Exact deterministic representation of Markovian ${ SIR}$ epidemics on networks with and without loops

Abstract

In a previous paper Sharkey et al. (Bull Math Biol doi:10.1007/s11538-013-9923-5, 2012) proved the exactness of closures at the level of triples for Markovian ${ SIR}$ (susceptible-infected-removed) dynamics on tree-like networks. This resulted in a deterministic representation of the epidemic dynamics on the network that can be numerically evaluated. In this paper, we extend this modelling framework to certain classes of networks exhibiting loops. We show that closures where the loops are kept intact are exact, and lead to a simplified and numerically solvable system of ODEs (ordinary-differential-equations). The findings of the paper lead us to a generalisation of closures that are based on partitioning the network around nodes that are cut-vertices (i.e. the removal of such a node leads to the network breaking down into at least two disjointed components or subnetworks). Exploiting this structural property of the network yields some natural closures, where the evolution of a particular state can typically be exactly given in terms of the corresponding or projected states on the subnetworks and the cut-vertex. A byproduct of this analysis is an alternative probabilistic proof of the exactness of the closures for tree-like networks presented in Sharkey et al. (Bull Math Biol doi:10.1007/s11538-013-9923-5, 2012). In this paper we also elaborate on how the main result can be applied to more realistic networks, for which we write down the ODEs explicitly and compare output from these to results from simulation. Furthermore, we give a general, recipe-like method of how to apply the reduction by closures technique for arbitrary networks, and give an upper bound on the maximum number of equations needed for an exact representation.

Appendices

Appendix A: the closed triangle network

The triangle network is a loop of three nodes numbered 1, 2 and 3, see Fig. 1b. The system dynamics are given by the following set of equations:

$$\begin{aligned} \dot{\langle I_1 \rangle }&= \tau \langle S_1I_2 \rangle + \tau \langle S_1I_3 \rangle - \gamma \langle I_1\rangle , \end{aligned}$$

(11)

$$\begin{aligned} \dot{\langle S_1 \rangle }&= -\tau \langle S_1I_2 \rangle - \tau \langle S_1I_3 \rangle ,\end{aligned}$$

(12)

$$\begin{aligned} \dot{\langle I_2 \rangle }&= \tau \langle I_1S_2 \rangle + \tau \langle S_2I_3\rangle - \gamma \langle I_2\rangle ,\end{aligned}$$

(13)

$$\begin{aligned} \dot{\langle S_2 \rangle }&= -\tau \langle I_1S_2 \rangle - \tau \langle S_2I_3\rangle , \end{aligned}$$

(14)

$$\begin{aligned} \dot{\langle I_3 \rangle }&= \tau \langle I_2S_3 \rangle + \tau \langle I_1S_3 \rangle - \gamma \langle I_3 \rangle ,\end{aligned}$$

(15)

$$\begin{aligned} \dot{\langle S_3 \rangle }&= -\tau \langle I_2S_3 \rangle - \tau \langle I_1S_3\rangle ,\end{aligned}$$

(16)

$$\begin{aligned} \dot{\langle I_1S_2 \rangle }&= -(\tau + \gamma ) \langle I_1S_2 \rangle - \tau \langle I_1S_2I_3 \rangle + \tau \langle S_1S_2I_3 \rangle ,\end{aligned}$$

(17)

$$\begin{aligned} \dot{\langle S_1I_2 \rangle }&= -(\tau + \gamma ) \langle S_1I_2 \rangle + \tau \langle S_1S_2I_3 \rangle - \tau \langle S_1I_2I_3 \rangle ,\end{aligned}$$

(18)

$$\begin{aligned} \dot{\langle I_2S_3 \rangle }&= -(\tau + \gamma ) \langle I_2S_3 \rangle + \tau \langle I_1S_2S_3 \rangle - \tau \langle I_1I_2S_3 \rangle ,\end{aligned}$$

(19)

$$\begin{aligned} \dot{\langle S_2I_3 \rangle }&= -(\tau + \gamma ) \langle S_2I_3 \rangle - \tau \langle I_1S_2I_3 \rangle + \tau \langle I_1S_2S_3 \rangle ,\end{aligned}$$

(20)

$$\begin{aligned} \dot{\langle I_1S_3 \rangle }&= -(\tau + \gamma ) \langle I_1S_3 \rangle - \tau \langle I_1I_2S_3 \rangle + \tau \langle S_1I_2S_3 \rangle ,\end{aligned}$$

(21)

$$\begin{aligned} \dot{\langle S_1I_3 \rangle }&= -(\tau + \gamma ) \langle S_1I_3 \rangle - \tau \langle S_1I_2I_3 \rangle + \tau \langle S_1I_2S_3 \rangle , \end{aligned}$$

(22)

$$\begin{aligned} \dot{\langle I_1S_2I_3 \rangle }&= -2(\tau + \gamma ) \langle I_1S_2I_3 \rangle + \tau \langle I_1S_2S_3 \rangle + \tau \langle S_1S_2I_3 \rangle ,\end{aligned}$$

(23)

$$\begin{aligned} \dot{\langle S_1I_2I_3 \rangle }&= -2(\tau + \gamma ) \langle S_1I_2I_3 \rangle + \tau \langle S_1S_2I_3 \rangle + \tau \langle S_1I_2S_3 \rangle , \end{aligned}$$

(24)

$$\begin{aligned} \dot{\langle I_1I_2S_3 \rangle }&= -2(\tau + \gamma ) \langle I_1I_2S_3 \rangle + \tau \langle I_1S_2S_3 \rangle + \tau \langle S_1I_2S_3 \rangle ,\end{aligned}$$

(25)

$$\begin{aligned} \dot{\langle S_1I_2S_3 \rangle }&= -(2\tau + \gamma )\langle S_1I_2S_3 \rangle , \end{aligned}$$

