1 Introduction

The recent COVID-19 pandemic demonstrated the necessity to develop and study effective models of epidemic dynamics Anderson and May (1991). Classical epidemic models are compartmental models leading to relatively simple low-dimensional ordinary differential equations (ODEs) Brauer and van den Driessche (2008); Diekmann and Heesterbeek (2000). These minimal ODE models can be derived from first principles Kiss et al. (2017) but often suffer from strong assumptions, such as a sufficiently high link density within the network of individuals. Within the past two decades, it became apparent that viewing the structure of the contagion process via a network science approach is crucial Pastor-Satorras and Vespignani (2001); Colizza et al. (2006); Durrett (2010); House and Keeling (2011); Pastor-Satorras et al. (2015); Thurner et al. (2020). During the COVID-19 pandemic it also became apparent that there exist two major effects controlling direct epidemic spreading in humans without an available vaccine or immediate medical treatment (non-pharmaceutical interventions): exogenous quarantine measures Maier and Brockmann (2020); Kucharski et al. (2020) and endogenous social self-distancing (or social avoidance) of existing contacts Giordano et al. (2020). Formally, measures can be considered on a finer scale, such as (digital) contact tracing Ferretti et al. (2020); Kretzschmar et al. (2020). Yet, most non-pharmaceutical interventions (NPI) can be grouped into external/exogenous ones leading effectively to quarantine-type effects, and intrinsic/endogenous ones within a population that lead effectively to a social re-organization of contact networks. There have been different modeling approaches with a trend towards large-scale black-box simulations Kerr et al. (2021). Our approach is complementary in the sense of aiming at mathematically tractable models to understand the impact of different effects in more detail. In particular, we stress that although our model design is motivated by COVID, the basic rules we present below are of generic interest for a wide variety of diseases.

In this work, we are interested in developing and analyzing effective, yet tractable mathematical network epidemic models to understand how to compare and balance the effects of quarantining and social self-distancing. Motivated by the COVID-19 pandemic Thurner et al. (2020), we start from a standard susceptible-infected-recovered (SIR) model on a complex network Kiss et al. (2017). Next, we use two well-established modelling approaches. First, we add a possible quarantine state, X, of the nodes Maier and Brockmann (2020); Peak et al. (2020) together with a transition rate, \(\kappa \), of infected individuals to transition to state X. Second, we use a social self-distancing rule of susceptible individuals trying to avoid contact with infected individuals leading to re-wiring of links Gross et al. (2006); Shaw and Schwartz (2008); Risau-Gusman and Zanette (2009) controlled by a re-wiring rate, w; we keep the population size and the total number of links fixed to account for the propensity to keep social contact. Note that re-wiring links makes the network fully adaptive Gross and Sayama (2009), i.e., there is dynamics on and of the network. The resulting model is a Markov chain on all possible node states and all possible edge configurations. It can be simulated on small to medium size networks, but becomes quickly computationally intractable on large networks. For this reason, we derive suitable ODE approximations Kiss et al. (2017) based on a moment-closure approximation Keeling (1999); Keeling et al. (1997); Gross et al. (2006); Kuehn (2016). This approximation technique can also account for the dynamically changing connectivity of the network. We obtain a hierarchy of models. A particular model is obtained after fixing a truncation level. In our analysis we focus on the second-order or pair-approximation moment-closure, which leads to a five-dimensional ODE. We compare medium-size direct network simulations with ODE simulations.

As a next step, we investigate the main questions associated with SIR-type models: (I) Does epidemic spreading happen, or does it die out immediately? (II) How big is the cumulative size of the epidemic outbreak? (III) What is the maximum size of the infected population during an epidemic? As expected, the first question (I) can be calculated directly using local analysis and we can express the epidemic threshold as a function of the re-wiring rate w and the quarantine transition rate \(\kappa \) in the approximating ODEs. Questions (II)-(III) are harder to address as network structure effects preclude the application of classical methods from mathematical epidemiology to calculate the SIR outbreak size and/or the maximum peak. We develop a new technical tool by viewing the outbreak size as a global observable and applying moment-closure methods and integral estimates on the level of this observable. This technique leads to an upper bound on the global outbreak size within suitable parameter regimes, which we cross-validate numerically.

