Various authors have presented extensions to the base temporal contact network models outlined above, accounting for factors that are believed to affect epidemic dynamics. Broadly, these include characteristics and dynamics of the underlying contact networks, infectious disease dynamics, and the complex interactions that can arise between them, all potentially influencing the epidemic threshold.
Social structure
Epidemiological contact networks are naturally driven by contacts between individuals and, whether the population under study is human, domestic animals, or wildlife, social structure largely dictates the nature of these contacts. In wildlife, it is believed that social structure has evolved in part to protect populations from spread of infectious disease (
Rozins et al. 2018;
Sah et al. 2017). In humans as well, widespread social and cultural norms often serve as a barrier to disease spreading (
Schaller 2011;
Fincher and Thornhill 2012). On the other hand, it is believed that pathogens have evolved in some cases to exploit social structure in order to further their survival. The effect of social structure on disease spreading has been studied extensively for static contact networks (
Salathé and Jones 2010;
Huang and Li 2007;
Wu and Liu 2008;
Stegehuis et al. 2016;
Liu and Hu 2005), and inclusion of social structure in temporal contact networks may provide crucial insights into real-world systems.
Nadini et al. (2018) recently explored the effect of explicit static community structure on disease spreading in activity-driven networks. In their model, each node is randomly assigned to a single community at time
t=0, with community sizes taken from a heavy-tailed distribution. When a node is activated, rather than forming connections with other nodes selected uniformly at random from the entire population, connections are formed with nodes selected uniformly at random from the same community or social group with probability
μ and from a different community or social group with probability 1−
μ. In this way, strength of the social structure is parameterized by
μ. The relation defining the epidemic threshold for these modular activity-driven networks has no closed-form solution, but some conclusions can be drawn from its form. As
μ→0, the modular structure of the network vanishes, and the threshold for both SIS and SIR dynamics is the same as in the activity-driven network without modularity (
Perra et al. 2012). As
μ→1, however, strong modularity induces a difference between the epidemic threshold for SIR and SIS dynamics. When modularity is strong, the effects appear to be driven by infected individuals having an increased probability of contact with members of their own social group or community. For SIR dynamics, these repeated contacts are increasingly with recovered individuals, in which case contagion spread will be reduced, inhibiting pathogen transmission and movement beyond the social group. For SIS dynamics, the movement of individuals from infected to susceptible status allows sustained transmission and the potential for pathogen endemicity depending on group size and pathogen extinction potential. Pathogen persistence in a social group provides greater probability of extra-group transmission and spread.
Though this model incorporates social structure and changing contacts over time, we must also recognize that social structure itself evolves, and, depending on the time scale on which changes occur relative to disease dynamics, may significantly inhibit or promote disease spreading. For example, with the population growth and recovery of the South American sea lion (
Otaria flavescens), significant changes in social composition and spatial distribution of colonies has been observed (
Grandi et al. 2008), modifications that will have important influence on pathogen transmission and persistence dynamics. Pathogen infection itself can also influence movement behaviors of individuals feeding back to modify dispersal behavior and social structure across the population, as observed in banded mongoose (
Mungos mungo) infected with the novel tuberculosis pathogen
Mycobacterium mungi (
Fairbanks et al. 2014).
Fundamental to temporal contact networks is the concept that each individual’s social contact patterns can change over time.
Holme and Liljeros (2015) examined the effects of these changes on spreading processes, finding that, for a set of empirical networks, exact times and order of contacts is less predictive of disease outbreaks than the beginning and end of contact and the overall intensity of contact during that period.
Real social networks include both strong ties, contacts that are made repeatedly, frequently, or for long periods, as well as weak ties, which are isolated, sporadic, or of short duration. Incorporating both types of contact patterns can be challenging, as strong ties typically require models to be non-Markovian, which tends to result in reduced analytic tractability.
Sun et al. (2015) studied the effects of strong and weak ties on disease spreading using activity-driven network models with memory. In their non-Markovian model, an active node with memory of
ni previously-contacted nodes will contact a new node with probability 1/(
ni+1) and a previously-contacted node with probability
ni/(
ni+1). Through numerical simulation, they found that for SIR dynamics the memory effect increased the epidemic threshold, while for SIS dynamics the memory effect lowered the epidemic threshold. This result was also reported by
Karsai et al. (2015), and is consistent with the effects of community structure on epidemic spreading, as described by
Nadini et al. (2018).
Most temporal contact network models assume that contacts occur as a Poisson process, but it has been observed that human behavior tends to result in events occurring in concentrated bursts alternating with long periods of inaction. This phenomenon is known as
burstiness (
Barabási 2005).