(26)

$$\begin{aligned} \dot{\langle S_1S_2I_3 \rangle }&= -(2\tau + \gamma ) \langle S_1S_2I_3 \rangle ,\end{aligned}$$

(27)

$$\begin{aligned} \dot{\langle I_1S_2S_3 \rangle }&= -(2\tau + \gamma ) \langle I_1S_2S_3 \rangle . \end{aligned}$$

(28)

Since this network has no cut-vertices, closures are not possible for any of the subsystems above. For instance,

$$\begin{aligned} \langle S_1S_2I_3\rangle = \frac{\langle I_1S_2 \rangle \langle S_2 I_3 \rangle }{\langle S_2 \rangle }, \quad \langle I_1S_2I_3 \rangle = \frac{\langle I_1S_2 \rangle \langle S_2 I_3 \rangle }{\langle S_2 \rangle }, \end{aligned}$$

are closures that do not hold, see Fig. 2. We note that the evaluation of one of the closures above (the first) requires an extra equation for $\langle S_1S_2\rangle$. This is given by,

$$\begin{aligned} \langle \dot{S_1S_2\rangle } = -2\tau \langle S_1S_2I_3\rangle . \end{aligned}$$

Depending on the closures that we wish to test, additional equations may be needed.

Appendix B: equations for the lollipop network

The lollipop network we consider has nodes numbered as shown in Fig. 1c. The equations describing the ${ SIR}$ model can be formulated as follows:

$$\begin{aligned} \dot{\langle I_1 \rangle }&= \tau \langle S_1I_2 \rangle + \tau \langle S_1I_3 \rangle + \tau \langle S_1I_4 \rangle - \gamma \langle I_1 \rangle , \end{aligned}$$

(29)

$$\begin{aligned} \dot{ \langle S_1 \rangle }&= -\tau \langle S_1I_2 \rangle - \tau \langle S_1I_3 \rangle - \tau \langle S_1I_4 \rangle ,\end{aligned}$$

(30)

$$\begin{aligned} \dot{\langle I_2 \rangle }&= \tau \langle I_1S_2 \rangle - \gamma \langle I_2 \rangle ,\end{aligned}$$

(31)

$$\begin{aligned} \dot{ \langle S_2 \rangle }&= -\tau \langle I_1S_2 \rangle , \end{aligned}$$

(32)

$$\begin{aligned} \dot{\langle I_3 \rangle }&= \tau \langle I_1S_3 \rangle + \tau \langle S_3I_4 \rangle - \gamma \langle I_3 \rangle ,\end{aligned}$$

(33)

$$\begin{aligned} \dot{\langle S_3 \rangle }&= -\tau \langle I_1S_3 \rangle - \tau \langle S_3I_4 \rangle ,\end{aligned}$$

(34)

$$\begin{aligned} \dot{\langle I_4 \rangle }&= \tau \langle I_1S_4 \rangle + \tau \langle I_3S_4 \rangle - \gamma \langle I_4 \rangle , \end{aligned}$$

(35)

$$\begin{aligned} \dot{\langle S_4 \rangle }&= -\tau \langle I_1S_4 \rangle - \tau \langle I_3S_4 \rangle , \end{aligned}$$

(36)

$$\begin{aligned} \dot{\langle I_1S_2 \rangle }&= -(\tau + \gamma ) \langle I_1S_2 \rangle + \tau \langle S_1S_2I_3 \rangle + \tau \langle S_1S_2I_4 \rangle ,\end{aligned}$$

(37)

$$\begin{aligned} \dot{\langle S_1I_2 \rangle }&= -(\tau + \gamma ) \langle S_1I_2 \rangle - \tau \langle S_1I_2I_3 \rangle - \tau \langle S_1I_2I_4 \rangle ,\end{aligned}$$

(38)

$$\begin{aligned} \dot{\langle I_1S_3 \rangle }&= -(\tau + \gamma ) \langle I_1S_3 \rangle + \tau \langle S_1I_2S_3 \rangle + \tau \langle S_1S_3I_4 \rangle - \tau \langle I_1S_3I_4 \rangle ,\end{aligned}$$

(39)

$$\begin{aligned} \dot{\langle S_1I_3 \rangle }&= -(\tau + \gamma ) \langle S_1I_3 \rangle - \tau \langle S_1I_2I_3 \rangle - \tau \langle S_1I_3I_4 \rangle +\tau \langle S_1S_3I_4 \rangle ,\end{aligned}$$

(40)

$$\begin{aligned} \dot{\langle I_1S_4 \rangle }&= -(\tau + \gamma ) \langle I_1S_4 \rangle + \tau \langle S_1I_2S_4 \rangle + \tau \langle S_1I_3S_4 \rangle - \tau \langle I_1I_3S_4 \rangle , \end{aligned}$$

(41)

$$\begin{aligned} \dot{\langle S_1I_4 \rangle }&= -(\tau + \gamma ) \langle S_1I_4 \rangle - \tau \langle S_1I_2I_4 \rangle - \tau \langle S_1I_3I_4 \rangle + \tau \langle S_1I_3S_4 \rangle , \end{aligned}$$

(42)

$$\begin{aligned} \dot{\langle S_3I_4 \rangle }&= -(\tau + \gamma ) \langle S_3I_4 \rangle + \tau \langle I_1S_3S_4 \rangle - \tau \langle I_1S_3I_4 \rangle ,\end{aligned}$$

(43)

$$\begin{aligned} \dot{\langle I_3S_4 \rangle }&= -(\tau + \gamma ) \langle I_3S_4 \rangle + \tau \langle I_1S_3S_4 \rangle - \tau \langle I_1I_3S_4 \rangle ,\end{aligned}$$

