Bounds on $R_0$ and final epidemic size when the next-generation matrix $M$ is only partially known

Abstract

We study a multitype SIR epidemic model where individuals are categorized into different types, and where infection spread is characterized by a next-generation matrix $M=\{m_{ij}\}$ with community fractions $\{π_j\}$ for the different types of individuals. We analyse two key quantities: the basic reproduction number $R_0$ and the final epidemic outcome of the different types $\{τ_i\}$. We consider the situation where $M$ is only partly known, through the row sums $\{r_i\}$ or the column sums $\{c_j\}$, and treat both a general $M$ and the special but common situation where $M$ is proportional to a contact matrix satisfying detailed balance. For a general $M$, which is partially observed through $\{r_i\}$ or $\{c_j\}$, we obtain sharp upper and lower bounds of $R_0$ and $\{τ_i\}$, but for the case where $M$ satisfies detailed balance the problem is harder: our obtained bounds for $R_0$ are narrower than the general case but still not sharp, and bounds for the final size are only obtained when there are two types of individual.

Figures (3)

• Figure 1: Plots showing regions in the $(r_1,r_2)$ plane of different behaviours of $\bar{\tau}(\theta)$ for different values of $\pi_1$. The solid red curve is $\hat{r}_2(r_1)$ and the blue curve is $\tilde{r}_2(r_1)$. The green line is $r_2=r_1$. Recall that $\bar{\tau}(\theta)$ is identically zero if $r_2 \le 1$ and the analysis assumes $r_2>r_1$.
• Figure 2: Plots showing lower and upper bounds for $R_0$ and $\bar{\tau}$, both under generic $M$ and under detailed balance restriction, knowing only the row sums, when there are $k=2$ types of individuals, $\pi_1=0.2$ and $r_1=1.6$.
• Figure 3: Different bounds for $R_0$ and the final size $\bar{\tau}$ in the Belgian study taking heterogeneity within age-groups into account.

Theorems & Definitions (38)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3: Known column sums
• Remark 3.1
• Theorem 3.4: Known row sums
• Remark 3.2
• Theorem 3.5: Known row sums
• Remark 3.3
• Remark 3.4
• Remark 3.5