I have the following ODE from a disease pandemic in a network ($i=1,2,..., n$): \begin{equation} \dot{x_i} = \beta s_i(1 -l_i)\sum_{j\in N} [A_{ij} (1-l_j)x_j] -(\gamma +\kappa) x_i \end{equation} \begin{equation} \dot{s_i} = - \beta s_i(1 -l_i)\sum_{j\in N} [A_{ij} (1-l_j) x_j] \end{equation} \begin{equation} \dot{r_i} = \gamma x_i \end{equation} \begin{equation} \dot{d_i} = \kappa x_i \end{equation} \begin{equation} x_i+ s_i + r_i +d_i =1 \end{equation} with the initial value point $(x_i(0), s_i(0), r_i(0), d_i(0))$ such that \begin{equation*} x_i(0) \geq 0, \ s_i(0) \geq 0, \ r_i(0)\geq 0, \ d_i(0)\geq 0, \ \text{and} \ x_i(0) + s_i(0) + r_i(0)+ d_i(0)= 1. \end{equation*} I drop the continuous-time $t \in [0, \infty)$; $x_i$ represents the probability for agent $i$ to be infected; $s_i$ agent $i$'s probability to be susceptible; $r_i$ agent $i$'s probability to recover; and $d_i$ agent $i$'s probability to die, with $x_i + s_i + r_i +d_i=1$ for all $i=1,..,n$. The social network structure $A$ is described by its adjacent matrix $A=[A_{ij}]$, with $A_{ii}=0$ for all $i=1,.., n$, and $A_{ij}\geq 0$ for all $i \neq j$. The variable $l_i$ is agent $i$'s probability of being sent into lockdown.
I want to (1) determine the basic reproduction number at the disease free equilibrium $E_0= (x_1,...,x_n, s_1,...s_n, r_1,...,r_n, d_1,..., d_n)$ where $x_i=0, s_i=1, r_i=0, d_i=0$ for each $i$; and (2) determine the final size of the epidemic.
- For (1), following the next generation matrix by Van den Driessche, P. & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29–48, I have: \begin{equation*} F= [\mathcal{F}_{ij} (E_0)],\ \mathcal{F}_{ij} (E_0) = \beta A_{ij}(1 -l_i) (1-l_j), \ \text{for} \ 1\leq i, j \leq n, \ \text{and} \end{equation*}
\begin{equation*} V^{-1}= diag (V_{ii}^{-1}), \ V_{ii} = \gamma + \kappa. \end{equation*} Then, $R_0 = \rho (FV^{-1})$, where $FV^{-1}= [M_{ij}]_{1\leq i, j \leq n}$, where $M_{ij}= \frac{\beta}{\gamma + \kappa} A_{ij} (1-l_i)(1-l_j)$. I know that, by definition, $R_0 = \rho (M):= \max \{|\lambda|: \lambda \ \text{is an eigenvalue of} \ M \}$. Given that the social network structure $A$ is undirected, it follows that $A_{ij}= A_{ji}$, so that $M_{ij}= M_{ji}$, for all $i$ and $j$. Hence, the next generation matrix $FV^{-1}$ is symmetric with a zero diagonal, since $M_{ii} = 0$ for each $i\in N$. Additionally, since all the values $A_{ij}$, $1-l_i$, and $1-l_j$ are real and non-negative, it follows that $FV^{-1}$ is a non-negative symmetric real matrix. Assuming that everything above is correct, my problem is on determining the largest eigenvalue of the matrix $M$.
- Form (2), I don't have any attempt to share since I am still getting myself familiar with the literature.
I appreciate any feedback, comment, or suggestions on addressing these two questions. Thank you.