A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase

Tom Britton; David Juher; Joan Saldaña

doi:10.1007/s11538-016-0227-4

Bulletin of Mathematical Biology

December 2016, Volume 78, Issue 12, pp 2427–2454 | Cite as

A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase

Authors
Authors and affiliations

Tom Britton
David Juher
Joan SaldañaEmail author

Original Article

First Online: 31 October 2016

Received: 22 March 2016

Accepted: 17 October 2016

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Abstract

This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate $\omega $ (and reconnect to non-infectious individuals with probability $\alpha $ or else simply drop the edge if $\alpha =0$), so-called preventive rewiring. The models are denoted SIR-$\omega $ and SEIR-$\omega $, and we focus attention on the early stages of an outbreak, where we derive the expressions for the basic reproduction number $R_0$ and the expected degree of the infectious nodes $E(D_I)$ using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-$\omega $ and SEIR-$\omega $ epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same $E(D_I)$ for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of $R_0$ computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with $\alpha > 0$ (perhaps unaware of being infected), the two approaches give different epidemic thresholds, with the branching process approximation being more in agreement with simulations.

Keywords

Network epidemic models Preventive rewiring Branching process Pair approximation

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Notes

Acknowledgments

T.B. is grateful to Riksbankens Jubileumsfond (Grant P12-0705:1) for financial support. The research of D.J. and J.S. has been partially supported by the research Grants MTM2011-27739-C04-03 and MTM2014-52402-C3-3-P of the Spanish government (MINECO). J.S. and D.J. are members of the research groups 2014SGR-1083 and 2014SGR-555 of the Generalitat de Catalunya, respectively.

Appendix

The Limit System for the SEIR-$\omega $ Model and Its Equilibria

The limit system for the local densities involved in the computation of $E(D_I)$ for the SEIR pairwise model when $2[SS]/[S] \rightarrow 2L/N$ (the mean degree), $[SE]/[S] \rightarrow 0$, $[SI]/[S] \rightarrow 0$, and $[SR]/[S] \rightarrow 0$, i.e., at the beginning of an epidemic, is

$$\begin{aligned} \frac{d}{\hbox {d}t} \left( \frac{[SI]}{[I]} \right)&= - \left( \beta + \omega _{SI} + \phi \frac{[E]}{[I]} \right) \frac{[SI]}{[I]} + \phi \frac{[SE]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{[SE]}{[I]} \right)&= \beta (\bar{q} - 1) \frac{[SI]}{[I]} - \left( \phi \left( 1 + \frac{[E]}{[I]} \right) + \omega _{SE} - \gamma \right) \frac{[SE]}{[I]} + \alpha \, \omega _{EI} \frac{[EI]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{[E]}{[I]} \right)&= \beta \frac{[SI]}{[I]} - \left( \phi \left( 1 + \frac{[E]}{[I]} \right) - \gamma \right) \frac{[E]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{[EI]}{[I]} \right)&= \beta \frac{[SI]}{[I]} - \left( \phi \left( 1+ \frac{[E]}{[I]} \right) + \omega _{EI} \right) \frac{[EI]}{[I]} + \phi \frac{2[EE]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{2[II]}{[I]} \right)&= 2 \phi \frac{[EI]}{[I]} - \left( \phi \frac{[E]}{[I]} + \gamma \right) \frac{2[II]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{[IR]}{[I]} \right)&= \phi \frac{[ER]}{[I]} + \gamma \frac{2[II]}{[I]} - \phi \frac{[E]}{[I]} \frac{[IR]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{2[EE]}{[I]} \right)&= - \left( \phi \left( 2 + \frac{[E]}{[I]} \right) - \gamma \right) \frac{2[EE]}{[I]} \\ \frac{d}{\hbox {d}t} \left( \frac{[ER]}{[I]} \right)&= - \left( \phi \left( 1+ \frac{[E]}{[I]} \right) - \gamma \right) \frac{[ER]}{[I]} + \gamma \frac{[EI]}{[I]} \end{aligned}$$

where $\bar{q} := E(\tilde{D})$. Note that only one positive term contains a rewiring rate, namely the last one in the second equation which has $\alpha \, \omega _{EI}$ as a prefactor. Since [SE] / [I] increases with $\alpha \, \omega _{EI}$ and, in turn, the first equation tells us that [SI] / [I] increases with [SE] / [I], the presence of such a positive term means that also $R_0$ will increase with $\alpha \, \omega _{EI}$. On the other hand, since the first three equations of the limit system are decoupled from the other five when $\alpha =0$, the same three equations will also govern the dynamics of [SI] / [I] when individuals simply drop connections to infectious neighbors ($\alpha =0$). In particular, this reflects the (expected) fact that whether exposed individuals break off with infectious neighbors ($\omega _{EI}>0$) or not ($\omega _{EI}=0$) does not affect the early dynamics of [SI] / [I] as long as they do not replace these links with new connections.

For a given value $\xi $ of $([E]/[I])^*$, the equilibrium equations (i.e., the equations obtained by making the right-hand side (rhs) of the previous system equal to 0) define a linear system for the remaining variables and, hence, the equilibrium can be easily expressed in terms of $\xi $. A simple inspection of the equations shows that there are two equilibria, $P_1=(0,0,0,0,0,0,0,0)$ and $P_2=(0,0,\gamma /\phi - 1,0,0,0,0,0)$, where the local densities around infectious nodes, $([SI]/[I])^*$, $(2[II]/[I])^*$, $([EI]/[I])^*$, and $([IR]/[I])^*$, are zero. Moreover, from the third equation, we see that there exists an equilibrium with $([SI]/[I])^* > 0$ if and only if $([E]/[I])^* > \gamma /\phi - 1$, which in turn implies that $(2[EE]/[I])^*=0$. Consequently, the first four equations at equilibrium do not depend on the last four.

