A monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling
Abstract
For a recently derived pairwise model of network epidemics with nonMarkovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks.
Keywords
Epidemic Network Infectious period Reproduction number Pairwise model NonMarkovian Integrodifferential equation Delay Final sizeAbbreviations
 CDF
cumulative distribution function
 EBCM
edgebased compartmental model
 ODE
ordinary differential equation
 SIR
susceptibleinfectedremoved
1 Introduction
Modelling epidemics on networks however increases the complexity of the models since the underlying population structure means that individuals are not interchangeable. Thus we must track which individuals are in each status rather than simply how many individuals are in each status. For example, in the most fundamental case of Markovian transmission and recovery, both time to infection and the time spent as infected and infectious is taken from exponential distributions with appropriate rates. Even for the purely Markovian case we need to deal with a continuous time Markov chain with a discrete state space with \(3^{N}\) elements, where three stands for the three possible states a node can be in (S, I and R) and N denotes the number of nodes in the network. Writing down evolution equations for the probability of the system being in any of these states is possible but impractical due to the high dimensionality of the system. Hence, in order to deal with this complexity one need to employ some ‘clever’ averaging.
Probabilistic methods, such as branching processes can be used to deal with the early growth and the asymptotic behaviour [1], with percolation theory also leading to good analytical treatment for the early growth and final size [14]. For the later dynamics, we generally need to derive a meanfield model, e.g. a low dimensional system of ODEs.
There are many well established ways to derive meanfield models. Perhaps the most compact method is the so called edge based compartmental model (EBCM) [17] which has been successfully used to capture SIR dynamics with arbitrary transmission and infection processes [24] on configurationlike networks. The EBCM provides an excellent approximation of the exact stochastic network epidemic, which becomes exact in some appropriate limits and conditions on the underlying network [4, 6].
Another powerful method to model epidemic spread on network is provided by the message passing approach [7] and this works for arbitrary transmission and recovery processes but at the expense of a system consisting of a large number of integrodifferential equations.
In addition, pairwise models have been successfully used to approximate stochastic epidemics on networks and represent a vast improvement on compartmental models. Pairwise models also have the advantage of being easy to understand and very intuitive when compared to the EBCM or the message passing model.
All the above are able to capture the time evolution of the epidemic while also offering insights about the epidemic threshold and final size. All these models have the same starting point and not surprisingly it can be shown that often these models are equivalent [10, 16, 24] and they simply represent different choices of how one averages and how the reduced state space is defined [10].
While dealing with the complexity and the modelling of contact structures, the dynamics of the disease needs to be accounted for appropriately. It is well known that the duration of the infectiousness has a major impact on whether an outbreak happens and how many people it affects as being a key parameter in the basic reproduction number. To highlight a recent example, in the WestAfrican ebola outbreak one crucial part of the intervention strategy was to reduce the length of the postmortem infectious period [2]. In this paper we bridge the gap by considering a model that can capture both the complexity of contact structure as well as the features of the disease itself. To do this we consider pairwise models with Markovian infection but arbitrary recovery process and we focus on the outbreak threshold derived from this model and its dependence on the choice of the recovery process. The paper is structured as follows. First, we introduce the pairwise model, the analytical final epidemic size relation followed by the newly introduced basic pairwise reproduction number \(\mathscr {R}_{0}^{p}\). The main result of the paper is on the relation between the variance in the distribution of the recovery process and the basic pairwise reproduction number. This is followed by some discussion of our results with respect to the concept of stochastic ordering, and the possible extension of our results to heterogeneous networks. We conclude with extensive numerical results and a discussion of our findings.
2 Methods

\([X](t)\) for the expected number of nodes in state X at time t,

\([XY](t)\) for the expected number of links connecting a node in state X to another in state Y, and

\([XYZ](t)\) for the expected number of triplets in state \(XYZ\),
2.1 NonMarkovian recovery
Above we assume that the infection process along S–I links is Markovian with transmission rate \(\tau>0\). The recovery part is considered to be nonMarkovian given by a random variable \(\mathscr {I}\), with a cumulative distribution function \(F_{\mathscr {I}}(a)\) and probability density function \(f_{\mathscr {I}}(a)\). We use the associated survival function \(\xi(a)=1F_{\mathscr {I}}(a)\) and hazard function \(h(a)=\frac{\xi'(a)}{\xi(a)}=\frac{f(a)}{\xi(a)}\). We note that \(\varphi(a)\) is the initial condition which gives the age of infection of individuals at time \(t=0\).