(44)

$$\begin{aligned} \dot{\langle S_1I_3I_4 \rangle }&= -2(\tau + \gamma ) \langle S_1I_3I_4 \rangle + \tau \langle S_1S_3I_4 \rangle +\tau \langle S_1I_3S_4 \rangle - \tau \langle S_1I_2I_3I_4 \rangle ,\end{aligned}$$

(45)

$$\begin{aligned} \dot{\langle S_1S_3I_4 \rangle }&= -(2\tau + \gamma ) \langle S_1S_3I_4 \rangle - \tau \langle S_1I_2S_3I_4 \rangle , \end{aligned}$$

(46)

$$\begin{aligned} \dot{\langle S_1I_3S_4 \rangle }&= -(2\tau + \gamma ) \langle S_1I_3S_4 \rangle - \tau \langle S_1I_2I_3S_4 \rangle , \end{aligned}$$

(47)

$$\begin{aligned} \dot{\langle I_1S_3I_4 \rangle }&= -2(\tau + \gamma ) \langle I_1S_3I_4 \rangle + \tau \langle S_1S_3I_4 \rangle + \tau \langle I_1S_3S_4 \rangle + \tau \langle S_1I_2S_3I_4 \rangle ,\end{aligned}$$

(48)

$$\begin{aligned} \dot{\langle I_1I_3S_4 \rangle }&= -2(\tau + \gamma ) \langle I_1I_3S_4 \rangle + \tau \langle S_1I_3S_4 \rangle + \tau \langle I_1S_3S_4 \rangle + \tau \langle S_1I_2I_3S_4\rangle , \end{aligned}$$

(49)

$$\begin{aligned} \dot{\langle I_1S_3S_4 \rangle }&= -(2\tau + \gamma ) \langle I_1S_3S_4 \rangle + \tau \langle S_1I_2S_3S_4 \rangle . \end{aligned}$$

(50)

This first group of equations consist of variables (e.g. configurations of states and subgraphs) which cannot be closed or further reduced. Naturally, this first set requires equations at the levels of triples and quadruples or full system size. Note that triples which are part of the triangle cannot be closed. However, the second group of equations, i.e.

$$\begin{aligned} \dot{\langle S_1I_2I_4 \rangle }&= -2(\tau + \gamma ) \langle S_1I_2I_4 \rangle + \tau \langle S_1I_2I_3S_4 \rangle - \tau \langle S_1I_2I_3I_4 \rangle , \end{aligned}$$

(51)

$$\begin{aligned} \dot{\langle S_1I_2I_3 \rangle }&= -2(\tau + \gamma ) \langle S_1I_2I_3 \rangle + \tau \langle S_1I_2S_3I_4 \rangle - \tau \langle S_1I_2I_3I_4 \rangle ,\end{aligned}$$

(52)

$$\begin{aligned} \dot{\langle S_1S_2I_3 \rangle }&= -(\tau + \gamma ) \langle S_1S_2I_3 \rangle + \tau \langle S_1S_2S_3I_4 \rangle - \tau \langle S_1S_2I_3I_4 \rangle ,\end{aligned}$$

(53)

$$\begin{aligned} \dot{\langle S_1S_2I_4 \rangle }&= -(\tau + \gamma ) \langle S_1S_2I_4 \rangle + \tau \langle S_1S_2I_3S_4 \rangle - \tau \langle S_1S_2I_3I_4 \rangle ,\end{aligned}$$

(54)

$$\begin{aligned} \dot{\langle S_1I_2S_3 \rangle }&= -(\tau + \gamma ) \langle S_1I_2S_3 \rangle - 2\tau \langle S_1I_2S_3I_4 \rangle , \end{aligned}$$

(55)

$$\begin{aligned} \dot{\langle S_1I_2S_4 \rangle }&= -(\tau + \gamma ) \langle S_1S_3I_4 \rangle - 2\tau \langle S_1I_2I_3S_4 \rangle , \end{aligned}$$

(56)

$$\begin{aligned} \dot{\langle S_1I_2I_3I_4 \rangle }&= -3(\tau + \gamma ) \langle S_1I_2I_3I_4 \rangle + \tau ( \langle S_1I_2I_3S_4 \rangle + \langle S_1I_2S_3I_4 \rangle ), \end{aligned}$$

(57)

$$\begin{aligned} \dot{\langle S_1I_2I_3S_4 \rangle }&= -(3\tau + 2\gamma ) \langle S_1I_2I_3S_4 \rangle , \end{aligned}$$

(58)

$$\begin{aligned} \dot{\langle S_1I_2S_3I_4 \rangle }&= -(3\tau + 2\gamma ) \langle S_1I_2S_3I_4 \rangle , \end{aligned}$$

(59)

$$\begin{aligned} \dot{ \langle S_1S_2I_3I_4 \rangle }&= -2(\tau + \gamma ) \langle S_1S_2I_3I_4 \rangle + \tau \langle S_1S_2S_3I_4 \rangle +\tau \langle S_1S_2I_3S_4 \rangle , \end{aligned}$$

(60)

$$\begin{aligned} \dot{\langle S_1S_2I_3S_4 \rangle }&= -(2\tau + \gamma ) \langle S_1S_2I_3S_4 \rangle , \end{aligned}$$

(61)

$$\begin{aligned} \dot{\langle S_1S_2S_3I_4 \rangle }&= -(2\tau + \gamma ) \langle S_1S_2S_3I_4 \rangle , \end{aligned}$$

(62)

$$\begin{aligned} \dot{\langle S_1I_2S_3S_4) \rangle }&= -(\tau + \gamma ) \langle S_1I_2S_3S_4 \rangle , \end{aligned}$$

(63)

can be closed by using the following identities:

$$\begin{aligned} \langle S_1I_2I_4 \rangle \langle S_1 \rangle&= \langle S_1I_2 \rangle \langle S_1 I_4 \rangle ,\end{aligned}$$

(64)

$$\begin{aligned} \langle S_1I_2I_3 \rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 I_3 \rangle ,\end{aligned}$$