From a public health perspective, we find that large parameter regimes in the \((w,\kappa )\)-plane show a linear, or almost linear relationship regarding the effects of quarantine versus social self-distancing leading to a bounded triangular region, within which the epidemic cannot be avoided, controlled, or contained efficiently. This shows that there is a balancing effect between strong quarantining and social self-distancing, i.e., the weakening of one measure necessitates the strengthening of the other and vice versa. On the one hand, this is an intuitive result, on the other, it arises without any major assumptions in broad parameter regimes in a complex fully-adaptive network epidemic model. In addition, it can be observed in network- and ODE simulations, and can be analytically treated both, locally and globally via nonlinear dynamics techniques. Hence, it seems advisable, when reducing or observing the reduction of mitigation measures, i.e., a decrease of w and/or \(\kappa \), to avoid entering the dangerous triangle region as there is no easy way to exit it by a simple small change.

Our contributions are threefold. First, on the level of modeling, we derive a new class of models combining two existing approaches, namely quarantining Maier and Brockmann (2020) and social self-distancing Gross et al. (2006). For this new class of models, we employ numerical simulations, moment closure, and bifurcation theory techniques. Second, we develop a new technical tool extending existing moment closure methods Keeling et al. (1997); Gross et al. (2006); Kuehn (2016) to the integral level of observables, that could have major implications even beyond epidemic modeling. Third, we derive clear applicable conclusions, most prominently the parametrically approximately linear relation for the effectiveness of exogenous versus endogenous NPIs in epidemic spreading.

The paper is structured as follows: In Sect. 2 we provide the detailed mathematical model for adaptive social self-distancing models with quarantine including moment-closure via pair approximation; in the appendix we develop more complicated higher-order moment-closure models. In Sect. 3 we present our main analytical and numerical results, including a new global moment-closure viewpoint for global observables. In Sect. 4, we provide a summary and outlook of how our approach can be extended to a wider set of applications.

2 Adaptive SIR-type Network Models

Here we compare two different, yet comparable, measures present in most epidemics: social self-distancing, i.e., nodes/agents avoid infected individuals simply due to the risk of acquiring the disease themselves, and external quarantine measures, which enforce the removal of infected nodes from the population. It is evident from data that both measures have played a key role during the COVID-19 pandemic Maier and Brockmann (2020); Kucharski et al. (2020); Giordano et al. (2020). To account for the complex social structure, we start with microscopic Markov process models of susceptible-infected-recovered (SIR) dynamics on general networks with N nodes, K undirected links, and node states S, I, and R. Then we add mitigation measures to the SIR model. The well-known basic SIR rules are:

  • (infection) infected I nodes infect susceptible S nodes along susceptible-infected SI links with a rate \(\beta >0\).

  • (recovery/death) infected I nodes become recovered R nodes at a rate \(\gamma >0\).

One way to model social self-distancing, as proposed in Gross et al. (2006); Shaw and Schwartz (2008), is the preference of the susceptible S nodes to avoid interactions with the infected I nodes:

  • (social self-distancing) SI links are re-wired at a rate \(w\ge 0\) to susceptible-susceptible SS links or to susceptible-recovered SR links (with equal proportion).

The self-distancing/re-wiring rule makes the network fully adaptive Gross and Sayama (2009) and allows for very general network topologies. The rule also takes into account that links are not lost, which mirrors the desire to keep as many social connections as possible and to optimally re-wire them to mitigate risk. Note that broken social connections can be re-established in principle once both individuals are susceptible but that this process is currently modeled using uniformly at random re-wiring; see also Sect. 4 for a broader discussion of possible extensions to the model.

While it is straightforward to simulate the resulting Markov process on any given network, the simulations become prohibitively expensive for large N. It is also straightforward to use the master equation for the resulting Markov process Norris (2006) and arrive at the following set of ODEs via standard techniques Kiss et al. (2017)

$$\begin{aligned} \begin{array}{lclcl} \dot{[S]}&{}=&{}\frac{{\text {d}}}{{\text {d}}t} [S] &{}=&{} -\beta [SI], \\ \dot{[I]}&{}=&{}\frac{{\text {d}}}{{\text {d}}t} [I] &{}=&{} \beta [SI] - \gamma [I], \\ \dot{[R]}&{}=&{}\frac{{\text {d}}}{{\text {d}}t} [R] &{}=&{} \gamma [I], \\ \dot{[SI]}&{}=&{} \frac{{\text {d}}}{{\text {d}}t} [SI] &{}=&{} -(\beta +\gamma +w)[SI] + \beta [SSI] -\beta [ISI], \\ \dot{[SS]}&{}=&{}\frac{{\text {d}}}{{\text {d}}t} [SS] &{}=&{} - \beta [SSI] + w\frac{[S]}{[R]+[S]} [SI], \end{array} \end{aligned}$$
(1)

