Network-Driven Global Stability Analysis: SVIRS Epidemic Model

TL;DR

This work analyzes a SVIRS epidemic model on a connected weighted network with graph-Laplacian diffusion to capture population mobility and vaccination dynamics. It derives a next-generation-based basic reproduction number $\mathcal{R}_0$ and shows the disease-free equilibrium exists for all parameters while an endemic equilibrium exists and is unique when $\mathcal{R}_0>1$, with global stability results obtained via Lyapunov functions on a reduced system. The authors prove that $\mathbf{D}$ is globally asymptotically stable for $\mathcal{R}_0<1$ and, under $\eta=0$, $\mathbf{E}$ is globally asymptotically stable for $\mathcal{R}_0>1$, using $L(t)$ and $L_*(t)$ to handle diffusion. Numerical simulations on the Minnesota road network validate the theoretical thresholds and demonstrate how mobility $\epsilon$ and vaccination parameters influence spatial spread and long-term outcomes.

Abstract

An epidemic Susceptible-Vaccinated-Infected-Removed-Susceptible (SVIRS) model is presented on a weighted-undirected network with graph Laplacian diffusion. Disease-free equilibrium always exists while the existence and uniqueness of endemic equilibrium have been shown. When the basic reproduction number is below unity, the disease-free equilibrium is asymptotically globally stable. The endemic equilibrium is asymptotically globally stable if the basic reproduction number is above unity. Numerical analysis is illustrated with a road graph of the state of Minnesota. The effect of all important model parameters has been discussed.

Figures (7)

• Figure 1: Flow chart of the model (\ref{['model']}) at node $x \in \mathcal{V}$.
• Figure 2: Sparsity of the Laplacian matrix corresponding the Minnesota road network. The graphical structure of the network is given in Fig. \ref{['fig:graph']}.
• Figure 3: The figure depicts the solution of infected individual $I(x, t)$, when the parameters are as follows: $\beta = 0.65/N, r =0.01, p = 0.002, \xi = 0.02, \eta = 0.03, \gamma = 0.11$. Furthermore, we assumed that the migration parameter rate or population mobility rate $\epsilon = 0.05$.
• Figure 4: The figure portrays that the solution of $I(x, t)$ while vaccination rate ($p$) is fixed at 0.02 and a fraction of newborn vaccinated is fixed at $r=0.001$. Moreover, $\xi = \eta = \sigma = 0.0$.
• Figure 5: The figure portraits that the solution of $I(x, t)$ while the population mobility rate is fixed at $\epsilon = 0.05$.

Theorems & Definitions (2)

• Lemma 2.1
• Theorem 2.2