(65)

$$\begin{aligned} \langle S_1S_2I_3\rangle \langle S_1 \rangle&= \langle S_1S_2\rangle \langle S_1 I_3\rangle ,\end{aligned}$$

(66)

$$\begin{aligned} \langle S_1S_2I_4\rangle \langle S_1 \rangle&= \langle S_1S_2\rangle \langle S_1 I_4\rangle ,\end{aligned}$$

(67)

$$\begin{aligned} \langle S_1I_2S_3\rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 S_3 \rangle ,\end{aligned}$$

(68)

$$\begin{aligned} \langle S_1I_2S_4 \rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 S_4\rangle ,\end{aligned}$$

(69)

$$\begin{aligned} \langle S_1I_2I_3I_4\rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 I_3 I_4 \rangle ,\end{aligned}$$

(70)

$$\begin{aligned} \langle S_1I_2I_3S_4 \rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 I_3 S_4\rangle ,\end{aligned}$$

(71)

$$\begin{aligned} \langle S_1I_2S_3I_4 \rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 S_3 I_4\rangle ,\end{aligned}$$

(72)

$$\begin{aligned} \langle S_1S_2I_3I_4 \rangle \langle S_1 \rangle&= \langle S_1S_2\rangle \langle S_1 I_3 I_4 \rangle ,\end{aligned}$$

(73)

$$\begin{aligned} \langle S_1S_2I_3S_4 \rangle \langle S_1 \rangle&= \langle S_1S_2\rangle \langle S_1 I_3 S_4 \rangle ,\end{aligned}$$

(74)

$$\begin{aligned} \langle S_1S_2S_3I_4 \rangle \langle S_1 \rangle&= \langle S_1S_2\rangle \langle S_1 S_3 I_4 \rangle ,\end{aligned}$$

(75)

$$\begin{aligned} \langle S_1I_2S_3S_4\rangle \langle S_1 \rangle&= \langle S_1I_2\rangle \langle S_1 S_3 S_4 \rangle . \end{aligned}$$

(76)

We note that there are two distinct types of closures. Namely, closures at the level of triples that are not part of the triangle and closures at the full system size. To complete the closed system we need the following extra equations for variables that are required by the closures. These new variables together with their equations are:

$$\begin{aligned} \dot{\langle S_1 S_2 \rangle }&= -\tau \langle S_1S_2 I_3 \rangle - \tau \langle S_1S_2 I_4 \rangle ,\end{aligned}$$

(77)

$$\begin{aligned} \dot{\langle S_1 S_3 \rangle }&= -\tau \langle S_1I_2 S_3 \rangle - 2\tau \langle S_1S_3 I_4 \rangle ,\end{aligned}$$

(78)

$$\begin{aligned} \dot{\langle S_1 S_4 \rangle }&= -\tau \langle S_1I_2S_4 \rangle - 2\tau \langle S_1I_3 S_4 \rangle ,\end{aligned}$$

(79)

$$\begin{aligned} \dot{\langle S_1 S_3 S_4 \rangle }&= -\tau \langle S_1I_2 S_3 S_4 \rangle . \end{aligned}$$

(80)

Substituting the closures given in Eqs. (64–76) into Eqs. (51–63) together with the set of equations that cannot be closed, Eqs. (29–50), and the extra variables induced by the closures, Eqs. (77– 80), will result in a system of 26 differential equations describing the system dynamics completely. Without closures, the system is fully specified by 35 equations. Strictly speaking, we can drop the equations for $\langle S_i \rangle$ if we are only interested in prevalence and then the equations in the full and RS drop to 31 and 23. Note that in the full system all $\langle S_i \rangle$s can be dropped, while in the RS we cannot drop $\langle S_1 \rangle$ as the closures rely on it.

Appendix C: equations for toast network

The evolution equations on the toast network labeled as in Fig. 1d are given by

$$\begin{aligned} \dot{\langle I_1 \rangle }&= \tau \langle S_1I_2 \rangle + \tau \langle S_1I_3 \rangle + \tau \langle S_1I_4 \rangle - \gamma \langle I_1 \rangle ,\end{aligned}$$

(81)

$$\begin{aligned} \dot{\langle S_1 \rangle }&= -\tau \langle S_1I_2 \rangle - \tau \langle S_1I_3 \rangle - \tau \langle S_1I_4 \rangle ,\end{aligned}$$

(82)

$$\begin{aligned} \dot{ \langle I_2 \rangle }&= \tau \langle I_1S_2 \rangle +\tau \langle S_2I_3 \rangle - \gamma \langle I_2 \rangle ,\end{aligned}$$

(83)

$$\begin{aligned} \dot{\langle S_2 \rangle }&= -\tau \langle I_1S_2 \rangle -\tau \langle S_2I_3 \rangle ,\end{aligned}$$

(84)

$$\begin{aligned} \dot{\langle I_3 \rangle }&= \tau \langle I_1S_3 \rangle + \tau \langle S_3I_4 \rangle + \tau \langle I_2S_3 \rangle - \gamma \langle I_3 \rangle ,\end{aligned}$$

(85)

$$\begin{aligned} \dot{\langle S_3 \rangle }&= -\tau \langle I_1S_3 \rangle - \tau \langle S_3I_4 \rangle - \tau \langle I_2S_3 \rangle ,\end{aligned}$$

(86)

$$\begin{aligned} \dot{\langle I_4 \rangle }&= \tau \langle I_1S_4 \rangle + \tau \langle I_3S_4 \rangle - \gamma \langle I_4 \rangle , \end{aligned}$$

(87)

$$\begin{aligned} \dot{\langle S_4 \rangle }&= -\tau \langle I_1S_4 \rangle - \tau \langle I_3S_4 \rangle , \end{aligned}$$