where \([S]=[S](t)\), \([I]=[I](t)\), \([R]=[R](t)\), \([SI]=[SI](t)\), \([SS]=[SS](t)\), \([SSI]=[SSI](t)\) and \([ISI]=[ISI](t)\) are expectation values of the number of susceptible, infected and recovered nodes, of SI-links and SS-links and of SSI- and ISI-triplet motifs. The ordinary differential equations (ODEs) (1) represent a variation of earlier models Shaw and Schwartz (2008); Gross et al. (2006). Note that we do not allow recovered individuals to pass back into the susceptible compartment. This is a reasonable assumption when the timescale at which immunity is lost is much larger than the characteristic epidemic timescales \(\beta ^{-1}\), \(\gamma ^{-1}\) and \(w^{-1}\). This also holds to some extend for the COVID-19 pandemic, where immunity is conjectured to last on average at the order of months, whereas the infection and recovery timescales are of the order of days. Although the ODEs (1) are actually exact in the mean-field limit Kiss et al. (2017) for any graph, they are not closed as we have not written down the equations for the SSI and ISI motifs. Although these equations could be derived, they would depend on fourth-order motifs, and so on Kuehn (2016); House and Keeling (2011). To avoid studying an infinite system of ODEs, we employ a standard moment-closure pair approximation Keeling et al. (1997); Keeling (1999); Kiss et al. (2017); Gross et al. (2006), assuming that

$$\begin{aligned}{}[ABC]\approx m(A,B)m(B,C)\frac{[AB][BC]}{[B]}\qquad \text {for }A,B,C\in \{S,I,R\}, \end{aligned}$$

where \(m(A,B) =2\) if \(A=B\) and \(m(A,B) =1\) if \(A\ne B\). With this closure, one obtains a system of four ODEs for the densities \(\rho _I=[I]/ N\) and \(\rho _R=[R]/ N\) of infected and recovered nodes and the per-node densities of susceptible-infected and susceptible-susceptible links:

$$\begin{aligned} \begin{array}{lcl} \frac{{\text {d}}}{{\text {d}}t} \rho _I &{}=&{} \beta \rho _{SI} - \gamma \rho _I, \\ \frac{{\text {d}}}{{\text {d}}t} \rho _R &{}=&{} \gamma \rho _I, \\ \frac{{\text {d}}}{{\text {d}}t} \rho _{SI} &{}=&{} -(\beta +\gamma +w) \rho _{SI} +\beta \rho _{SI}\frac{ 2 \rho _{SS}-\rho _{SI} }{1-\rho _I-\rho _R}, \\ \frac{{\text {d}}}{{\text {d}}t} \rho _{SS} &{}=&{} -2\beta \frac{\rho _{SI} \rho _{SS}}{1-\rho _I-\rho _R} +w\left[ \frac{1-\rho _I-\rho _R}{1-\rho _{I}}\right] \rho _{SI}. \end{array} \end{aligned}$$
(2)

Here we made use of node conservation and the notation \(\rho _{AB}\approx [AB]/N\) to emphasize that we are working with approximate per-node densities after moment-closure has been applied. As such, the Eq.  (2) only cover the aspect of social self-distancing and take into account the complex adaptive network structure via a second-order closure (Fig. 1).

Fig. 1
figure 1

Sample paths for the adaptive SIRX model (thin line) and the Pair Approximation from (4) (thick line). In a we depict the disease prevalence (\(\rho _I\)) and in b we depict the cumulative size of the recovered compartment (\(\rho _R\)). The dynamical parameters are given by a recovery rate of \(\gamma =0.025\) and an infection rate of \(\beta =0.005\). The intervention parameters for the quarantine and re-wiring rates are \(\kappa = w = 0.0025\). The release rate from the quarantined compartment is \(\delta = 0.001\). For the simulation we sampled from an Erdõs-Rényi ensemble of size \(N=2000\) with mean degree \(\mu =15\). We initialized \(1\%\) of nodes as infected \(\rho _I(0) = 0.01\) and \(\rho _{SI}(0) = \mu \rho _I(0)\)

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We take into account quarantine effects, such as in the modelling of COVID-19 in Maier and Brockmann (2020). There, network structure was not considered. Quarantine effects lead to certain features of epidemic spreading that cannot be captured by classical SIR models. We denote the quarantined compartment by X. The rules we use are:

  • (quarantine) infected I nodes are quarantined into a state X at a rate \(\kappa \ge 0\).

  • (recovery of X) quarantined nodes are released into the recovered compartment R at rate \(\delta >0\).