3 The pairwise reproduction number and infectious times
The intuitive derivation for \(\mathscr {R}^{p}_{0}\) follows from considering the rate at which new S–I links are created. From (10d), and focusing on the single positive term on the right hand side, it follows that S–I links are created at rate \(\frac{\tau(n1)}{n}\frac{[SS]}{[S]}\) which at time \(t=0\) and with a vanishingly small initial number of infected nodes reduces to \(\tau(n1)\). Now, multiplying this by the average lifetime of an S–I link, which is \(\frac{1\mathscr {L}[f_{\mathscr {I}}](\tau )}{\tau}\) [10], gives the desired threshold value in the limit of \([S] \rightarrow N\) at \(t=0\).
Notice that while \(\mathscr {R}_{0}\) depends on the expected value only, see (4), the pairwise reproduction number (14) uses the complete density function, thus the average length of the infectious period itself does not determine exactly the reproduction number. As a consequence, for an epidemic we have to know as precisely as possible the shape of the distribution. We shall analyse how the basic reproduction number (14), which is not only an epidemic threshold but also determines the final size via (15), depends on the variance of the recovery time distribution. In [21], using gamma, lognormal and uniform distributions we showed that within each of those distribution families, once the mean infectious period is fixed, smaller variance in the infectious period gives a higher reproduction number and consequently a more severe epidemic. Next we generalize this result without restricting ourselves to special distributions.
4 Main result: relationship between the variance and the reproduction number
In this section we give some simple conditions which may guarantee that smaller variance induces higher pairwise reproduction number. We consider a random variable \(\mathscr {I}\) corresponding to recovery times with probability density functions \(f_{\mathscr {I}}(t)\), cumulative distribution function \(F_{\mathscr {I}}(t)=\int_{0}^{t} f_{\mathscr {I}}(s) \,ds\) and we shall use the integral function of the CDF \(\mathscr {F}_{\mathscr {I}}(t) := \int_{0}^{t} F_{\mathscr {I}}(s) \,ds\). Clearly, \(\frac{d^{2}}{dt^{2}}\mathscr {F}_{\mathscr {I}}(t)=\frac{d}{dt}F_{\mathscr {I}}(t)=f_{\mathscr {I}}(t)\). Moreover, \(F_{\mathscr {I}}(0)=\mathscr {F}_{\mathscr {I}}(0)=0\).
Theorem 1
Proof
Remark 1
While one can easily construct a specific example for which the technical condition (19) does not hold, it is satisfied by all epidemiologically meaningful distributions, since extremely long infectious periods do not occur in epidemics. It trivially holds for distributions with compact support, and even for power law distributions with finite variance.
Corollary 1
Assume that the conditions of Theorem 1hold. Then the infectious period distribution with smaller variance induces a larger epidemic outbreak.
Proof
5 Relation to stochastic ordering and the work of Wilkinson and Sharkey
Theorem 2
Assume that\(\mathscr {I}_{1} \leq_{cx} \mathscr {I}_{2}\), and the technical condition (19) holds. Then, \(\mathscr {R}_{0,\mathscr {I}_{1}}^{p}>\mathscr {R}_{0,\mathscr {I}_{2}}^{p}\)holds.
Proof
From the convexity of \(\phi(x)=x\) and \(\phi(x)=x\), (17) follows, and the convexity of \(\phi(x)=x^{2}\) yields (18). From the convexity of \(\phi_{a}(x)=(xa)_{+}\), Theorem 3.A.1 in [23] deduced that \(\mathscr {I}_{1} \leq_{cx} \mathscr {I}_{2}\) if and only if \(\mathscr {F}_{\mathscr {I}_{1}}(t)\leq \mathscr {F}_{\mathscr {I}_{2}}(t)\) for all \(t>0\). Now instead of the strict inequality of (23), we have less or equal, but from (18) the two functions are not identical, hence analogously to the proof of Theorem 1 we can conclude (24), which completes the proof. □
Remark 2
(One can deduce Theorems 1 and 2 using [27])
In [27], the authors found the monotonic relationship between the variability of infectious periods and the final epidemic size, in a more general context of stochastic epidemics, that includes the pairwise models, by the means of convex ordering. According to Theorem 3.A.1b in [23], our condition (20) implies that the two distributions considered in Theorem 1 are convex ordered, and then the main conclusion of Theorem 1 follows from combining [27] with the argument of Corollary 1 (monotonicity relationship between the pairwise reproduction number and the final epidemic size). This also shows that Theorem 2 can be derived even without the technical condition, via [27].