(88)

$$\begin{aligned} \dot{\langle I_1S_2 \rangle }&= -(\tau + \gamma ) \langle I_1S_2 \rangle + \tau \langle S_1S_2I_3 \rangle + \tau \langle S_1S_2I_4 \rangle - \tau \langle I_1S_2I_3\rangle ,\end{aligned}$$

(89)

$$\begin{aligned} \dot{\langle S_1I_2 \rangle }&= -(\tau + \gamma ) \langle S_1I_2 \rangle + \tau \langle S_1S_2I_3 \rangle - \tau \langle S_1I_2I_3 \rangle - \tau \langle S_1I_2I_4\rangle ,\end{aligned}$$

(90)

$$\begin{aligned} \dot{\langle I_1S_3 \rangle }&= -(\tau + \gamma ) \langle I_1S_3 \rangle + \tau \langle S_1I_2S_3 \rangle + \tau \langle S_1S_3I_4 \rangle - \tau \langle I_1S_3I_4 \rangle - \tau \langle I_1I_2S_3 \rangle ,\nonumber \\ \end{aligned}$$

(91)

$$\begin{aligned} \dot{\langle S_1I_3 \rangle }&= -(\tau + \gamma ) \langle S_1I_3 \rangle - \tau \langle S_1I_2I_3 \rangle - \tau \langle S_1I_3I_4 \rangle +\tau \langle S_1S_3I_4 \rangle + \tau \langle S_1I_2S_3\rangle ,\nonumber \\ \end{aligned}$$

(92)

$$\begin{aligned} \dot{\langle I_1S_4 \rangle }&= -(\tau + \gamma ) \langle I_1S_4 \rangle + \tau \langle S_1I_2S_4 \rangle + \tau \langle S_1I_3S_4 \rangle - \tau \langle I_1I_3S_4 \rangle , \end{aligned}$$

(93)

$$\begin{aligned} \dot{\langle S_1I_4 \rangle }&= -(\tau + \gamma ) \langle S_1I_4 \rangle - \tau \langle S_1I_2I_4 \rangle - \tau \langle S_1I_3I_4 \rangle + \tau \langle S_1I_3S_4 \rangle , \end{aligned}$$

(94)

$$\begin{aligned} \dot{\langle S_3I_4 \rangle }&= -(\tau + \gamma ) \langle S_3I_4 \rangle + \tau \langle I_1S_3S_4 \rangle - \tau \langle I_1S_3I_4\rangle - \tau \langle I_2S_3I_4 \rangle ,\end{aligned}$$

(95)

$$\begin{aligned} \dot{\langle I_3S_4 \rangle }&= -(\tau + \gamma ) \langle I_3S_4 \rangle + \tau \langle I_1S_3S_4\rangle - \tau \langle I_1I_3S_4 \rangle + \tau \langle I_2S_3S_4 \rangle ,\end{aligned}$$

(96)

$$\begin{aligned} \dot{\langle S_2I_3 \rangle }&= -(\tau + \gamma ) \langle S_2I_3 \rangle + \tau \langle I_1S_2S_3 \rangle - \tau \langle I_1S_2I_3 \rangle + \tau \langle S_2S_3I_4 \rangle ,\end{aligned}$$

(97)

$$\begin{aligned} \dot{\langle I_2S_3\rangle }&= -(\tau + \gamma ) \langle I_2S_3 \rangle + \tau \langle I_1S_2S_3 \rangle - \tau \langle I_1I_2S_3 \rangle - \tau \langle I_2S_3I_4\rangle ,\end{aligned}$$

(98)

$$\begin{aligned} \dot{\langle S_1I_2I_4 \rangle }&= -2(\tau + \gamma ) \langle S_1I_2I_4 \rangle + \tau \langle S_1I_2I_3S_4 \rangle - \tau \langle S_1I_2I_3I_4 \rangle + \tau \langle S_1S_2I_3I_4\rangle ,\nonumber \\ \end{aligned}$$

(99)

$$\begin{aligned} \dot{\langle S_1I_2I_3 \rangle }&= -2(\tau + \gamma ) \langle S_1I_2I_3 \rangle + \tau \langle S_1I_2S_3I_4 \rangle - \tau \langle S_1I_2I_3I_4 \rangle \nonumber \\&+\,\, \tau \langle S_1I_2S_3 \rangle + \tau \langle S_1S_2I_3 \rangle ,\end{aligned}$$

(100)

$$\begin{aligned} \dot{\langle S_1I_3I_4 \rangle }&= -2(\tau + \gamma )\langle S_1I_3I_4\rangle + \tau \langle S_1S_3I_4\rangle +\tau \langle S_1I_3S_4 \rangle \nonumber \\&-\,\, \tau \langle S_1I_2I_3I_4 \rangle + \tau \langle S_1I_2S_3I_4\rangle ,\end{aligned}$$

(101)

$$\begin{aligned} \dot{ \langle S_1S_2I_3 \rangle }&= -(2\tau + \gamma ) \langle S_1S_2I_3 \rangle + \tau \langle S_1S_2S_3I_4 \rangle - \tau \langle S_1S_2I_3I_4 \rangle ,\end{aligned}$$

(102)

$$\begin{aligned} \dot{ \langle S_1S_2I_4 \rangle }&= -(\tau + \gamma ) \langle S_1S_2I_4 \rangle + \tau \langle S_1S_2I_3S_4 \rangle - 2\tau \langle S_1S_2I_3I_4 \rangle ,\end{aligned}$$

(103)

$$\begin{aligned} \dot{ \langle S_1I_2S_3 \rangle }&= -(2\tau + \gamma ) \langle S_1I_2S_3 \rangle - 2\tau \langle S_1I_2S_3I_4 \rangle , \end{aligned}$$

(104)

$$\begin{aligned} \dot{ \langle S_1S_3I_4 \rangle }&= -(2\tau + \gamma ) \langle S_1S_3I_4 \rangle - 2\tau \langle S_1I_2S_3I_4 \rangle , \end{aligned}$$