Fig. 2
figure 2

Rules for the adaptive SIRX model. The two left most rules are classical SIR with infection between infected I (red) and susceptible S (green) nodes at rate \(\beta \) and removal/recovery to R nodes at rate \(\gamma \). The middle rule is the adaptive re-wiring due to social self-distancing at rate w, where re-wiring can take place to susceptible or recovered nodes. The rightmost rules are quarantining to state X (blue) at rate \(\kappa \) and later removal to R at rate \(\delta \) (Color figure online)

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In particular, we consider quarantining and rewiring only for the infected compartment. This is a simplification of the present model and is not pursued like this in many real contact tracing efforts. The expected release time from the X-compartment is \(\langle T\rangle = 1/\delta \), since the rates are Poissonian. However, \(\delta \) does not have an effect on the amount of nodes in the infected compartment in this model and for any positive \(\delta \) the amount of nodes in the recovered compartment for \(t\rightarrow \infty \) is also independent of \(\delta \). All the rules are summarized graphically in Fig. 2. With these quarantine rules, we obtain the non-closed moment equations:

$$\begin{aligned} \begin{array}{lcl} \frac{{\text {d}}}{{\text {d}}t} [S] &{}=&{} -\beta [SI], \\ \frac{{\text {d}}}{{\text {d}}t} [I] &{}=&{} \beta [SI] - (\gamma + \kappa )[I], \\ \frac{{\text {d}}}{{\text {d}}t} [R] &{}=&{} \gamma [I] +\delta [X], \\ \frac{{\text {d}}}{{\text {d}}t} [X] &{}=&{} \kappa [I] - \delta [X], \\ \frac{{\text {d}}}{{\text {d}}t} [SI] &{}=&{} -(\beta +\gamma +w)[SI] + \beta [SSI] -\beta [ISI] - \kappa [SI], \\ \frac{{\text {d}}}{{\text {d}}t} [SS] &{}=&{} - \beta [SSI] + w\frac{[S]}{[R]+[S]} [SI]. \end{array} \end{aligned}$$
(3)

More complicated variants of the rules are discussed in

appendix A. We see that the parameters appear linearly in the equations, so that any one of them can be used to re-scale the time, e.g. \(t\mapsto \gamma t\). This leaves four effective dynamical parameters \(\beta , w, \kappa \) and \(\delta \). The last one does not affect the infected compartment or the recovered compartment at infinity and is therefore not part of the subsequent analysis.

Using a moment-closure pair approximation, we get the closed system

$$\begin{aligned} \begin{array}{lcl} \frac{{\text {d}}}{{\text {d}}t}{\rho _S}&{}=&{}-\beta \rho _{SI},\\ \frac{{\text {d}}}{{\text {d}}t}{\rho _{I}}&{}= &{} \beta \rho _{SI}-(\kappa +\gamma )\rho _{I},\\ \frac{{\text {d}}}{{\text {d}}t}{\rho _{R}}&{}=&{} \gamma \rho _I +\delta (1-\rho _{S}-\rho _{I}-\rho _{R}),\\ \frac{{\text {d}}}{{\text {d}}t}{\rho _{SI}}&{}=&{} -(\beta +\gamma +w+\kappa )\rho _{SI}+\beta \rho _{SI}\frac{2\rho _{SS}-\rho _{SI}}{\rho _S},\\ \frac{{\text {d}}}{{\text {d}}t}{\rho _{SS}}&{}=&{} - 2\beta \frac{\rho _{SI}\rho _{SS}}{\rho _{S}}+w\frac{\rho _S}{\rho _S+\rho _{R}}\rho _{SI}, \end{array} \end{aligned}$$
(4)

that allows us to compare the effects of social self-distancing and quarantine. We compare this model with full network simulations in Fig. 1. The results show excellent agreement for the vast majority of sample runs for a large part of the parameter space, when \(w,\kappa >0\); see also

appendix B for additional comparisons, where even for the singular cases \(w=0\) or \(\kappa =0\) excellent agreement is observed.

Fig. 3
figure 3

Comparison of the critical infection rate. We depict the rate \(\beta \) at which the epidemics surpasses a threshold of \(r_\infty =0.05\), which we take as a proxy for the critical point \(\beta _c\). We compare the simulations (left) with the mean-field analysis from the pair approximation (right). The parameters are as before, except for the size \(N=500\)

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3 Results

In contrast to SIS or SIRS, an epidemic eventually dies out for a standard SIR model. Hence, three questions arise:

  1. (I)

    Given an initial density of infected I(0) sufficiently close to the disease-free state, does the epidemic spread, or does it die out almost immediately?

