Minimizing cumulative infections in SIS epidemic models over networks via an edge deletion algorithm

Abstract: In this paper, we investigate the discrete SIS (Susceptible-Infected-Susceptible) models. We focus on minimizing epidemic spreading over networks by extending an existing edge deletion algorithm to the SIS model. To achieve this, we employ the mean-field approximation to linearize the network dynamics into a deterministic SIS model. We analytically demonstrate that the total number of infections is upper-bounded by a super-modular function, thereby ensuring the efficiency of the edge-deletion approach. To evaluate the proposed method, we conduct experiments on synthetic Erdos-Renyi networks and the real-world dataset collected from BBC Pandemic Haslemere app. Numerical simulations validate our theoretical results, confirming that both configurations converge to the stable, disease-free equilibrium.

• The paper develops a greedy edge-deletion algorithm that minimizes a supermodular upper bound on cumulative infections after linearizing SIS dynamics with heterogeneous edge transmission rates.
• The analysis proves disease-free convergence when the spectral radius of the post-deletion transition matrix is below one, while simulations show the method outperforms random removal and max-degree heuristics on Erdős–Rényi and Haslemere networks.
• The approach offers polynomial-time optimization through repeated linear-system solves, but its effectiveness is limited by asymptotic approximations, the stability requirement, and the lack of an explicit guarantee for the true infection objective.

Overview

This paper addresses influence minimization in discrete-time SIS epidemic processes on networks. The authors, Dung and Ly, extend edge-deletion techniques previously developed for SIR models to the SIS setting, where the key obstacle is that infection counts are non-monotone over time because recovered individuals can be reinfected. Their approach linearizes the stochastic Markov-chain SIS dynamics via mean-field approximation into a deterministic model (D-SIS), establishes a spectral condition guaranteeing convergence to the disease-free equilibrium, and proves that a cumulative infection quantity is upper-bounded by a monotone supermodular function of the removed edge set. This surrogate converts an NP-hard combinatorial problem into one amenable to greedy optimization with polynomial-time per-iteration cost.

Model and problem formulation

The underlying process is a Markov chain SIS model on an undirected graph $G=(V,E)$ with $N$ nodes, where each node is either Susceptible or Infected. Infection along edge $(i,j)$ at time $t$ is governed by a random variable $\beta_{ij}(t)$, and recovery by $\delta_i(t)$; both are assumed i.i.d. and nonnegative (Assumption 1). The exact discrete recurrence involves the product $\prod_j(1-\beta_{ij}(t)I_i(t))$, which the authors approximate via mean-field expectations and the small-parameter expansion $(1-a)(1-b)\approx(1-a-b)$ for $a,b\ll1$. Under Assumption 2 ($hD_i<1$ and $N$0), this yields the deterministic system

$N$1

where $N$2, $N$3, $N$4, and $N$5 is the sampling parameter.

Two quantities are introduced: the increased number of infections $N$6 after removing edge set $N$7 (with $N$8), and the cumulative infection variation $N$9. The optimization problem asks for a subset $(i,j)$0 with $(i,j)$1 minimizing $(i,j)$2 over a candidate edge set $(i,j)$3. The paper states, without full proof but citing an analogous argument, that this problem is NP-hard.

A notable modeling choice distinguishes this work from prior formulations such as Pare et al.: rather than a common infection rate scaled by adjacency ($(i,j)$4), each edge carries its own expected transmission rate $(i,j)$5, allowing heterogeneous per-edge parameters.

Convergence analysis

Letting $(i,j)$6 denote the state transition matrix, the dynamics satisfy $(i,j)$7 entrywise. Under irreducibility of $(i,j)$8 (Assumption 3) together with the earlier assumptions, the paper proves that if the spectral radius satisfies $(i,j)$9, then $t$0, so the network converges asymptotically to the disease-free equilibrium $t$1. This result parallels Theorem 1 of Pare et al., but the proof technique—bounding through the entrywise inequality rather than their argument—is presented as an alternative derivation adapted to heterogeneous edge rates.

The practical significance is that the spectral condition defines the regime in which mitigation is meaningful: if $t$2, the deterministic approximation admits an endemic equilibrium and the cumulative-infection bound developed next does not apply.

Supermodular upper bound and the edge deletion algorithm

The central theoretical contribution is Theorem 2: if $t$3 for the post-deletion transition matrix $t$4, then

$t$5

The proof chains three inequalities: induction on the linear bound gives a geometric-series sum approximated by $t$6; the increment bound $t$7 controls the cumulative variation; and nonnegativity of both $t$8 and $t$9 allows interchange of norm and summation. A caveat worth noting is that the geometric-sum step is stated as an approximation ("$\beta_{ij}(t)$0") justified by $\beta_{ij}(t)$1; the bound is therefore asymptotic in character rather than an exact finite-time inequality as written.