Remark 3
(An example when [27] can not be applied but our methodology works)
Let \(\mathscr {I}_{1} \sim\operatorname{Exp}(1)\) (exponential distribution with parameter 1). Then \(f_{\mathscr {I}_{1}}(t)=e^{t}\), \(F_{\mathscr {I}_{1}}(t)=1e^{t}\), \(\mathscr {F}_{\mathscr {I}_{1}}(t)=t1+e^{t}\), \(\mathbb{E}(\mathscr {I}_{1})=1\), \(\operatorname{Var}(\mathscr {I}_{1})=1\), and \(\mathscr {L}[f_{\mathscr {I}_{1}}](\tau)=1/(1+\tau)\).
Let \(\mathscr {I}_{2}\) be the discrete random variable that takes the value \(1u>0\) with probability 0.5, and the value \(1+u\) with probability 0.5, where \(0< u<1\). Then, we have \(\mathbb{E}(\mathscr {I}_{2})=1\), \(F_{\mathscr {I}_{2}}(t)=0\) for \(t<1u\), 0.5 for \(1u \leq t <1+u\) and 1 for \(1+u \leq t\). Furthermore, \(\operatorname{Var}(\mathscr {I}_{2})=u^{2}<1\), \(\mathscr {F}_{\mathscr {I}_{2}}(t)=0.5(t(1u))\) on \([1u,1+u]\), and \(\mathscr {L}[f_{\mathscr {I}_{2}}](\tau)=0.5( e^{\tau (1u)}+ e^{\tau(1+u)})\).
Then, for \(0< t<1u\) we have \(\mathscr {F}_{\mathscr {I}_{1}}(t)>0=\mathscr {F}_{\mathscr {I}_{2}}(t)\). However, at \(t=1\), we have \(\mathscr {F}_{\mathscr {I}_{1}}(1)= e^{1} <0.5 u =\mathscr {F}_{\mathscr {I}_{2}}(1)\), whenever \(u>2 e^{1} \approx0.736\). In light of Theorem 3.A.1b in [23], in this case the random variables \(\mathscr {I}_{1}\), \(\mathscr {I}_{2}\) are not convex ordered, thus [27] does not apply.
For sufficiently large τ, we have \(\mathscr {L}[f_{\mathscr {I}_{1}}](\tau)>\mathscr {L}[f_{\mathscr {I}_{2}}](\tau)\), hence \(\mathscr {R}_{0,\mathscr {I}_{1}}^{p}<\mathscr {R}_{0,\mathscr {I}_{2}}^{p}\), and the discrete random variable, which has the smaller variance, generates a larger epidemic outbreak. The pairwise reproduction number approach can be applied even in situations that are not covered by the convex order approach, as this simple example illustrates.
6 Implications for heterogeneous degree distributions
In a ConfigurationModel network, given a random S–I link, we expect the susceptible individual to have degree k with probability proportional to \(k[S_{k}]\) where \([S_{k}]\) is the number of susceptible individuals with degree k.
Note that in \(\mathscr {R}_{0}^{p}\), the terms capturing the distribution of infection durations separate from the terms capturing the distribution of degrees. The ordering of \(\mathscr {R}_{0}^{p}\) as the infection duration distribution changes is independent of the degree distribution. So the ordering of \(\mathscr {R}_{0}^{p}\) is the same as found in the regular networks. The final size depends monotonically on the Laplace transform of \(f_{\mathscr {I}}\), and so the results about the ordering of final sizes in regular networks carry over to heterogeneous networks as well.
7 Discussion
The role of the shape of the distribution of infectious periods in disease spread has been in the interest of modellers for some time [25]. Our previous works already indicated that for pairwise models of network epidemic, not only the mean, but higher order properties of the distribution of the recovery times have an impact on the outcome of the epidemic. We derived useful threshold quantities for nonMarkovian recovery in [11]. In [21], we showed that for particular distribution families (typically two parameter families such as gamma, lognormal, and uniform distribution), smaller variance leads to higher reproduction number within the same family when the mean is fixed. Our new result in this study allows us to make comparisons between distributions of different kinds. To show the usefulness of Theorem 1, as an example, we consider \(\mathscr {I}_{1} \sim\operatorname{Exp}(\gamma)\) and \(\mathscr {I}_{2} \sim\operatorname{Fixed} (\frac{1}{\gamma} )\), i.e. \(f_{\mathscr {I}_{1}}(t)=\gamma e^{\gamma t}\), \(t\geq0\) and \(f_{\mathscr {I}_{2}}(t)=\delta (t\frac{1}{\gamma} )\), where \(\delta(t)\) denotes the Dirac delta function. Clearly, we obtain \(\mathscr {F}_{\mathscr {I}_{1}}(t)=t+\frac{1}{\gamma}e^{\gamma t}\frac{1}{\gamma}\) and \(\mathscr {F}_{\mathscr {I}_{2}}(t)=(t\frac {1}{\gamma})_{+}\), thus there is no \(t_{0}>0\), such that \(\mathscr {F}_{\mathscr {I}_{1}}(t_{0})=\mathscr {F}_{\mathscr {I}_{2}}(t_{0})\). Since \(\mathbb{E}(\mathscr {I}_{1})=\mathbb{E}(\mathscr {I}_{2})=\frac {1}{\gamma}\), \(\frac{1}{\gamma^{2}}=\operatorname{Var}(\mathscr {I}_{1})>\operatorname{Var}(\mathscr {I}_{2})=0\) and the other conditions of Theorem 1 are satisfied, we find \(\mathscr {R}_{0,\mathscr {I}_{1}}^{p}<\mathscr {R}_{0,\mathscr {I}_{2}}^{p}\).