  2. (II)

    How big is the cumulative size of the epidemic outbreak \(R_\infty \) (we use \(r_\infty \) for the corresponding density)? \(R_\infty \) measures the total number of nodes at \(t\rightarrow \infty \) in the recovered compartment R (respectively \(r_\infty \) is the corresponding density).

  3. (III)

    What is the maximum size of the epidemic, \(\hat{[I]}:=\max _tI(t)\), i.e., what is the height of the highest peak?

To answer (I), the local calculation near the disease-free state is relatively simple if we have a closed ODE model. For example, consider the adaptive SIR model without quarantining (2) and use the disease-free state \(\rho _*\) with

$$\begin{aligned} \rho _I=\rho _R=\rho _{SI}=0\quad \text {and} \quad \rho _{SS}=\mu /2, \end{aligned}$$

i.e., also all links are of type SS. Here \(\mu \) is the average degree of the network, which equals 2K/N for a network with K edges. Linearizing the vector field at \(\rho _*\), we find that for

$$\begin{aligned} \beta <\beta ^{\text {adp}}_c=\frac{\gamma +w}{\mu -1}, \end{aligned}$$
(5)

an epidemic dies out exponentially fast. A very similar calculation for the full model (4) reveals:

$$\begin{aligned} \beta <\beta ^{\text {qurt+adp}}_c=\frac{\gamma +w+\kappa }{\mu -1}, \end{aligned}$$
(6)

as the critical threshold for the infection rate. We see that on a local level near \(\rho _*\), the effects of self-distancing and quarantine are comparable as the rates of both processes lower the critical threshold in a linear way.

Fig. 4
figure 4

Comparison of the overall size of the epidemic \(r_\infty \) for a fixed \(\beta =0.0025\) in the \((\kappa ,w)\)-parameter plane. a Simulations. b Mean-field analysis from the pair approximation. We also indicate the critical curve, as calculated from (6). Again \(N=500, \mu =15, \gamma =0.025\), \(\delta =0.01\) and \(\rho _I(0)=0.01\)

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These results are

consistent with the numerics. Figure 3 shows the rate \(\beta \) at which we record an epidemic outbreak/growth for direct network simulation, as well as for numerical integration of the mean-field ODEs. The linear level-set structured in the \((w,\kappa )\)-diagram expected from (6) is clearly visible on the network simulation and the ODE integration levels. This answers questions (I), and means that a combination of measures is particularly effective to contain a disease early on. Since it is unrealistic to assume that social self-distancing happens effectively in the situation of a new disease, our SIRX model suggests that one has to compensate and focus more on quarantine measures of infected individuals in the early phase.

However, the local structure near the disease-free state only yields a partial picture of SIR models. In fact, one often does observe epidemic outbreaks for SIR-type dynamics. For this case, we study \(r_\infty \) to answer our second question (II). Figure 4 shows \(r_\infty \) for a range of values near the epidemic transition in the \((w,\kappa )\)-plane. We compare numerical simulations that estimate \(\langle \lim _{t\rightarrow \infty }R(t)/N\rangle \) with the pair approximation \(\rho _{R,\infty }:=\lim _{t\rightarrow \infty }\rho _R(t)\) of Eq. (4).

The numerical results indicate another linear relation between the parameters w and \(\kappa \). In general, it is impossible to get an exact formula for the cumulative size of the epidemic outbreak for an arbitrary model for SIR-type dynamics on complex networks. Yet, we can arrive at an implicit formula starting with in our adaptive SIRX model (3). We denote the expected final number of recovered individuals by \(R_\infty :=\lim _{t\rightarrow \infty }[R](t)\) and write it as

$$\begin{aligned} R_\infty =R_\infty -[R](0)=\int _0^\infty \dot{[R]}~{\text {d}}t, \end{aligned}$$

where we used \([R](0)=0\) and we have omitted the argument t of the last integrand for brevity. Now we use the differential equation for [R] and insert it to get

$$\begin{aligned} R_\infty =\gamma \int _0^\infty [I]~{\text {d}}t + \delta \int _0^\infty [X]~{\text {d}}t. \end{aligned}$$

