Exact solutions and bounds for network SIR and SEIR models using a rooted-tree approximation.

Area of Science:

• Network science
• Mathematical epidemiology
• Computational dynamics

Background:

• Understanding contagion dynamics on complex networks is crucial for public health.
• Existing models often face computational challenges for large-scale network analysis.
• Markovian SIR and SEIR models are standard frameworks for disease spread simulation.

Purpose of the Study:

• To develop an efficient and accurate approximate model for contagion dynamics on networks.
• To provide exact solutions for specific network structures and bounded solutions for general networks.
• To analyze susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) dynamics.

Main Methods:

• Development of a new node-based approximate model for contagion.
• Mathematical analysis of the model's accuracy on tree graphs.
• Derivation of linear differential equations for node-state probabilities.
• Extension of analysis to SEIR models with multiple latent states.

Main Results:

• The approximate model is proven exact for Markovian SIR and SEIR dynamics on tree graphs with a single infection source.
• The model provides upper bounds on susceptible node probabilities for more general networks.
• Explicit closed-form solutions are derived for SIR models on tree graphs.
• A cooperative system of differential equations is presented for general network analysis.

Conclusions:

• The developed node-based model offers an efficient approach to studying network contagion.
• The model provides exactness in simplified cases and valuable bounds in complex scenarios.
• This work advances the mathematical understanding and computational modeling of epidemic spread.