Zino et al. (2018) examined the effect of burstiness in contact patterns on disease spreading in temporal networks. To do this, they altered the activity-driven network model by replacing the standard Poisson activation process for each node with a Hawkes process (
Hawkes 1971). Four time invariant parameters were assigned to control to each node’s Hawkes process: (i) the jump
Ji>0, which defines the strength of the self-excitement effect, (ii) the forgetting rate
γi>0, which defines how quickly excitement is forgotten, (iii) an initial activity rate
ai(0)>0, and (iv) a background activity rate,
\(\hat {a}_{i}>0\). It is easily seen that if, ∀
i,
Ji=0 and
\(a_{i}(0)=\hat {a}_{i}\), the result is the standard activity-driven network model with Poisson activation process. In the interest of analytic tractability,
Zino et al. assume the same Hawkes process for all nodes, therefore assigning the same jump and forgetting rate to each node. Under these assumptions, when
J<
γ, the epidemic threshold is given by
$$ \lambda_{c} = \frac{1-\frac{J}{\gamma}} {\left<\hat{a}\right>+\sqrt{\left<\hat{a}^{2}\right> + \frac{J^{2}}{2\gamma}\left<\hat{a}\right>}} $$
(9)
When
J=0, Equation
9 reduces to the epidemic threshold for the SIS model for standard activity-driven networks (
Perra et al. 2012). The above epidemic threshold is less than the epidemic threshold for the standard activity-driven network. Estimates of the epidemic threshold for this model using Monte Carlo numerical simulations find that the epidemic threshold is also reduced under SIR dynamics. The key result here is that models that do not account for burstiness may significantly overestimate the epidemic threshold.
In real-world social interactions, it is possible to observe coordinated bursts of contacts. For example, wildlife may gather at a common water source during a draught, temporarily placing them in close contact. To date, the effects of this phenomenon remain unexplored.
Contacts may also exhibit periodic patterns, e.g., due to seasonal changes in behavior and social interaction (
Enright and Kao 2018). The effect this has on disease spreading has not been explicitly examined in temporal networks, though most temporal network models appear to be capable of extension by simply repeating the time period. Indeed, this may be a strength of approaches such as the infection propagator of
Valdano et al. (2015b), which assumes periodic boundary conditions.
Node set changes
Temporal network models typically assume a closed population, but in most populations demographic changes such as births, deaths, emigration, and immigation are likely to occur on a time scale commensurate with the disease spreading process. Indeed, it has long been known from compartmental models that demographic changes can significantly affect epidemic dynamics (
Keeling and Rohani 2007).
Guerra et al. (2012) and
Demirel et al. (2017) investigated the interaction between continuous network growth and disease spreading. Network growth occurred through preferential attachment (
Barabási and Albert 1999), a process known to result in the creation of scale-free networks. Scale-free topologies are known to have no epidemic threshold in the thermodynamic limit (
Pastor-Satorras and Vespignani 2001). A real example of this phenomenon was studied in
Rushmore et al. (2014), where it was found that highly central individuals in primate social contact networks also tend to be larger-bodied individuals who just happen to encounter more pathogens. However, under SIR dynamics, nodes having high degree as a result of preferential attachment are also more likely to become infected and then removed (assuming the rate of recovery is fast enough), thus raising the epidemic threshold.
Disease dynamics
While significant attention has been given to how network topology and dynamics affect disease spread, comparatively little progress has been made in understanding the impacts of individual heterogeneity on disease dynamics. Individuals can vary in their relative susceptibility, infectivity, latency, and/or duration of the infectious period in real-world populations. Furthermore, these individual parameters may change over time, as these parameters might be associated with aging, changes in reproductive status, or in response to medical treatment. Coinfection can also introduce additional heterogeneity where one pathogen may induce partial immunity (
Dietz 1979) or, alternatively, increased susceptibility to another pathogen infecting the same host.
Darbon et al. (2019) examined the importance of accounting for variation in infection duration, reduction or extension of which could result in fewer or greater secondary infections, respectively. They calculated the epidemic threshold for three real-world networks using the infection propagator approach of
Valdano et al. (2015b), concluding that failing to account for this type of heterogeneity could result in significant mis-estimation of the epidemic threshold.
Behavioral response to pathogen spreading: adaptive networks
It is well known that contact patterns may change over time in response to disease spreading. Humans and animals are known to avoid contact with infected individuals or reduce their interactions overall in response to awareness of disease, a phenomenon known as
social distancing or
protective sequestration (
Reluga 2010). Indeed, quarantine or reducing exposure to a pathogen by reducing contact with others is well established as a public health intervention. Individuals may also have reason to increase their interactions in order to achieve exposure to a pathogen, thus promoting disease spread, sometimes in the interest of preventing future infection (
Henry 2005;
Lopes et al. 2016;
Ezenwa et al. 2016;
Cole 2006;
Aleman et al. 2009). Individual behavior can also be influenced by the clinical response to pathogen infection. For example, symptoms from infection such as lethargy may temporally reduce an individual’s contacts with others. Pathogens themselves may increase interactions in infected individuals in ways that promote their spread (
Poulin 2010;
Lefèvre et al. 2009).