(105)

$$\begin{aligned} \dot{ \langle I_1S_3I_4 \rangle }&= -2(\tau + \gamma ) \langle I_1S_3I_4 \rangle + \tau \langle S_1I_2S_3I_4 \rangle + \tau \langle S_1S_3I_4 \rangle \nonumber \\&+\,\, \tau \langle I_1S_3S_4 \rangle - \tau \langle I_1I_2S_3I_4 \rangle , \end{aligned}$$

(106)

$$\begin{aligned} \dot{ \langle S_1I_2S_4 \rangle }&= -(\tau + \gamma ) \langle S_1I_2S_4 \rangle - 2\tau \langle S_1I_2I_3S_4 \rangle +\tau \langle S_1S_2I_3S_4 \rangle , \end{aligned}$$

(107)

$$\begin{aligned} \dot{ \langle S_1I_3S_4 \rangle }&= -(2\tau + \gamma ) \langle S_1I_3S_4 \rangle - \tau \langle S_1I_2I_3S_4 \rangle + \tau \langle S_1I_2S_3S_4 \rangle , \end{aligned}$$

(108)

$$\begin{aligned} \dot{ \langle I_1I_3S_4 \rangle }&= -2(\tau + \gamma ) \langle I_1I_3S_4 \rangle + \tau \langle S_1I_3S_4 \rangle + \tau \langle I_1S_3S_4 \rangle \nonumber \\&+\,\, \tau \langle S_1I_2I_3S_4 \rangle + \tau \langle I_1I_2S_3S_4 \rangle ,\end{aligned}$$

(109)

$$\begin{aligned} \dot{ \langle I_1S_3S_4 \rangle }&= -(2\tau + \gamma ) \langle I_1S_3S_4 \rangle + \tau \langle S_1I_2S_3S_4 \rangle - \tau \langle I_1I_2S_3S_4 \rangle \end{aligned}$$

(110)

$$\begin{aligned} \dot{ \langle I_1S_2I_3 \rangle }&= -2(\tau + \gamma ) \langle I_1S_2I_3 \rangle + \tau \langle S_1S_2I_3I_4 \rangle + \tau \langle I_1S_2S_3I_4 \rangle \nonumber \\&+\,\,\tau \langle S_1S_2I_3\rangle +\tau \langle I_1S_2S_3 \rangle , \end{aligned}$$

(111)

$$\begin{aligned} \dot{ \langle I_1I_2S_3 \rangle }&= -2(\tau + \gamma ) \langle I_1I_2S_3 \rangle + \tau \langle S_1I_2S_3I_4 \rangle + \tau \langle I_1I_2S_3I_4 \rangle \nonumber \\&+\,\,\tau \langle S_1I_2S_3 \rangle +\tau \langle I_1S_2S_3\rangle , \end{aligned}$$

(112)

$$\begin{aligned} \dot{\langle I_2S_3I_4 \rangle }&= -2(\tau + \gamma ) \langle I_2S_3I_4 \rangle + \tau \langle I_1S_2S_3I_4 \rangle + \tau \langle I_1I_2S_3S_4 \rangle - \tau \langle I_1I_2S_3I_4 \rangle , \nonumber \\ \end{aligned}$$

(113)

$$\begin{aligned} \dot{\langle I_2S_3S_4 \rangle }&= -(\tau + \gamma ) \langle I_2S_3S_4 \rangle + \tau \langle I_1S_2S_3S_4 \rangle - 2\tau \langle I_1I_2S_3S_4 \rangle ,\end{aligned}$$

(114)

$$\begin{aligned} \dot{\langle S_2S_3I_4 \rangle }&= -(\tau + \gamma )\langle S_2S_3I_4 \rangle + \tau \langle I_1S_2S_3S_4 \rangle - 2\tau \langle I_1S_2S_3I_4 \rangle , \end{aligned}$$

(115)

$$\begin{aligned} \dot{\langle I_1S_2S_3 \rangle }&= -(2\tau + \gamma ) \langle I_1S_2S_3 \rangle + \tau \langle S_1S_2S_3I_4 \rangle - \tau \langle I_1S_2S_3I_4 \rangle , \end{aligned}$$

(116)

$$\begin{aligned} \dot{\langle S_1I_2I_3I_4 \rangle }&= -3(\tau + \gamma ) \langle S_1I_2I_3I_4 \rangle + \tau ( \langle S_1I_2I_3S_4 \rangle \nonumber \\&+\,\, 2\langle S_1I_2S_3I_4 \rangle + \langle S_1S_2I_3I_4 \rangle ), \end{aligned}$$

(117)

$$\begin{aligned} \dot{\langle S_1I_2I_3S_4 \rangle }&= -(3\tau + 2\gamma ) \langle S_1I_2I_3S_4 \rangle + \tau (\langle S_1S_2I_3S_4 \rangle +\langle S_1I_2S_3S_4 \rangle ), \end{aligned}$$

(118)

$$\begin{aligned} \dot{\langle S_1I_2S_3I_4 \rangle }&= -(4\tau + 2\gamma ) \langle S_1I_2S_3I_4 \rangle , \end{aligned}$$

(119)

$$\begin{aligned} \dot{\langle S_1S_2I_3I_4 \rangle }&= -(3\tau + 2\gamma ) \langle S_1S_2I_3I_4 \rangle + \tau \langle S_1S_2S_3I_4 \rangle +\tau \langle S_1S_2I_3S_4 \rangle , \end{aligned}$$

(120)

$$\begin{aligned} \dot{\langle S_1S_2I_3S_4 \rangle }&= -(3\tau + \gamma ) \langle S_1S_2I_3S_4 \rangle ,\end{aligned}$$