Proposition 1 then establishes that $\beta_{ij}(t)$2 is non-increasing and supermodular in $\beta_{ij}(t)$3. The argument uses the Sherman–Morrison formula to show that the marginal decrease of $\beta_{ij}(t)$4 upon adding an edge removal shrinks as $\beta_{ij}(t)$5 grows, exploiting nonnegativity of $\beta_{ij}(t)$6 and an integral representation of the rank-one update. Since $\beta_{ij}(t)$7 is entrywise nonnegative and monotone in $\beta_{ij}(t)$8, the product inherits monotone supermodularity.

This structure licenses a greedy algorithm: at each of $\beta_{ij}(t)$9 rounds, remove the candidate edge $\delta_i(t)$0 maximizing the marginal drop $\delta_i(t)$1. Each round requires evaluating $\delta_i(t)$2 for all remaining candidates, which involves solving linear systems with the matrix $\delta_i(t)$3; the supermodularity guarantees the greedy solution carries the standard near-optimality guarantees available for monotone submodular minimization via surrogate bounds, though the paper does not state an explicit approximation ratio relative to the true objective $\delta_i(t)$4.

Experimental evaluation

Simulations compare the proposed Edge Deletion Algorithm (EDA) against two baselines: random edge removal and max-degree heuristic. Two networks are used:

Setting	Nodes	Edges	Candidate set $\delta_i(t)$5	Removals $\delta_i(t)$6	Recovery rate	Infection rate
Erdős–Rényi ($\delta_i(t)$7)	700	5568	2500	500	$\delta_i(t)$8	$\delta_i(t)$9
Haslemere (BBC Pandemic)	—	1162 (after degree cap at 15)	520	210	$\prod_j(1-\beta_{ij}(t)I_i(t))$0	$\prod_j(1-\beta_{ij}(t)I_i(t))$1

Both use $\prod_j(1-\beta_{ij}(t)I_i(t))$2 and five seed nodes with initial infections in $\prod_j(1-\beta_{ij}(t)I_i(t))$3 (ER) or $\prod_j(1-\beta_{ij}(t)I_i(t))$4 (Haslemere). For the real-world Haslemere contact network from the BBC Pandemic dataset, preprocessing caps node degree at 15 to limit the influence of hubs—a deliberate deviation from the raw data that the authors justify as producing a more realistic setting, though it also removes exactly the high-degree structure that max-degree heuristics exploit.

The reported outcome is that EDA achieves the highest reduction in infections on both networks across all rounds of edge removal, outperforming random and max-degree baselines. Separately, convergence experiments confirm that infection trajectories decay to zero on both networks when $\prod_j(1-\beta_{ij}(t)I_i(t))$5, consistent with Theorem 1—even without any edge deletions under the chosen parameter regimes.

Limitations and open questions

Several constraints qualify the results. First, the entire framework rests on the spectral condition $\prod_j(1-\beta_{ij}(t)I_i(t))$6 holding after every deletion; the algorithm does not address how to proceed when this fails or when the pre-deletion network is already supercritical. Second, the upper-bound proof relies on the small-transmission approximation of the product term and on an asymptotic geometric-sum argument, so the tightness of $\prod_j(1-\beta_{ij}(t)I_i(t))$7 as a proxy for actual cumulative infections is not quantified. Third, no explicit approximation-ratio guarantee connecting the greedy solution under $\prod_j(1-\beta_{ij}(t)I_i(t))$8 to the optimum of the true NP-hard objective $\prod_j(1-\beta_{ij}(t)I_i(t))$9 is provided—the claim of "effectively solvable" rests on supermodularity of the surrogate, not on a proven bound for the original problem. Fourth, experiments use randomly drawn homogeneous ranges for $(1-a)(1-b)\approx(1-a-b)$0 and $(1-a)(1-b)\approx(1-a-b)$1 rather than empirically estimated epidemiological parameters, and the degree-capped Haslemere network may understate performance differences against degree-based heuristics. Finally, the model assumes static topology and i.i.d. transmission events; time-varying graphs and correlated dynamics remain outside the analysis, which the authors themselves identify as directions for future work.

Conclusion

This paper adapts edge-deletion mitigation from SIR to SIS epidemics by constructing a monotone supermodular upper bound on cumulative infections under a spectral stability condition, enabling greedy selection of edges whose removal most reduces epidemic size. The theory is supported by simulations on synthetic Erdős–Rényi graphs and the Haslemere contact network, where the proposed algorithm dominates random and max-degree baselines. The main open issues are tightening the gap between the surrogate and the true objective, providing finite-time (rather than asymptotic) bounds, and extending the analysis to dynamic topologies and supercritical regimes.