Gross et al. (2006) examined epidemic dynamics on discrete-time adaptive networks under SIS disease dynamics. In their model, at each time step, susceptible nodes disconnect from each adjacent infected node with probability
ω (the re-wiring rate) and form a new link with another susceptible node selected uniformly at random. Thus individuals protectively modify their contact patterns in response to local knowledge of the disease spreading process. This rewiring process introduces spreading dynamics that are not found in static networks, including bistability characterized by two thresholds: the
persistence threshold, which is the minimum transmissibility for an already-established disease to remain endemic, and the
invasion threshold, which is equivalent to the epidemic threshold that we examine herein (
Marceau et al. 2010). Under this adaptive rewiring behavior,
Gross et al. find that the epidemic threshold is characterized by
$$ \beta_{c} = \frac{\omega}{\left< k\right>\left[1-\exp{(-\omega/\mu)}\right]}. $$
(10)
Risau-Gusman and Zanette (2009) consider a model based on
Gross et al. (2006), but rewiring at each time step with a node selected uniformly at random from the entire population, irrespective of the target node’s disease status. Under this arguably more realistic model, the epidemic threshold is characterized by
$$ \beta_{c} = \frac{\omega + \mu}{\mu\left< k\right> - \mu}. $$
(11)
One key dynamic observed by
Gross et al. is that, while rewiring acts as a barrier to a disease becoming epidemic, over time it also induces formation of closely-connected communities comprised solely of susceptible individuals. Because they are more densely connected, these communities therefore have a lower average epidemic threshold than the entire network. The model of
Risau-Gusman and Zanette
does not exhibit this effect, as new connections are not formed preferentially with susceptible individuals, but rather without regard to infection status. We note that (
Shaw and Schwartz 2008) made an important early contribution to this area by studying the fluctuating dynamics of the SIRS model on adaptive networks.
Rizzo et al. (2014) investigated the effects of decreased activity rate in infected individuals in activity-driven networks. They define activity rate multipliers
ηS and
ηI for susceptible and infected individuals respectively, where
ηI<
ηS, then found the epidemic threshold to be given by
$$ \lambda_{c}=\frac{2}{m\left(\left(\eta_{S}+\eta_{I}\right)\left< x\right> + \sqrt{\left(\eta_{S}-\eta_{I}\right)^{2}\left< x\right>^{2}+4\eta_{S} \eta_{I}\left< x^{2}\right>}\right)}. $$
(12)
Thus activity reduction for infected individuals increases the epidemic threshold. When
ηS=
ηI=
η, there is no difference in activity rates between susceptible and infected individuals, and the epidemic threshold matches that found by
Perra et al. (2012) (Eq.
6).
Kotnis and Kuri (2013) used activity-driven networks to investigate the effects of social distancing in response to global awareness of infectious disease spreading. Their model considers two base activity rates,
ah for healthy individuals and
ainf for infecteds, where
ah≥
ainf. Thus infected individuals are assumed to have a potentially lower activity rate as a result of either being aware of their infected state or due to clinical symptoms of infection. Susceptible individuals are assigned an activity rate
asus(
I)=
ahe−δ·I(t), incorporating a risk perception factor
δ that causes activity rates to decrease as the number of infecteds
I(
t) increases. The resulting epidemic threshold for the SIS model is given by
$$ \lambda_{c} = \frac{1}{m \left(a_{h} e^{-\delta I} + a_{\text{inf}}\right)}. $$
(13)
Rizzo et al. (2014) used a similar activity-driven network model to demonstrate that the epidemic threshold increases when susceptibles reduce their activity rate in response to global awareness of infection.
In reality, individuals may not be informed of the global prevalence of infection within the population, and may only be aware of infecteds within their local neighborhood in the network.
Hu et al. (2018) considered this case for activity-driven networks using an SAIS model, including an Alert state to represent individual awareness of risk and therefore preventative behavior, based on the number of infected and alert neighbors. Alert individuals adopt a preventative behavior for
h time steps, and then return to their normal behavior. This duration
h can be used to represent temporary preventative behavior such as wearing masks (
h=1) or a more permanent intervention such as vaccination (
h=
∞). The authors find that when the duration of the preventative behavior is short, risk awareness has no effect on the epidemic threshold, but longer durations serve to increase the epidemic threshold.
Moinet et al. (2018) examined the effects of awareness in activity-driven networks with memory. To account for adoption of preventative behavior due to awareness as a result of contact with infected individuals, the probability of infection spreading from an infected individual to a susceptible individual
i is specified by
βi(
t)=
β exp[−
δnI(
i)
ΔT], where
δ represents the strength of the awareness and
nI(
i)
ΔT is the proportion of contacts with infected individuals within the time interval [
t−
ΔT,
t]. They report that the epidemic threshold for both SIS and SIR dynamics is unaffected by awareness in activity-driven networks without memory, but awareness increases the epidemic threshold in activity-driven networks with memory. It should be noted, though, that this effect is only seen in finite networks and does not appear to be present in the limit of large networks.