(121)

$$\begin{aligned} \dot{\langle S_1S_2S_3I_4 \rangle }&= -(2\tau + \gamma ) \langle S_1S_2S_3I_4 \rangle , \end{aligned}$$

(122)

$$\begin{aligned} \dot{\langle S_1I_2S_3S_4 \rangle }&= -(2\tau + \gamma ) \langle S_1I_2S_3S_4 \rangle , \end{aligned}$$

(123)

$$\begin{aligned} \dot{\langle I_1S_2S_3S_4 \rangle }&= -(3\tau + \gamma ) \langle I_1S_2S_3S_4 \rangle ,\end{aligned}$$

(124)

$$\begin{aligned} \dot{\langle I_1S_2S_3I_4 \rangle }&= -(3\tau + 2\gamma ) \langle I_1S_2S_3I_4 \rangle +\tau \langle I_1S_2S_3S_4 \rangle + \tau \langle S_1S_2S_3I_4 \rangle , \end{aligned}$$

(125)

$$\begin{aligned} \dot{\langle I_1I_2S_3S_4 \rangle }&= -(3\tau + 2\gamma ) \langle I_1I_2S_3S_4 \rangle +\tau \langle I_1S_2S_3S_4 \rangle + \tau \langle S_1I_2S_3S_4 \rangle , \end{aligned}$$

(126)

$$\begin{aligned} \dot{\langle I_1I_2S_3I_4 \rangle }&= -3(\tau + \gamma ) \langle I_1I_2S_3I_4 \rangle +\tau \langle I_1S_2S_3I_4 \rangle + \tau \langle I_1I_2S_3S_4 \rangle + 2\tau \langle S_1I_2S_3I_4 \rangle .\nonumber \\ \end{aligned}$$

(127)

Appendix D: equations for the star-triangle network

In this section we write down the system of differential equations that are an exact representation of the ${ SIR}$ epidemic on the star triangle-network, see Fig. 1g. To simplify the notation, nodes within triangle $i$, i.e. $t_i^1$ and $t_i^2$, are now denoted by $i_1$ and $i_2$. The relevant equations are:

$$\begin{aligned} \dot{\langle S_1 \rangle }&= - \tau \sum _{j=1}^M\sum _{k=1}^2 \langle S_1I_{j_k} \rangle ,\end{aligned}$$

(128)

$$\begin{aligned} \dot{ \langle I_{1} \rangle }&= + \tau \sum _{j=1}^M\sum _{k=1}^2 \langle S_1I_{j_k} \rangle -\gamma \langle I_1 \rangle ,\end{aligned}$$

(129)

$$\begin{aligned} \dot{ \langle S_{i_1} \rangle }&= - \tau \langle I_1S_{i_1} \rangle - \tau \langle S_{i_1}I_{i_2} \rangle ,\end{aligned}$$

(130)

$$\begin{aligned} \dot{ \langle I_{i_1} \rangle }&= \tau \langle I_1S_{i_1} \rangle + \tau \langle S_{i_1}I_{i_2} \rangle - \gamma \langle I_{i_1} \rangle ,\end{aligned}$$

(131)

$$\begin{aligned} \dot{ \langle S_{i_2} \rangle }&= - \tau \langle I_1S_{i_2} \rangle - \tau \langle I_{i_1}S_{i_2} \rangle ,\end{aligned}$$

(132)

$$\begin{aligned} \dot{ \langle I_{i_2} \rangle }&= \tau \langle I_1S_{i_2} \rangle + \tau \langle I_{i_1}S_{i_2} \rangle - \gamma \langle I_{i_2} \rangle ,\end{aligned}$$

(133)

$$\begin{aligned} \dot{ \langle S_{i_1}I_{i_2} \rangle }&= -(\tau +\gamma ) \langle S_{i_1}I_{i_2} \rangle + \tau \langle I_1S_{i_1}S_{i_2} \rangle - \tau \langle I_1S_{i_1}I_{i_2} \rangle , \end{aligned}$$

(134)

$$\begin{aligned} \dot{ \langle I_{i_1}S_{i_2} \rangle }&= -(\tau +\gamma ) \langle I_{i_1}S_{i_2} \rangle + \tau \langle I_1S_{i_1}S_{i_2} \rangle - \tau \langle I_1I_{i_1}S_{i_2} \rangle , \end{aligned}$$

(135)

$$\begin{aligned} \dot{ \langle I_1S_{i_1} \rangle }&= -(\tau +\gamma ) \langle I_1S_{i_1} \rangle + \tau \langle S_1S_{i_1}I_{i_2} \rangle - \tau \langle I_1S_{i_1}I_{i_2} \rangle \nonumber \\&+\,\, \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_1} \rangle \langle S_1I_{j_k} \rangle }{ \langle S_1 \rangle }, \end{aligned}$$

(136)

$$\begin{aligned} \dot{ \langle I_1S_{i_2} \rangle }&= -(\tau +\gamma ) \langle I_1S_{i_2} \rangle + \tau \langle S_1I_{i_1}S_{i_2} \rangle - \tau \langle I_1I_{i_1}S_{i_2} \rangle \nonumber \\&+\,\, \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_2} \rangle \langle S_1I_{j_k} \rangle }{ \langle S_1 \rangle }, \end{aligned}$$

(137)

$$\begin{aligned} \dot{ \langle S_1 I_{i_1} \rangle }&= - (\tau +\gamma ) \langle S_1I_{i_1} \rangle -\tau \langle S_1 I_{i_1} I_{i_2} \rangle + \tau \langle S_1 S_{i_1} I_{i_2} \rangle \nonumber \\&-\,\, \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1 I_{i_1} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle },\end{aligned}$$

