A Class of Pairwise Models for Epidemic Dynamics on Weighted Networks

Abstract

In this paper, we study the SIS (susceptible–infected–susceptible) and SIR (susceptible–infected–removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with stochastic network simulations to evaluate the impact of different weight distributions on epidemic thresholds and dynamics in general. For the SIR model, the basic reproductive ratio R ₀ is computed, and we show that (i) for both network models R ₀ is maximised if all weights are equal, and (ii) when the two models are ‘equally-matched’, the networks with a random weight distribution give rise to a higher R ₀ value. The models with different weight distributions are also used to explore the agreement between the pairwise and simulation models for different parameter combinations.

Appendices

Appendix A: Reducing the Weighted Pairwise Models to the Un-weighted Equivalents

We start from the system

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where m=1,2,…,M. To close this system of equations at the level of pairs, we use the approximations

$$[ABC]_{mn}=\frac{k-1}{k}\frac{[AB]_{m}[BC]_{n}}{[B]}. $$

To reduce these equations to the standard pairwise model for un-weighted networks, we use the fact that $\sum_{m=1}^{M} {[AB]_{m}} = [AB]$ for A,B∈{S,I} and aim to derive the evolution equation for [AB]. Assuming that all weights are equal to W, the following relation holds:

where the summations of the triples can be resolved as follows:

Using the same argument for all other triples, the pairwise model for weighted networks with all weights being equal (without loss of generality W=1) reduces to the classic pairwise model, that is

A similar argument holds for the pairwise model on weighted networks with SIR dynamics.

Appendix B: Proof of Theorem 1

We illustrate the main steps needed to complete the proof of Theorem 1. This revolves around starting from the inequality itself and showing via a series of algebraic manipulations that it is equivalent to a simpler inequality that holds trivially. Upon using that p ₁ k=k ₁, p ₂ k=k ₂, and p ₂+p ₁=1, the original inequality can be rearranged to give

$$ \sqrt{\bigl[(k_1-1)r_1-(k_2-1)r_2 \bigr]^2+4k_1k_2r_1r_2} \le(k_1-1)r_1+(k_2-1)r_2+2r_1p_2+2r_2p_1. $$

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Based on the assumptions of the theorem, the right-hand side is positive, and thus this inequality is equivalent to the one where both the left- and right-hand sides are squared. Combined with the fact that p ₂=1−p ₁, after a series of simplifications and factorisations this inequality can be recast as

$$ 4p_1(1-p_1) \bigl(r_1^2+r_2^2 \bigr)+8kp_1(1-p_1)r_1r_2 \le 4kp_1(1-p_1) \bigl(r_1^2+r_2^2 \bigr)+8p_1(1-p_1)r_1r_2, $$

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which can be further simplified to

$$ 4p_1(1-p_1) (r_1-r_2)^2(k-1) \ge0, $$

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which holds trivially and thus completes the proof. We note that in the strictest mathematical sense the condition of the theorem should be (k ₁−1)r ₁+(k ₂−1)r ₂+2r ₁ p ₂+2r ₂ p ₁≥0. This holds if the current assumptions are observed since these are stronger but follow from a practical reasoning whereby for the network with fixed weight distribution, a node should have at least one link with every possible weight type.

Appendix C: Proof of Theorem 2

First, we show that $R_{0}^{1}$ is maximised when w ₁=w ₂=W. $R_{0}^{1}$ can be rewritten to give

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Maximising this given the constraint w ₁ p ₁+w ₂(1−p ₁)=W can be achieved by considering $R_{0}^{1}$ as a function of the two weights and incorporating the constraint into it via the Lagrange multiplier method. Hence, we define a new function f(w ₁,w ₂,λ) as follows:

Finding the extrema of this functions leads to a system of three equations

Expressing λ from the first two equations and equating these two expressions yields

$$ \frac{(k-1)\tau\gamma}{(\tau{w_{1}}+\gamma)^2} = \frac{(k-1)\tau\gamma }{(\tau{w_{2}}+\gamma)^2}. $$

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Therefore,

$$ w_{1} = w_{2} = W, $$

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and it is straightforward to confirm that this is a maximum.

Performing the same analysis for $R_{0}^{2}$ is possible but it is more tedious. Instead, we propose a more elegant argument to show that $R_{0}^{2}$ under the constraint of constant total link weight achieves its maximum when w ₁=w ₂=W. The argument starts by considering $R_{0}^{2}$ when w ₁=w ₂=W. In this case, and using that r ₂=r ₁=r=τW/(τW+γ) we can write:

However, it is known from Theorem 1 that $R_{0}^{2} \leq R_{0}^{1}$, and we have previously shown that $R_{0}^{1}$ under the present constraint achieves its maximum when w ₁=w ₂=W, and its maximum is equal to (k−1)r. All the above can be written as

$$ R_0^2 \leq R_{0}^1 \leq(k-1)r. $$

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Now taking into consideration that $R^{2\ast}_{0}=(k-1)r$, the inequality above can be written as

$$ R_0^2 \leq R_{0}^1 \leq(k-1)r =R^{2\ast}_0, $$

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and this concludes the proof.