We obtained two integrals, which would suffice to calculate \(r_\infty \). Using the same idea as for [R], we find for [I] and [X] that

$$\begin{aligned} 0-[I](0)= & {} \int _0^\infty \dot{[I]}~{\text {d}}t \;=\,\beta \int _0^\infty [SI]~{\text {d}}t - (\kappa +\gamma )\int _0^\infty [I]~{\text {d}}t\\ 0= & {} \int _0^\infty \dot{[X]}~{\text {d}}t ~ \;=\, \kappa \int _0^\infty [I]~{\text {d}}t -\delta \int _0^\infty [X]~{\text {d}}t, \end{aligned}$$

as there cannot be any quarantined infected nodes in the beginning or at the end of the epidemic. Using the ODE for [I], we get

$$\begin{aligned} R_\infty =\gamma \int _0^\infty [I] ~{\text {d}}t +\kappa \int _0^\infty [I] ~{\text {d}}t = \beta \int _0^\infty [SI]~{\text {d}}t + [I](0). \end{aligned}$$

Several crucial observations are evident in the formula. The procedure generically does not terminate on the infinite network level as in generic cases, we expect that all motifs could eventually occur. This means that without further assumptions only an infinite series expression for \(R_\infty \) is obtained or we could obtain upper or lower bounds. Still, the infinite series and particularly upper bounds are very informative as they display the influence of the different parameters and link them to higher-order network motifs. Indeed, at the next step, the expected number of links comes into play. We get

$$\begin{aligned} 0-[SI](0)&=\int _0^\infty \dot{[SI]}~{\text {d}}t\nonumber \\&= -(\beta +\gamma +w+\kappa )\int _0^\infty [SI]~{\text {d}}t + \beta \int _0^\infty [SSI]~{\text {d}}t -\beta \int _0^\infty [ISI]~{\text {d}}t, \end{aligned}$$

and we obtain

$$\begin{aligned} R_\infty = [I](0) + \frac{\beta }{\beta +\gamma +w+\kappa }[SI](0) + \frac{\beta ^2}{\beta +\gamma +w+\kappa }\int _0^\infty \Big ( [SSI] - [ISI]\Big ) ~{\text {d}}t.\nonumber \\ \end{aligned}$$
(7)

The infinite sum will yield new motif terms involving infected nodes at every step at the time \(t=0\). This demonstrates the importance of the network structure. For example, a highly connected first cluster of infected nodes yields a large number of SI-links and thereby a large final outbreak size. We could continue this procedure to obtain an infinite series formally but this does not give any concrete quantitative approximations. Instead, we aim for an upper bound of the total number of infected/recovered.

Fig. 5
figure 5

Approximations of \(r_\infty \). In a we show three approximations of \(r_\infty \) for the adaptive SIRX model via the Pair Approximation (4), the implicit Eq. (9) and via repeated simulation of the stochastic dynamics. The implicit equation is an inequality in \(r_\infty \) and depends on the initial conditions, which are here chosen in agreement with the other approximations, namely \(\rho _I(0) = 0.01\) and \(\rho _{SI}(0) = \mu \rho _I(0)\). The inequality is achieved by a Hölder bound, which requires a positivity condition on \({\dot{\rho }}_{SI}-2{\dot{\rho }}_{SS}\ge 0\) at all times. In b we show the shaded regions where the positivity condition holds for a range of infection rate \(\beta =0.003\) (blue, upper set of lines), \(\beta =0.0025\) (black, middle set of lines) and \(\beta =0.002\) (red, lower set of lines). For each infection rate we show the boundary for a set of initial conditions to illustrate the dependence on \(\rho _I(0)\) and \(\rho _{SI}(0)\). For the initial SI-link density we choose a mean-field scenario (dashed line) with \(\rho _{SI}(0)=\mu \rho _{S}(0)\), a scenario (dash-dotted line) with dis-proportionally many initial SI-links \(\rho _{SI}(0)=6\mu \rho _{S}(0)\) and a scenario (dotted) with very few SI-links \(\rho _{SI}(0)=(\mu /5) \rho _{S}(0)\). The mean-field transition lines (solid lines) are seen at the respective infection rates in the respective colours. They all lie within the positivity region. All parameters are as before, in particular \(\mu =15\) (Color figure online)

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A new technical step is that we directly impose the moment-closure pair approximation directly on (7), using the approximate densities \(\rho _I, \rho _R, \rho _{SI}\) and \(\rho _{SS}\) of the closed equations (4). We get an approximation for \(r_\infty \) in terms of \(\rho _{R,\infty }:=\lim _{t\rightarrow \infty }\rho _R(t)\),

$$\begin{aligned} r_\infty \approx \rho _{R,\infty }&= \rho _I(0) + \frac{\beta }{\beta +\gamma +w+\kappa }\rho _{SI}(0)\nonumber \\&\quad + \frac{\beta }{\beta +\gamma +w+\kappa }\int _0^\infty \beta \frac{\rho _{SI}}{\rho _S}\Big ( 2\rho _{SS} - \rho _{SI}\Big ) ~{\text {d}}t. \end{aligned}$$
(8)

