Elementary proof of convergence to the mean-field model for the SIR process

Abstract

The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (IMA J Appl Math, doi:10.1093/imamat/hxw010, 2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.

Appendix

Lemma 4 Consider the initial value problems $x'=f_1(t,x)$, $x(0)=x_1$ and $x'=f_2(t,x)$, $x(0)=x_2$ with solutions $\varphi _1(t)$ and $\varphi _2(t)$ respectively. If $f_1$ is Lipschitz in x with constant L and $\left||f_1(t,x)-f_2(t,x)\right||\le M$, then $\left||\varphi _1(t)-\varphi _2(t)\right||\le (\left||x_1-x_2\right||+M/L)e^{Lt}-M/L$.

Proof

Using the integral form of the differential equations,

$$\begin{aligned} \left||\varphi _1(t)-\varphi _2(t)\right|| = \left|| x_1-x_2+\int _0^t f_1(u,\varphi _1(u)) du - \int _0^t f_2(u,\varphi _2(u)) du\right||. \end{aligned}$$

Using the triangle inequality,

$$\begin{aligned} \left||\varphi _1(t)-\varphi _2(t)\right|| \le \left||x_1-x_2\right||+\left||\int _0^t f_1(u,\varphi _1(u)) du - \int _0^t f_2(u,\varphi _2(u)) du\right||. \end{aligned}$$

Using the integral form of the triangle inequality,

$$\begin{aligned} \left||\varphi _1(t)-\varphi _2(t)\right|| \le \left|| x_1-x_2\right|| + \int _0^t \left||f_1(u,\varphi _1(u)) - f_2(u,\varphi _2(u))\right|| du. \end{aligned}$$

Applying the triangle inequality again,

$$\begin{aligned}&\le \left|| x_1-x_2\right||+\int _0^t \bigl (\left||f_1(u,\varphi _1(u))-f_1(u,\varphi _2(u))\right|| + \left||f_1(u,\varphi _2(u))-f_2(u,\varphi _2(u))\right||\bigr ) du. \end{aligned}$$

Applying the assumptions of this lemma,

$$\begin{aligned} \left||\varphi _1(t)-\varphi _2(t)\right|| \le \left|| x_1-x_2\right|| + \int _0^t (L \left||\varphi _1(u)-\varphi _2(u)\right|| + M) du. \end{aligned}$$

Applying a specialized form of Gronwall’s inequality then proves the claim.

Lemma 5

(Gronwall’s Inequality) Suppose for $t\ge 0$, that $\theta (t)$ is a continuous nonnegative function; $L,M\ge 0$; and $\theta (t)\le \theta (0)+\int _0^t (L\theta (u)+M) du$. Then for $t\ge 0$, $\theta$ is bounded by the solution to the initial value problem $x'=Lx+M$, $x(0)=\theta (0)$:

$$\begin{aligned} \theta (t) \le (\theta (0)+M/L)e^{Lt}-M/L. \end{aligned}$$

Proof

This result can be found in any graduate ODE text such as Lemma 6.2 in Hale (2009).