Appendix D: The R ₀-Like Threshold R

Let us start from the evolution equation for [I](t),

where $\lambda_{1}= \frac{[SI]_{1}}{[I]}$ and $\lambda_{2}=\frac {[SI]_{2}}{[I]}$, and let R be defined as

$$ R = \frac{\tau{w_{1}}{\lambda_{1}}+\tau{w_{2}}{\lambda _{2}}}{\gamma}. $$

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Following the method outlined by Keeling (1999) and Eames (2008), we calculate the early quasi-equilibrium values of λ _1,2 as follows:

$$\begin{aligned} \dot{\lambda}_{1} &= 0\quad {\Leftrightarrow}\quad [\dot{SI}]_{1}[I] = \dot {[I]}[SI]_{1}, \\ \dot{\lambda}_{2} &= 0\quad {\Leftrightarrow}\quad [\dot{SI}]_{2}[I] = [\dot{I}]][SI]_{2}. \end{aligned} $$

Upon using the pairwise equations and the closure, consider $[\dot {S}I]_{1}[I] = [\dot{I}][SI]_{1}$:

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Using the classical closure

$$\begin{aligned} {[ABC]}_{12}&=\frac{k-1}{k}\frac{[AB]_{1}[BC]_{2}}{[B]} , \\ {[ABC]}_{21}&=\frac{k-1}{k}\frac{[AB]_{2}[BC]_{1}}{[B]}, \end{aligned} $$

and making the substitution: [SI]₁=λ ₁[I], [SI]₂=λ ₂[I], [I]≪1, [S]≈N, [SS]₁≈kNp ₁, [SS]₂≈kN(1−p ₁) together with γR=τw ₁ λ ₁+τw ₂ λ ₂, we have

$$({\tau}w_{1}\lambda_{1}+{\tau}w_{2} \lambda_{2})kp_{1}-({\tau}w_{1} \lambda_{1}+{\tau}w_{2}\lambda_{2})p_{1}-({ \tau}w_{1}\lambda_{1}+{\tau}w_{2} \lambda_{2})\lambda_{1}-\tau{w_{1}} \lambda_{1}=0, $$

which can be solved for λ ₁ to give

$$\lambda_{1} = \frac{\gamma{(k-1)}{p_{1}}{R}}{\tau{w_{1}}+\gamma{R}}. $$

Similarly, λ ₂ can be found as

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Substituting the expressions for λ _1,2 into the original equation for R yields

$$R = \frac{A+B+\sqrt{(A+B)^2+4\tau^2w_{1}w_{2}(k-2)}}{2\gamma}, $$

where A=τw ₁[(k−1)p ₁−1] and B=τw ₂[(k−1)p ₂−1]. If we define

$$R_{1} = \frac{\tau{w_{1}}[(k-1)p_{1}-1]}{\gamma},\quad \mbox{and} \quad R_{2} = \frac{\tau {w_{2}}[(k-1)p_{2}-1]}{\gamma}, $$

the expression simplifies to

$$R = \frac{R_{1}+R_{2}+\sqrt{(R_{1}+R_{2})^2+4R_{1}R_{2}Q}}{2}, $$

where $Q = \frac{(k-2)}{[(k-1)p_{1}-1][(k-1)p_{2}-1]}$.

Substituting the modified closure

into Eq. (24) and making further substitution: [SI]₁=λ ₁[I], [SI]₂=λ ₂[I], [I]≪1, [S]≈N, [SS]₁≈k ₁ N, [SS]₂≈k ₂ N, we have

$$({\tau}w_{1}\lambda_{1}+{\tau}w_{2} \lambda_{2})k_{1}-({\tau}w_{1} \lambda_{1}+{\tau}w_{2}\lambda_{2}) \lambda_{1}-2\tau w_1 \lambda_1 =0 \quad {\Longrightarrow}\quad \lambda_{1} = \frac{\gamma{k_{1}}{R}}{2\tau {w_{1}}+\gamma{R}}. $$

Similarly, the equation $[\dot{SI}]_{2}{[I]} = [\dot{I}][SI]_{2}$ yields

$$\lambda_{2}= \frac{\gamma{k_{2}}{R}}{2\tau{w_{2}}+\gamma{R}}. $$

Substituting these expressions for λ _1,2 into Eq. (23), we have

If we define

$$R_{1} = \frac{\tau{w_{1}}(k_{1}-2)}{\gamma},\qquad R_{2} = \frac{\tau{w_{2}}(k_{2}-2)}{\gamma}, $$

the above expression for R simplifies to

$$ R = \frac{R_{1}+R_{2}+\sqrt{(R_{1}+R_{2})^2+4R_{1}R_{2}(Q-1)}}{2}, $$

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where

$$Q = \frac{k_{1} k_{2}}{(k_{1}-2)(k_{2}-2)}. $$