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A minimal model for multigroup adaptive SIS epidemics

Massimo A. Achterberg, Mattia Sensi, Sara Sottile

TL;DR

The paper advances adaptive epidemic modeling by generalizing the aNIMFA framework to a multigroup network of communities, capturing both local and global awareness through adaptive link dynamics. It derives a Next Generation Matrix-based basic reproduction number $R_0$, establishes boundedness and global stability of the disease-free state for $R_0<1$, and proves the existence of endemic equilibria when $R_0>1$. Numerical simulations reveal complex dynamics, including limit cycles in a two-community setting and the differential effectiveness of breaking intra- versus inter-community links across network topologies, as well as the interplay between local and global information in shaping adaptation. The work provides a flexible, extensible platform for studying adaptive contact patterns in SIS-type epidemics and suggests promising directions for extending to other compartmental models and seasonality, with publicly available code for replication.

Abstract

We propose a generalization of the adaptive N-Intertwined Mean-Field Approximation (aNIMFA) model studied in Achterberg and Sensi (2023) to a heterogeneous network of communities. In particular, the multigroup aNIMFA model describes the impact of both local and global disease awareness on the spread of a disease in a network. We obtain results on existence and stability of the equilibria of the system, in terms of the basic reproduction number $R_0$. Assuming individuals have no reason to decrease their contacts in the absence of disease, we show that the basic reproduction number $R_0$ is equivalent to the basic reproduction number of the NIMFA model on static networks. Based on numerical simulations, we demonstrate that with just two communities periodic behaviour can occur, which contrasts the case with only a single community, in which periodicity was ruled out analytically. We also find that breaking connections between communities is more fruitful compared to breaking connections within communities to reduce the disease outbreak on dense networks, but both strategies are viable to networks with fewer links. Finally, we emphasize that our method of modelling adaptivity is not limited to SIS models, but has huge potential to be applied in other compartmental models in epidemiology.

A minimal model for multigroup adaptive SIS epidemics

TL;DR

The paper advances adaptive epidemic modeling by generalizing the aNIMFA framework to a multigroup network of communities, capturing both local and global awareness through adaptive link dynamics. It derives a Next Generation Matrix-based basic reproduction number , establishes boundedness and global stability of the disease-free state for , and proves the existence of endemic equilibria when . Numerical simulations reveal complex dynamics, including limit cycles in a two-community setting and the differential effectiveness of breaking intra- versus inter-community links across network topologies, as well as the interplay between local and global information in shaping adaptation. The work provides a flexible, extensible platform for studying adaptive contact patterns in SIS-type epidemics and suggests promising directions for extending to other compartmental models and seasonality, with publicly available code for replication.

Abstract

We propose a generalization of the adaptive N-Intertwined Mean-Field Approximation (aNIMFA) model studied in Achterberg and Sensi (2023) to a heterogeneous network of communities. In particular, the multigroup aNIMFA model describes the impact of both local and global disease awareness on the spread of a disease in a network. We obtain results on existence and stability of the equilibria of the system, in terms of the basic reproduction number . Assuming individuals have no reason to decrease their contacts in the absence of disease, we show that the basic reproduction number is equivalent to the basic reproduction number of the NIMFA model on static networks. Based on numerical simulations, we demonstrate that with just two communities periodic behaviour can occur, which contrasts the case with only a single community, in which periodicity was ruled out analytically. We also find that breaking connections between communities is more fruitful compared to breaking connections within communities to reduce the disease outbreak on dense networks, but both strategies are viable to networks with fewer links. Finally, we emphasize that our method of modelling adaptivity is not limited to SIS models, but has huge potential to be applied in other compartmental models in epidemiology.
Paper Structure (14 sections, 5 theorems, 34 equations, 5 figures, 1 table)

This paper contains 14 sections, 5 theorems, 34 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The multigroup aNIMFA model
  3. Analytical results
  4. Boundedness of the feasible region
  5. Disease-Free Equilibrium
  6. Basic Reproduction Number
  7. Stability of the DFE
  8. Endemic Equilibria
  9. Numerical simulations
  10. Case study 1: Periodic behaviour
  11. Case study 2: Internal and external connections
  12. Case study 3: Local versus global awareness
  13. Conclusions
  14. Data Availability Statement

Key Result

Lemma 1

Consider a solution of system eq_animfa starting at $y_i(0) \in [0,1]$ and $z_{ij}(0) \in [0,1]$ for all $i$ and $j$. Recall that $f_{\textnormal{br,ij}}(y_i,y_j,\Bar{y}),f_{\textnormal{cr,ij}}(y_i,y_j,\Bar{y})\geq 0$ for all $y_i,y_j,\Bar{y}\in [0,1]$ and all $i,j$. Then, $y_i(t),z_{ij}(t) \in [0,1

Figures (5)

  • Figure 1: Interaction within and between two connected communities $i$ and $j$. Solid lines: change of state of individuals within one community; dashed lines: inter-community infections. Notice that the position of the Susceptible and Infected compartments are switched from the top and the bottom row.
  • Figure 2: Case study 1: effect of asymmetric response between two communities. Parameters are listed in Table \ref{['tab:param1']}. (a) local and average prevalence; (b) link densities. We observe a quick convergence towards a stable limit cycle, representing endemicity of the disease alternating between high and low prevalence in the population.
  • Figure 3: Case study 2: influence of breaking within or between communities on the prevalence and link densities. For each figure we use $n=20$ and the values of parameters were obtained by sampling, respectively: $\beta_{ij} \sim U([0,0.25]), \delta_i = 1, \zeta_{ij} \sim U([0,2]), \xi_{ij} \sim U([0,1])$ for all $i,j$, where $U$ denotes the uniform distribution, resulting in $R_0 = 2.53$, with the exception that we used for (a), $c=4$, for (b), $c=1$ and for (c), $c=0.25$.
  • Figure 4: Case study 2: impact of the parameter $c$ on the steady-state prevalence $\bar{y}_\infty$ and the peak prevalence $y_p$ in the (a) complete graph and (b) cycle graph. Parameters are the same as Figure \ref{['fig_cs3']}, except for (b), we have chosen a different sampling for $\beta_{ij} \sim U([0,1])$, where $U$ denotes the uniform distribution, such that $R_0 = 1.83$. The horizontal axis is shown on a logarithmic scale.
  • Figure 5: Case study 3: balancing between local and global awareness of the disease (a) within communities and (b) between communities showing the peak prevalence (red triangles) and the steady-state prevalence (green circles). All curves are declining for increasing $c$ values, but the steady-state prevalence within communities is non-monotonic. Simulations are based on an Barabási-Albert graph with $m_0=3, m=2$, with the values of parameters sampling: $\delta_i = 1, \zeta_{ij} \sim U([0, 2]), \xi_{ij} \sim U([0, 1])$, $y_i(0)=0$ for all $i \neq 1$ and $y_1(0)=0.2$ and $z_{ij}(0)=0$ for all $i,j$, where $U$ denotes the uniform distribution. Furthermore, (a) $n=20$ nodes, $\beta_{ij} \sim U([0,1])$ such that $R_0 = 2.93$ and (b) $n=50$ nodes, $\beta_{ij} \sim U([0, 0.8])$ such that $R_0 = 2.60$.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3: Lemma 2 in van2008further
  • Corollary 4
  • Theorem 5
  • proof
  • Conjecture 6