Using \(\beta \rho _{SI}=-\dot{\rho }_{S}\) and applying the logarithmic derivative gives

$$\begin{aligned} \rho _{R,\infty }\approx K_0 + \frac{\beta }{\beta +\gamma +w+\kappa }\int _0^\infty -\frac{{\text {d}}}{{\text {d}}t}(\ln (\rho _S))\Big ( 2\rho _{SS} - \rho _{SI}\Big ) ~{\text {d}}t, \end{aligned}$$

where \(K_0:=\rho _I(0) + \frac{\beta }{\beta +\gamma +w+\kappa }\rho _{SI}(0)>0\). Integration by parts and \(\ln (x) = -|\ln (x)|\) for \(x\in (0,1]\) yields

$$\begin{aligned} \rho _{R,\infty }= & {} K_0 + \frac{\beta }{\beta +\gamma +w+\kappa }\left( - \left. \ln (\rho _S)\Big ( 2\rho _{SS} - \rho _{SI}\Big )\right| _{0}^{\infty }\right. \\&\left. - \int _0^\infty |\ln (\rho _S)|\Big ( 2\dot{\rho }_{SS} - \dot{\rho }_{SI}\Big ) ~{\text {d}}t. \right) \end{aligned}$$

Next, we use \(|\ln (\rho _S(t))|\ge |\ln (\rho _{S,\infty })| = -\ln (1-\rho _{R,\infty })\) and assume that \(2{\dot{\rho }}_{SS} - {\dot{\rho }}_{SI}\le 0\) for all times (see the computations below for further justification of this assumption). Then we obtain the bound

$$\begin{aligned} \rho _{R,\infty }\le & {} \; K_0 + \frac{\beta }{\beta +\gamma +w+\kappa }\left( - \left. \ln (\rho _S)\Big ( 2\rho _{SS} - \rho _{SI}\Big )\right| _{0}^{\infty }\right. \\&\left. + \ln \Big (1- \rho _{R,\infty }\Big ) \int _0^\infty \Big ( 2\dot{\rho }_{SS} - \dot{\rho }_{SI}\Big ) ~{\text {d}}t \right) \\= & {} \; K_0 + \frac{\beta }{\beta +\gamma +w+\kappa }\bigg ( - \ln \Big (1-\rho _{R,\infty }\Big ) 2\rho _{SS}(\infty ) + \ln \Big (1-\rho _I(0)\Big )\\&\quad \, \Big ( 2\rho _{SS}(0) - \rho _{SI}(0)\Big ) +\ln \Big (1- \rho _{R,\infty }\Big ) \Big ( 2\rho _{SS}(\infty )-2\rho _{SS}(0) +\rho _{SI}(0)\Big )\bigg )\\= & {} \; K_0 + \frac{\beta }{\beta +\gamma +w+\kappa }\ln \left[ \frac{1-\rho _I(0)}{1-\rho _{R,\infty }} \right] \left( 2\rho _{SS}(0) -\rho _{SI}(0)\right) \end{aligned}$$

Simplifying yields the desired upper bound

$$\begin{aligned} \rho _{R,\infty } \le \; \rho _I(0) + \frac{\beta }{\beta +\gamma +w+\kappa }\left[ \rho _{SI}(0) + \left( 2\rho _{SS}(0) -\rho _{SI}(0)\right) \ln \left[ \frac{1-\rho _I(0)}{1-\rho _{R,\infty }} \right] \right] .\nonumber \\ \end{aligned}$$
(9)

The bound (9) is a transcendental inequality in \(\rho _{R,\infty }\). Regarding our assumption

$$\begin{aligned} {\dot{\rho }}_{SI}\ge 2{\dot{\rho }}_{SS}, \end{aligned}$$
(10)

we find that it holds numerically for a broad ranges of parameters. In Fig. 5(b) we show the domains of validity for the positivity assumption in the \((w,\kappa )\)-plane for a range of infection rates and initial conditions. The assumption holds in a neighbourhood around the critical transition.

