A Geometric Formulation of Transmission Dynamics: Social Risk Metrics in Hilbert Spaces and Stability Analysis

1. Introduction

Nonlinear dynamical systems arise in a wide variety of scientific contexts, ranging from ecological and biological interactions to socio-economic processes. Among the classical models used to describe interacting populations are the Lotka–Volterra equations, introduced independently by Alfred J. Lotka and Vito Volterra in the 1920s to model predator–prey interactions [1]. These equations constitute a paradigmatic example of nonlinear coupled differential systems whose solutions often exhibit oscillatory or cyclic behavior. The Lotka–Volterra framework has since been extended to numerous fields, including economics, epidemiology, and social dynamics.

The study of nonlinear dynamics through linear operators acting on observables has gained increasing attention in recent decades. In particular, Koopman operator theory provides a powerful framework in which nonlinear flows can be represented by a linear evolution in a space of observable functions [3]. This perspective has proven useful for spectral analysis, modal decomposition, and stability studies of nonlinear systems.

Despite their conceptual simplicity, nonlinear dynamical systems can display highly complex global behavior. One of the central challenges in the analysis of such systems lies in understanding how large-scale collective structures emerge from the interaction of elementary dynamical modes. In recent years, Koopman operator theory has emerged as a powerful approach to address this difficulty. Instead of analyzing the nonlinear evolution of the state variables directly, the Koopman framework studies the evolution of observable functions defined on the state space. Remarkably, this evolution is governed by a linear operator acting on an infinite-dimensional space of observables, even when the underlying system itself is nonlinear [8]. This linear representation allows the use of spectral theory and functional analysis tools to investigate nonlinear dynamics.

The purpose of this work is to study the aggregation of dynamical modes generated by nonlinear systems of Lotka–Volterra type and to analyze the resulting structure within a Hilbert space of observables. More specifically, we consider a countable family of observables associated with individual dynamical modes and construct an aggregated observable as a weighted superposition of these components. Under suitable square-summability conditions on the coefficients, we prove that the sequence of partial aggregations converges in a Hilbert space. This provides a well-defined aggregate state that represents the collective dynamics of the system.

The aggregation of observables naturally leads to a Hilbert space structure, where square summable combinations provide a convenient functional framework. Hilbert spaces play a central role in functional analysis and operator theory [6], allowing the use of spectral methods and geometric tools for the study of dynamical systems.

Once the aggregated observable is defined, its time evolution can be naturally described using the Koopman semigroup associated with the underlying dynamical flow. This yields a linear representation of the aggregated dynamics in the space of observables, allowing the use of operator-theoretic tools to analyze stability and structural properties. One of the central results of the paper shows that the geometry induced by the aggregated dynamics admits a canonical class of metrics generated by coercive operators. Although multiple such metrics may exist, they are all equivalent and therefore induce the same stability topology on the system.

This observation has an important conceptual consequence: the qualitative stability properties of the aggregated dynamics do not depend on a particular choice of metric within this natural class. In other words, the geometry associated with the aggregated observable dynamics is intrinsically determined by the Hilbert structure of the observable space.

To illustrate the applicability of the framework, we introduce a social risk functional defined as a quadratic observable associated with the induced metric. This example demonstrates how application-oriented quantities can be derived directly from the geometric structure generated by aggregated nonlinear dynamics.

The paper is organized as follows. Section 2 introduces the generating dynamical modes and reviews the relevant properties of Lotka–Volterra systems. Section 3 constructs the Hilbert space aggregation of observables and establishes convergence of the partial sums. Section 4 describes the Koopman representation of the aggregated dynamics. Section 5 establishes the equivalence class of metrics induced by coercive operators and proves the metric stability result. Finally, Section 6 presents the social risk functional as an application of the developed framework.

2. State of the Art

The mathematical study of interacting populations and nonlinear coupled dynamics has long been shaped by Lotka–Volterra type systems, which remain among the most classical and influential models of nonlinear interaction. Originally introduced in the context of predator–prey dynamics, Lotka–Volterra equations have since served as canonical prototypes for oscillatory interaction, coexistence, and nonlinear balance mechanisms in a wide variety of applications.

In parallel, the operator-theoretic viewpoint initiated by Koopman established a fundamentally different perspective on nonlinear dynamical systems. Instead of tracking the evolution of state variables directly, Koopman theory studies the linear evolution of observables in an infinite-dimensional function space. This shift has become increasingly important in recent years, especially through the development of spectral analysis, modal decompositions, and data-driven approximations of nonlinear flows [3].