Details of all the distributions of the infection times used for the explicit stochastic network simulations
Distribution  Parameters  Mean  Variance 

Fixed  3/2  3/2  0 
Uniform  U(1,2)  3/2  1/12 = 0.08(3) 
Gamma  scale = 0.5, shape = 3  3/2  0.75 
Exponential  2/3  3/2  9/4 = 2.25 
Lognormal  σ = 1, μ = ln(3/2)−1/2  3/2  3.866 
Weibull  scale = 1, shape = 0.6014  3/2  6.914 
Several observations can be made. In Figs. 1, 3 and 5 one can note that the epidemic threshold depends heavily on the distribution of the infectious period. While all distributions have the same mean, they differ in terms of their variance. In fact, the variance of the distributions are ordered as shown in Table 1. Based on Theorem 1 and Corollary 1 we know that for a fixed transmission rate τ and for infectious period distributions with the same mean, the distribution with the higher variance will lead to a smaller \(\mathscr {R}_{0}^{p}\) and hence smaller attack rate. This confirms that the ordering of the variances in Table 1 is reflected accurately in all attack rate versus τ plots. Moreover, the insets in Figs. 1, 3 and 5 shows that the final epidemic size relation in terms of \(\mathscr {R}_{0}^{p}\) is universal, independently of how the infectious periods are distributed. For the truncated scalefree networks in Fig. 5, the attack rate behaves differently but the general analytical final epidemic size relation remains extremely accurate. Obviously high degree heterogeneity leads to large variance and this makes the value of \(\mathscr {R}_{0}^{p}\) to be large and above threshold even for small values of τ.
8 Conclusion
Figures 2, 4 and 6 show the initial growth of the epidemic. The relation between variance and attack rate seems to translate into a straightforward association between variance and initial growth rate. Namely, distributions with higher variance leads to slower initial growth. This is not always the case since \(\mathscr {R}_{0}^{p}\) is a generation rather than time based measure. However, here the mean of the distributions and the transmission rates are identical and thus the ordering seems to carry through.
We can offer an intuitive explanation of our result. A key factor determining how many infections occur is the proportion of SI edges that eventually transmit. If we have M edges where M is large, with an average infection duration D and transmission rate τ, then the expected number of transmission events to occur is \(D M \tau\), but only the first transmission event per edge has any impact. Those edges in which the infection duration is longer will tend to have more transmission events, while those with shorter duration are more likely to have no transmission events. Increasing the variance in duration tends to increase the concentration of the transmission events into a smaller set of edges, resulting in fewer successful transmissions, and conversely, decreasing the variance redistributes some transmission events from edges which have already transmitted to those edges which have not.
As next steps one could consider the extension of \(\mathscr {R}_{0}^{p}\) and the final size formula for epidemics where both the infection and transmission processes are nonMarkovian. Such results already exist [24] but there an EBCM was used. It would also be appropriate to explore the applicability of this newly introduced pairwise reproduction number given that it lent itself to derive a number of analytical results and it fits with the network and contact concepts. In particular one would explore how could this be measured in practice and how does its value translate into control measures.
Notes
Acknowledgements
This article is a significantly extended version of Röst G., Kiss I. Z., Vizi Z., Variance of Infectious Periods and Reproduction Numbers for Network Epidemics with NonMarkovian Recovery, Progress in Industrial Mathematics at ECMI 2016.
Availability of data and materials
Not applicable. The article contains no data.
Authors’ contributions
The study was conceived by GR. The main result and its proof was found by ZV. Network epidemic methodology was provided by IZK. Numerical studies were done by JCM. All authors contrituted to the writing of the manuscript. All authors read and approved the final manuscript.
Funding
ZV was supported by the EUfunded Hungarian grant EFOP3.6.216201700015. GR was supported by Hungarian National Research Fund Grant NKFI FK 124016 and MSCAIF 748193. JCM was supported by Global Good.
Competing interests
None of the authors have any competing interest.
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