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Transmission Neural Networks: Approximate Receding Horizon Control for Virus Spread on Networks

Shuang Gao, Peter E. Caines

TL;DR

The paper addresses controlling virus spread on networks when the full Markovian SIS model has a state space of size $2^n$. It leverages Transmission Neural Networks (TransNNs) to approximate infection dynamics with upper-bound guarantees and derives a receding horizon control framework that uses TransNN predictions to reduce computation significantly compared to dynamic programming, while offering competitive performance relative to TransNN-based optimal control. The contributions include establishing upper bounds on infection probabilities, formulating MDP and TransNN-based control schemes, and developing a receding horizon variant with demonstrated computational efficiency and effectiveness in numerical experiments. This work provides a scalable approach for networked epidemic control and highlights TransNNs as a practical tool for high-dimensional MDP-like problems in epidemiology and beyond.

Abstract

Transmission Neural Networks (TransNNs) proposed by Gao and Caines (2022) serve as both virus spread models over networks and neural network models with tuneable activation functions. This paper establishes that TransNNs provide upper bounds on the infection probability generated from the associated Markovian stochastic Susceptible-Infected-Susceptible (SIS) model with 2^n state configurations where n is the number of nodes in the network, and can be employed as an approximate model for the latter. Based on such an approximation, a TransNN-based receding horizon control approach for mitigating virus spread is proposed and we demonstrate that it allows significant computational savings compared to the dynamic programming solution to Markovian SIS model with 2^n state configurations, as well as providing less conservative control actions compared to the TransNN-based optimal control. Finally, numerical comparisons among (a) dynamic programming solutions for the Markovian SIS model, (b) TransNN-based optimal control and (c) the proposed TransNN-based receding horizon control are presented.

Transmission Neural Networks: Approximate Receding Horizon Control for Virus Spread on Networks

TL;DR

The paper addresses controlling virus spread on networks when the full Markovian SIS model has a state space of size . It leverages Transmission Neural Networks (TransNNs) to approximate infection dynamics with upper-bound guarantees and derives a receding horizon control framework that uses TransNN predictions to reduce computation significantly compared to dynamic programming, while offering competitive performance relative to TransNN-based optimal control. The contributions include establishing upper bounds on infection probabilities, formulating MDP and TransNN-based control schemes, and developing a receding horizon variant with demonstrated computational efficiency and effectiveness in numerical experiments. This work provides a scalable approach for networked epidemic control and highlights TransNNs as a practical tool for high-dimensional MDP-like problems in epidemiology and beyond.

Abstract

Transmission Neural Networks (TransNNs) proposed by Gao and Caines (2022) serve as both virus spread models over networks and neural network models with tuneable activation functions. This paper establishes that TransNNs provide upper bounds on the infection probability generated from the associated Markovian stochastic Susceptible-Infected-Susceptible (SIS) model with 2^n state configurations where n is the number of nodes in the network, and can be employed as an approximate model for the latter. Based on such an approximation, a TransNN-based receding horizon control approach for mitigating virus spread is proposed and we demonstrate that it allows significant computational savings compared to the dynamic programming solution to Markovian SIS model with 2^n state configurations, as well as providing less conservative control actions compared to the TransNN-based optimal control. Finally, numerical comparisons among (a) dynamic programming solutions for the Markovian SIS model, (b) TransNN-based optimal control and (c) the proposed TransNN-based receding horizon control are presented.
Paper Structure (11 sections, 4 theorems, 48 equations, 6 figures)

This paper contains 11 sections, 4 theorems, 48 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Virus Spread Dynamics Over Networks
  3. Approximation by TransNNs
  4. Infection Probability Upper Bound by TransNNs
  5. Controlling Virus Spread based on TransNNs
  6. MDP Solution for Markovian SIS Dynamics
  7. Optimal Control based on TransNNs
  8. Receding Horizon Control based on TransNNs
  9. Complexity
  10. Numerical Results
  11. Conclusion

Key Result

proposition 1

Assume (A1) and (A2) hold. Given the state configuration at time $k$ denoted by ${X(k)= x} \in \{0,1\}^n$, the transition probability to a state configuration $q\in\{0,1\}^n$ is given by where

Figures (6)

  • Figure 1: A network example with 5 nodes (left) and the transmission probabilities (right) among nodes.
  • Figure 2: Control actions (right) generated from the MDP control, and actual state realizations (left) the under such control actions.
  • Figure 3: Control actions (right) generated from TransNN-based optimal control and the infection probabilities (left) in the TransNN model under such control actions.
  • Figure 4: States of TransNNs (left) under the TransNN-based optimal control actions and the adjoint states (right). Brown squares (left) represent $s_i(0)= \infty$ which corresponds to $p_i(0)=1$. Although $s_i(0)$ may be $+\infty$ for some $i \in [n]$, the cost and the state $s_i(k)$ for all $k>0$ and all $i\in [n]$ always remain bounded.
  • Figure 5: Control actions (right) generated from TransNN-based receding horizon control (see Section \ref{['subsec:rhc']}) and one actual state realization (left) the under such control law.
  • ...and 1 more figures

Theorems & Definitions (7)

  • proposition 1: Conditional Probability of Infection ShuangPeterTransNNControl25
  • remark 1: Discussions on Assumption (A4)
  • proposition 2
  • proof
  • proposition 3
  • remark 2
  • proposition 4: ShuangPeterTransNNControl25