Note that our analysis is in sharp contrast to the classical three-dimensional SIR ODE model, where an exact implicit functional relation for \(r_\infty \) can be obtained. Therefore, having an upper bound available such as (9) helps us to study the parameter dependencies. The same linear combination of the two parameters w and \(\kappa \) appears as in the local bifurcation case near the epidemic threshold. Now however, they occur via an inverse. The same conclusions as for the local epidemic spreading near the outbreak threshold are valid: we need a linear mix of quarantine and social self-distancing to keep the total number of infected \(r_\infty \) under control. Figure 5(a) shows a comparison of the upper bound for with numerical simulations of the full network as well as simulations of the pair approximation ODEs. Both capture the main trend well that occurs when the infection rate is increased. When the total infected population is around the 10 percent level, our approximations show that employing a combination of quarantine and social-distancing might be effective in practice, while going beyond this level, a very steep increase of the total number of infected occurs.

Fig. 6
figure 6

Comparison of the maximal disease prevalence. a Simulation. b Mean-field from the Pair Approximation. We also indicate the critical curve, as calculated from (6). We can see here a slight deviation, which can be explained by the fact that the simulations are random processes. The average of sample path maxima over many sample paths is not the same as the maximum of the average of sample paths. The former overestimates the expectation value of \(\rho _I\). Again \(N=500, \mu =15, \gamma =0.025, \beta =0.0025, \delta =0.01\), \(\rho _I(0) = 0.01\) and \(\rho _{SI}(0) = \mu \rho _I(0)\)

Full size image

Finally, we answer question (III) regarding the maximum peak \(\hat{[I]}\) of the number of infected, i.e., \(\hat{[I]}/N\) is the maximal fraction across the entire duration of the epidemic. Figure 6 shows the corresponding results comparing direct network simulations in Fig. 6(a) with mean-field approximations in Fig. 6(b). The structure of the results is familiar in the sense that a linear dependence between w and \(\kappa \) emerges for our studied parameter ranges. Therefore, one can conclude that using a well-tuned combination of quarantine and social self-distancing outside of a triangular region in \((w,\kappa )\)-space is likely to be not only effective in preventing outbreaks, or reducing the total number of infected during epidemic, but also to prevent high peaks. This conclusion is also very robust over a wide variety of parameter ranges and we provide further numerical results in Appendix C to support this claim.

4 Conclusion & Outlook

In this work, we provided three contributions. First, we developed a new type of adaptive network models that include two of the most important epidemic control measures: quarantine and social self-distancing. We derived mean-field models via pair approximation; even more detailed approximation schemes are discussed in the appendix. Second, we analyzed the new model via a numerical combination of direct network simulations and mean-field ODEs, which show excellent agreement. We focused on three questions regarding (I) the epidemic threshold, (II) the total number of infected individuals, and (III) the maximum peak of the epidemic. In all three cases, we demonstrated for a broad range that the parameters controlling quarantine and social-distancing enter in a comparable linear combination to control the epidemic spread. This has the practical implication that a suitable combination of these two measures outside of a well-defined triangular region in parameter space is the best choice as one cannot expect either measure to be executed perfectly in practice. Third, on a technical level, we have shown a new way to provide estimates for the total infected population during an epidemic by using pair approximation and integral estimates directly on the level of the final infected number observable. This provides a new technical tool for broad classes of epidemic models on networks since one can now aim to employ moment-closure on many other observables directly. Yet, it remains an open conjecture, whether one can give an analytical proof of the parametrically linear trade-off relationship between quarantine and social-distancing on a global dynamical systems level. We anticipate that this problem is challenging as it would require full control of nonlinear dynamics in phase space, which is usually difficult to achieve – even for simple dynamical systems.

Many generalizations of the presented model are possible. For example, one could try to use slightly different rules for the link dynamics allowing for link deletion Ball et al. (2019); Tufekci and Wilson (2013). Yet, we conjecture that the same analysis principles we have developed here still apply. Furthermore, it would be desirable to not only consider the mitigation of the epidemic itself but also whether quarantine and social self-distancing can help us or are detrimental to detect early-warning signs for large epidemic outbreaks O’Regan and Drake (2013); Widder and Kuehn (2016); Brett et al. (2018). This line of research has already been started in recent years for epidemics on adaptive networks Kuehn et al. (2015); Horstmeyer et al. (2018) but the interplay between pre-epidemic mitigation measures and warning signs has remained unexplored. Another important generalization would be to consider non-Markovian network epidemic models Van Mieghem and Van de Bovenkamp (2013); Sherborne et al. (2018); Clauß and Kuehn (2022). This could account for behavioral changes based upon historical NPI data or allow for re-establishing a social network that has been broken due to self-distancing.