as the number of new diagnosed cases per day, is influenced by the variance in node degrees across the network (Fig 4B): In the exponential and scale-free network, the disease initially spreads rapidly as a small number of high-degree nodes become infected. Following this phase, the outbreak shifts from spreading among densely connected individuals to those with fewer connections. This transition eventually drives the epidemic below the reproductive threshold R₀, which is dependent on the number of susceptible-infected (S-I) contacts (Fig 4C). In contrast, the disease spreads more slowly in the Poisson and uniform networks, as these lack many high-degree nodes or superspreaders. However, the final outbreak size is larger in these networks (Fig 4D), as the proportion of low-degree agents (or dead ends) is smaller.

For the ‘aware’ populations, the impact of spontaneous contact reduction in response to infection incidence is more pronounced in networks with slower disease spread (Fig 4E). Again, in the exponential network, the outbreak rapidly infects most agents who can transmit the disease. Meanwhile, in the low-variance networks (uniform and Poisson), contact reduction in response to incidence is more effective, decreasing the average degree of infected agents. As a result, the final outbreak sizes become comparable across these three network types (Fig 4D), with the largest reduction in cases observed in the Poisson network. In this network, the reduction in mean degree over time is primarily driven by the awareness mechanism (Fig 4E). In contrast, the scale-free network is mainly affected by the early diagnosis of high-contact agents at the onset of the outbreak, with the contact reduction due to awareness contributing only marginally to the overall connectedness of the network.

Population-wide simultaneous behaviour reduction

Lastly, we use HAS to study population-wide outbreak containment strategy within an SIDRS model. The size of the system is with initially infected agents. In this example, we set the diagnosis, infection and recovery rates to be homogeneous, given as , and , respectively. Rates are given per day which means that all agents are expected to test once every twenty days for infection and infected agents remain infectious for an average of ten days to mimic COVID-like dynamics [63]. Contacts are exponentially distributed with parameter 4, shifted by 1 to remove zero contact agents, yielding an average of 5 contacts per day. Recovered agents are becoming susceptible again, on average after three months [63]. The model follows SIS-type dynamics [64]: after exponential growth the epidemic eventually reaches an equilibrium and becomes endemic. In our SIDRS model, we evaluate a population-wide outbreak containment strategy that depends on the ten-day incidence rate. If this rate reaches a critical value all agents must reduce their contacts to the bare minimum, i.e. 1% of the initial value (). This measure is lifted once the ten-days incidence rate drops below a second critical value . This strategy was, for example, used in many European countries from 2020 onwards to regulate opening and closing of public places during the COVID pandemic.

We evaluated pairs of (, ) in a discrete grit to identify parametrizations for these three limit cases (Fig 5A). Both and are given as percentage of the population, with , , and . The discrete grid is constructed with increments of 0.05 for each parameter. We run 100 simulation for each parameter pair (, ). There are three limit cases based on the choices of and (Fig 5A): if is very low, the pathogen will eventually die out (Fig 5B). If is higher than the endemic equilibrium, the system may reach the endemic point (Fig 5C). If the endemic point is not contained within the interval [, ), the outbreak state becomes periodic (Fig 5D). Considering the societal impact of contact reduction, a periodic state could be seen as the least favorable outcome, as it results in numerous cases and frequent behavior adjustments. Depending on the severity of this hypothetical disease, policymakers should consider either implementing a single, extended lockdown to ensure thorough eradication by not lifting restrictions prematurely, or enacting a brief lockdown to flatten the curve and prevent simultaneous infections, while reopening quickly enough to allow the outbreak to reach an endemic state.

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Fig 5. Limit cases for population-wide containment strategy.

The system contains agents. The spreading process is SIDRS and diagnosis, infection and recovery rates are given as , and , respectively. Recovered agents are becoming susceptible again on average after three months. Contacts are exponentially distributed with parameter 4, shifted by 1 to remove zero contact agents. The containment measure is activated when the ten-day incidence rate exceeds a certain threshold and deactivated when it falls below another threshold. Contacts are reduced with factor when the measure is active. Different combinations of these thresholds result in three distinct limit cases. A. Values of (, ) and resulting limit case that predominantly occurs with the parameter set up. The dotted lines denote regions where the limit cases cannot be clearly distinguished. B. Representative trajectory of a pathogen extinction outcome, C. Representative trajectory of periodic alternation between the thresholds, or D. realization of endemic equilibrium. The number of diagnosed is shown in blue, and the total cases are shown in red. Shaded areas indicate when the containment measure is active.

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