Dynamics of epidemic spreading on connected graphs

Area of Science:

• Mathematical Biology
• Epidemiology
• Network Science

Background:

• Understanding epidemic dynamics on complex networks is crucial for public health interventions.
• Existing models often simplify network structures or compartmental dynamics.

Purpose of the Study:

• To introduce a novel mathematical model for epidemic spreading on connected graphs.
• To analyze the system's properties and derive the final infected population.
• To develop and validate a numerical scheme for the proposed model.

Main Methods:

• A partial differential equation-ordinary differential equation (PDE-ODE) system.
• Incorporation of the SIR (Susceptible-Infected-Recovered) model at each vertex.
• Heat equations on edges with Robin boundary conditions for inter-vertex transmission.
• A semi-implicit finite difference numerical scheme.

Main Results:

• The model captures essential epidemic dynamics on graph structures.
• A closed-form expression for the final size of the epidemic is derived.
• The numerical scheme preserves key properties like solution uniqueness, positivity, and population conservation.
• Numerical simulations demonstrate the model's behavior on various connected graphs.

Conclusions:

• The proposed PDE-ODE model provides a robust framework for studying epidemics on networks.
• The developed numerical method is accurate and stable for simulating epidemic spread.
• This work offers a valuable tool for predicting and managing infectious disease outbreaks in connected populations.