(138)

$$\begin{aligned} \dot{ \langle S_1 I_{i_2} \rangle }&= - (\tau +\gamma ) \langle S_1I_{i_2} \rangle - \tau \langle S_1 I_{i_1} I_{i_2} \rangle + \tau \langle S_1 I_{i_1} S_{i_2} \rangle \nonumber \\&-\,\, \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1I_{i_2} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle },\end{aligned}$$

(139)

$$\begin{aligned} \dot{ \langle S_1 S_{i_1} \rangle }&= - \tau \langle S_1S_{i_1}I_{i_2} \rangle - \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_1} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle }, \end{aligned}$$

(140)

$$\begin{aligned} \dot{ \langle S_1 S_{i_2} \rangle }&= - \tau \langle S_1I_{i_1}S_{i_2} \rangle - \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_2} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle },\end{aligned}$$

(141)

$$\begin{aligned} \dot{ \langle I_1S_{i_1}S_{i_2} \rangle }&= -(2\tau +\gamma ) \langle I_1S_{i_1}S_{i_2} \rangle + \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_1}S_{i_2} \rangle \langle S_1I_{j_k} \rangle }{ \langle S_1 \rangle }, \end{aligned}$$

(142)

$$\begin{aligned} \dot{ \langle I_1S_{i_1}I_{i_2} \rangle }&= -2(\tau +\gamma ) \langle I_1S_{i_1}I_{i_2} \rangle + \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_1}I_{i_2} \rangle \langle S_1I_{j_k} \rangle }{ \langle S_1 \rangle } \nonumber \\&\quad + \tau \langle I_1S_{i_1}S_{i_2} \rangle + \tau \langle S_1S_{i_1}I_{i_2} \rangle , \end{aligned}$$

(143)

$$\begin{aligned} \dot{ \langle I_1I_{i_1}S_{i_2} \rangle }&= -2(\tau +\gamma ) \langle I_1I_{i_1}S_{i_2} \rangle + \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1I_{i_1}S_{i_2} \rangle \langle S_1I_{j_k} \rangle }{ \langle S_1 \rangle } \nonumber \\&\quad + \tau \langle I_1S_{i_1}S_{i_2} \rangle + \tau \langle S_1I_{i_1}S_{i_2} \rangle , \end{aligned}$$

(144)

$$\begin{aligned} \dot{ \langle S_1 S_{i_1} I_{i_2} \rangle }&= - (2 \tau + \gamma ) \langle S_1 S_{i_1} I_{i_2} \rangle - \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_1} I_{i_2} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle },\end{aligned}$$

(145)

$$\begin{aligned} \dot{ \langle S_1 I_{i_1} S_{i_2} \rangle }&= - (2 \tau + \gamma ) \langle S_1 I_{i_1} S_{i_2} \rangle - \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1I_{i_1} S_{i_2} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle },\end{aligned}$$

(146)

$$\begin{aligned} \dot{ \langle S_1I_{i_1}I_{i_2} \rangle }&= -2(\tau +\gamma ) \langle S_1I_{i_1}I_{i_2} \rangle + \tau \langle S_1S_{i_1}I_{i_2}\rangle +\tau \langle S_1I_{i_1}S_{i_2}\rangle \nonumber \\&- \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1I_{i_1}I_{i_2} \rangle \langle S_1I_{j_k} \rangle }{ \langle S_1 \rangle }, \end{aligned}$$

(147)

$$\begin{aligned} \dot{ \langle S_1 S_{i_1} S_{i_2} \rangle }&= - \tau \sum _{j=1, j \ne i}^M\sum _{k=1}^2\frac{ \langle S_1S_{i_1} S_{i_2} \rangle \langle S_1 I_{j_k} \rangle }{ \langle S_1 \rangle }, \end{aligned}$$

(148)

where, in Eqs. (136–148) we have used the following closures:

$$\begin{aligned} \langle S_1S_{i_1}S_{i_2}I_{k_j} \rangle \langle S_1 \rangle&= \langle S_1S_{i_1}S_{i_2} \rangle \langle S_1I_{k_j} \rangle ,\\ \langle S_1S_{i_1}I_{k_j} \rangle \langle S_1 \rangle&= \langle S_1S_{i_1} \rangle \langle S_1I_{k_j}\rangle ,\\ \langle S_1S_{i_2}I_{k_j}\rangle \langle S_1 \rangle&= \langle S_1S_{i_2}\rangle \langle S_1I_{k_j}\rangle ,\\ \langle S_1S_{i_1}I_{i_2}I_{k_j} \rangle \langle S_1 \rangle&= \langle S_1S_{i_1}I_{i_2}\rangle \langle S_1I_{k_j}\rangle , \end{aligned}$$

and

$$\begin{aligned} \langle S_1I_{i_1}S_{i_2}I_{k_j} \rangle \langle S_1 \rangle = \langle S_1I_{i_1}S_{i_2}\rangle \langle S_1I_{k_j}\rangle , \end{aligned}$$

for $i, k=1, 2, \ldots , M$, $i \ne k$, and $j=1, 2$.

These closures are of two main type, namely:

• 1. Closure of a triple which is not a triangle $(i\ne j)$:

  $$\begin{aligned} \langle X_{i_l}S_1Y_{j_k} \rangle \langle S_1 \rangle =\langle X_{i_l}S_1 \rangle \langle S_1Y_{j_k}\rangle , \end{aligned}$$

  (149)

  where $l,k=1,2$, $i,j=1,\ldots ,M$ and $X, Y$ are either $S$ or $I$ in some particular combination.

• 2. Closure of a quadruple containing a triangle:

  $$\begin{aligned} \langle X_{i_1}Y_{i_2}S_1Z_{j_k} \rangle \langle S_1 \rangle =\langle X_{i_1}Y_{i_2}S_1\rangle \langle S_1Z_{j_k}\rangle , \end{aligned}$$

  (150)

  where $k=1,2$, $i,j=1,\ldots ,M$ and $X, Y, Z$ are either $S$ or $I$ in some particular combination.