Optimal control of a kinetic model describing social interactions on a graph

TL;DR

The paper develops a kinetic model for agents migrating on a graph and exchanging a viral load, and introduces two control channels: mobility control via a controlled transition matrix $P^u$ and in-node control of interactions, to minimize the node-weighted mean $\rho_i m_i$. It extends the basic binary-exchange model to a separate infection-healing framework, proving that eradication is achievable under appropriate controls and parameter regimes, and derives optimality conditions and a basic reproduction number $\mathcal{R}_0$ under control. Through analytical results (Metzler-system stability) and extensive numerical experiments, including a real-world Lombardy mobility network, the work demonstrates how the interplay between mobility and local interventions shapes the epidemic outcome and can significantly reduce or eradicate infection under feasible costs. These findings offer a principled, scalable approach to policy design for networked populations where both movement and local interactions can be targeted. The methodology and insights are relevant for epidemiology, network science, and applied kinetic theory, with potential adaptations to other social-dynamics on graphs.

Abstract

In this paper we introduce the optimal control of a kinetic model describing agents who migrate on a graph and interact within its nodes exchanging a physical quantity. As a prototype model, we consider the spread of an infectious disease on a graph, so that the exchanged quantity is the viral-load. The control, exerted on both the mobility and on the interactions separately, aims at minimising the average macroscopic viral-load. We prove that minimising the average viral-load weighted by the mass in each node is the most effective and convenient strategy. We consider two different interactions: in the first one the infection (gain) and the healing (loss) processes happen within the same interaction, while in the second case the infection and healing result from two different processes. With the appropriate controls, we prove that in the first case it is possible to stop the increase of the disease, but paying a very high cost in terms of control, while in the second case it is possible to eradicate the disease. We test numerically the role of each intervention and the interplay between the mobility and the interaction control strategies in each model.

Figures (9)

• Figure 1: Left to right: example of evolution in time of number of agents and average viral load in the case $\nu_1^i = \nu_2^i = 1/2$ for all $i \in \mathcal{I}$. The numerical test shows accordance with Proposition \ref{['prop2']}. Refer to Section \ref{['sec:num1']} for more details about the simulation.
• Figure 2: Left to right: evolution in time of number of agents (divided by $\rho_i^c$) and average viral load, in absence of mobility ($\chi = 0$). We see that, in those nodes where the initial mass fraction $\rho_i$ is greater than the associated critical value $\rho_i^c$, the average viral load grows exponentially, while in the other nodes vanishes. Refer to Section \ref{['sec:num2']} for additional details about the simulation.
• Figure 3: Evolution in time of number of agents (left) and average viral load (right): uncontrolled case. The infection grows exponentially on the graph.
• Figure 4: Columns: evolution in time of number of agents (left) and average viral load (right). Top row: effects of controlling the agents' mobility alone. Agents leave node number 3 and distribute in the remaining vertices. The overall viral load is decreased by a factor of 3 with respect to the uncontrolled scenario. Bottom row: effects of node isolation. Without in-node interventions and preventing highly infectious agents to distribute in nodes with lower load, the infection can grow dramatically by several order of magnitudes.
• Figure 5: Columns: evolution in time of number of agents (left) and average viral load (right). Top row: effects of partial in-node interventions. Even if limited to the early stages of infection, when it is still spreading to a comparatively low levels, control policies that slow in-node interactions between agent have a meaningful impact when compared to interventions on the mobility alone, with a reduction of about 30% in the average viral load values. Bottom row: both in-node interactions and mobility are controlled fully. The infection ceases spreading after the average viral load reaching a value nearly three orders of magnitudes lower than the uncontrolled scenario.

Theorems & Definitions (32)

• Proposition 1
• Remark 2
• Proposition 3
• proof
• Remark 4
• Proposition 5
• Lemma 6
• proof
• Proposition 7
• proof