Expressing $([SI]/[I])^*$, $([SE]/[I])^*$ and $([EI]/[I])^*$ in terms of $\xi $, and replacing them into the second equation, it follows that there will be an equilibrium with positive local densities around infectious nodes if there exists a solution $\xi ^* > 0$ of the equation

$$\begin{aligned} (\bar{q} -1) (1+\xi ) \phi + (\bar{q}+\alpha -1) \, \omega _{EI}= & {} \frac{\phi ^2}{\beta } \left( \xi + \frac{ \phi + \omega _{SE} - \gamma }{\phi } \right) \left( \xi + \frac{\omega _{SI} + \beta }{\phi } \right) \nonumber \\&\quad \left( \xi + \frac{\phi + \omega _{EI}}{\phi } \right) \end{aligned}$$

(20)

such that $\xi ^* > \gamma /\phi -1$ if $\gamma > \phi $. Note that, for $\omega _{EI} = 0$, Eq. (20) becomes a quadratic equation in $\xi $ with a positive solution $\xi ^*$ given by (see Llensa et al. 2014)

$$\begin{aligned} \xi ^* = \frac{1}{2\phi }\left( \gamma - \beta - \phi - \omega _{SI} - \omega _{SE} + \sqrt{(\gamma + \beta - \phi + \omega _{SI} - \omega _{SE})^2 + 4 \phi \beta (\bar{q} -1)} \right) \end{aligned}$$

(21)

with the proviso that $\phi \beta (\bar{q} -1) > (\phi + \omega _{SE} - \gamma )(\beta +\omega _{SI})$. Moreover, in this case ($\omega _{EI}=0$), when $\gamma > \phi $ it follows that $\xi ^* > \gamma /\phi -1$ if $\phi \beta (\bar{q} -1) > \omega _{SE} (\gamma + \beta + \omega _{SI} - \phi )$.

If $\gamma \ge \phi $ and $\omega _{EI}>0$, a necessary and sufficient condition for the existence of a unique solution $\xi ^*$ of (20) such that $\xi ^* > \gamma /\phi -1$ is that the straight line defined by the left-hand side (lhs) of (20) intersects the vertical line $\xi =\gamma /\phi -1$ above the intersection with this vertical line of the cubic polynomial defined by the rhs of (20). This polynomial has two negative roots, $\xi _1$ and $\xi _2$, and the third root $\xi _3$ can be positive or negative depending on the sign of $\gamma - \phi - \omega _{SE}$. The resulting condition on the parameters is

$$\begin{aligned} \phi \beta ( (\bar{q} - 1) \gamma + (\bar{q}+\alpha -1) \, \omega _{EI}) > \omega _{SE} (\gamma + \beta + \omega _{SI} - \phi )(\gamma + \omega _{EI}). \end{aligned}$$

Note that this condition is fulfilled when $\omega _{SE}=0$ which was, indeed, what we assumed in the model if $\omega _{EI}>0$.

If $\gamma < \phi $, then $\xi _3 < 0$ and a sufficient condition for the existence of a $\xi ^* > 0$ is that the intersection of the lhs of (20) with the y-axis is above the intersection with this axis of the rhs of (20). This condition leads to

$$\begin{aligned} \phi \beta ( (\bar{q} - 1) \phi + (\bar{q}+\alpha -1) \, \omega _{EI}) > (\phi - \gamma + \omega _{SE}) (\omega _{SI} + \beta ) (\phi + \omega _{EI}). \end{aligned}$$

Since we are assuming $\omega _{EI}>0$, then $\omega _{SE}=0$, and from this inequality it follows a simpler sufficient condition for the existence of $\xi ^*>0$, namely $(\bar{q}-2) \beta > \omega _{SI}$.

Relationship between the basic reproduction numbers $R^{BA}_0$ and $R_0^{PA}$ for the SEIR-$\omega $ model

From (18), it follows that the only positive value of $\xi ^*$ for which $R_0^{PA} = 1$ is $\xi ^* = \gamma /\phi $. After replacing $\xi ^*$ by this value, expression (20) can be rewritten as

$$\begin{aligned} \frac{\phi \beta }{(\phi + \omega _{SE})(\gamma +\beta +\omega _{SI})} \left( E(\tilde{D}) -1 + \frac{\alpha \, \omega _{EI}}{\gamma +\phi +\omega _{EI}} \right) =1. \end{aligned}$$

Comparing this expression and that of $R_0$ given by (5), it follows that $R_0^{PA} = 1 \Leftrightarrow R^{BA}_0 = 1$ if $\alpha \, \omega _{EI}=0$, and that $R_0^{PA} = 1 \Rightarrow R^{BA}_0 > 1$ (and $R^{BA}_0 = 1 \Rightarrow R_0^{PA} < 1$) if $\alpha \, \omega _{EI}>0$. So, for $\alpha >0$, both approximations lead to the same epidemic threshold when $\omega _{EI}=0$.

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Authors and Affiliations

Tom Britton
- 1
David Juher
- 2
Joan Saldaña
- 2
Email author

1.Department of MathematicsStockholm UniversityStockholmSweden
2.Departament d’Informàtica, Matemàtica Aplicada i EstadísticaUniversitat de GironaCataloniaSpain

A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase

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Acknowledgments

References

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