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

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 $\{\pi_j\}$ 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 $\{\tau_i\}$. 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 $\{\tau_i\}$, 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.