We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node u of the graph has an initial value ξu(0). In the first process, which we call the NodeModel, at each time step t ≥ 0, a random node u and a random sample of k of its neighbours υ1, υ2, ... , υk are selected. Then, u updates its current value ξu(t) to [EQUATION], where α ∈ (0, 1) and k ≥ 1 are parameters of the process. In the second process, called the EdgeModel, at each step a random pair of adjacent nodes (u, υ) is selected, and then node u updates its value equivalently to the NodeModel with k = 1 and υ as the selected neighbour.

We consider a population protocol version of the SIR model. In every round, an individual is chosen uniformly at random. If the individual is susceptible, then it becomes infected w.p. $\beta I_t/N$, where $I_t$ is the number of infections at time $t$ and $N$ is the total number of individuals. If the individual is infected, then it recovers w.p. $\gamma$, whereas, if the individual is already recovered, nothing happens. We prove sharp bounds on the probability of the disease becoming pandemic vs extinguishing early (dying out quickly). The probability of extinguishing early, $\Pr{\mathcal{E}_{ext}}$, is typically neglected in prior work since most use (deterministic) differential equations. Leveraging on this, using $\Pr{\mathcal{E}_{ext}}$, we proceed by bounding the expected size of the population that contracts the disease $\mathbf{E}\left[R_\infty\right]$. Prior work only calculated $\mathbf{E}\left[R_\infty | \overline{\mathcal{E}_{ext}}\right]$, or obtained non-closed form solutions. We then study the two-country model also accounting for the role of $\Pr{\mathcal{E}_{ext}}$. We assume that both countries have different infection rates $\beta^{(i)}$, but share the same recovery rate $\gamma$. In this model, each round has two steps: First, an individual is chosen u.a.r. and travels w.p. $p_{travel}$ to the other country. Afterwards, the process continues as before with the respective infection rates. Finally, using simulations, we characterise the influence of $p_{travel}$ on the total number of infections. Our simulations show that, depending on the $\beta^{(i)}$, increasing $p_{travel}$ can decrease or increase the expected total number of infections $\mathbf{E}\left[R_\infty\right]$.