A major line of recent research has focused on the approximation of Koopman operators through computational and data-driven methodologies. Among these, dynamic mode decomposition (DMD) and its variants have become central tools for extracting coherent structures and reduced-order representations from complex time-dependent systems [9]. Extended and nonlinear variants, including EDMD and related lifting strategies, further connect nonlinear dynamics with linear representations in higher-dimensional observable spaces. Recent work has also compared Koopman-based linearization with Carleman-type approaches for nonlinear ordinary differential equations, including benchmark examples such as the Lotka–Volterra model [10].

At the same time, modern surveys emphasize that the Koopman framework is not restricted to numerical approximation, but also provides a conceptual bridge between nonlinear dynamics, spectral theory, and operator methods in functional spaces [8,11]. This has motivated applications in model reduction, control, system identification, and nonlinear modal analysis.

Despite these advances, most of the existing literature focuses either on the spectral approximation of nonlinear dynamics, on data-driven identification of Koopman operators, or on modal decompositions in finite or truncated settings. By contrast, less attention has been given to the problem of countable aggregation of nonlinear generating modes in a Hilbert-space setting, together with the induced geometric structure of the resulting observable dynamics.

The present work is motivated by this gap. We consider a countable family of Lotka–Volterra-generated observables and study their aggregation in a Hilbert space through square-summable coefficients. This leads to a well-posed limit observable whose evolution is governed by a Koopman semigroup. Our main contribution is then to show that the resulting observable space admits a canonical class of equivalent metrics induced by coercive operators, so that the stability topology of the aggregated dynamics is independent of the particular choice of compatible metric. As an illustration, this metric structure is later used to define a social risk functional.

3. Koopman Representation of the Aggregated Dynamics

Let E be a state space and let ${\left({\phi }_t\right)}_{t\ge 0}$ be the flow generated by a family of Lotka–Volterra systems on E. Let H be a real Hilbert space of observables $g:E\to \mathbb{R}$ such that, for every $t\ge 0$, the Koopman operator

$${\mathcal{K}}_{t}g:=g\circ {\phi }_t$$

is well defined from H into itself.

Assume that ${\left({\mathcal{K}}_t\right)}_{t\ge 0}$ is a strongly continuous semigroup on H, i.e.,

• ${\mathcal{K}}_0=I$,
• ${\mathcal{K}}_{t+s}={\mathcal{K}}_t{\mathcal{K}}_s$ for all $t,s\ge 0$,
• for every $g\in H$,

  $$\underset{t\to 0^{+}}{lim}{∥{\mathcal{K}}_{t}g-g∥}_H=0.$$

Let ${\left\{u_k^0\right\}}_{k\ge 1}\subset H$ be a family of initial observables, and let ${\left(a_k\right)}_{k\ge 1}\in {\ell }^2$. For each $n\ge 1$, define

$$X_n^0:={\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0.$$

Assume that $X_n^0\to X_0$ in H as $n\to \infty$.

For each $t\ge 0$, define the evolved partial aggregation

$$X_n\left(t\right):={\mathcal{K}}_t{X}_n^0.$$

Theorem 1 

(Koopman representation of the aggregated dynamics). Under the assumptions above, for every $t\ge 0$ the sequence ${\left(X_n\left(t\right)\right)}_{n\ge 1}$ converges in H to

$$X\left(t\right):={\mathcal{K}}_t{X}_0.$$

Moreover,

$$X\left(t\right)=\underset{n\to \infty }{lim}{\displaystyle \sum _{k=1}^n}{a}_k{\mathcal{K}}_t{u}_k^0\mathrm{in}H.$$

Hence, the aggregation procedure and the Koopman evolution commute in the limit:

$${\mathcal{K}}_t\left(\underset{n\to \infty }{lim}{\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0\right)=\underset{n\to \infty }{lim}{\displaystyle \sum _{k=1}^n}{a}_k{\mathcal{K}}_t{u}_k^0.$$

Proof. 

Since ${\left({\mathcal{K}}_t\right)}_{t\ge 0}$ is a strongly continuous semigroup on H, each operator ${\mathcal{K}}_t:H\to H$ is linear and bounded for fixed $t\ge 0$. Because $X_n^0\to X_0$ in H, continuity of ${\mathcal{K}}_t$ yields

$${\mathcal{K}}_t{X}_n^0\to {\mathcal{K}}_t{X}_0\mathrm{in}H.$$

By definition, ${\mathcal{K}}_t{X}_n^0=X_n\left(t\right)$, hence

$$X_n\left(t\right)\to X\left(t\right):={\mathcal{K}}_t{X}_0\mathrm{in}H.$$

On the other hand, by linearity of ${\mathcal{K}}_t$,

$$X_n\left(t\right)={\mathcal{K}}_t\left({\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0\right)={\displaystyle \sum _{k=1}^n}{a}_k{\mathcal{K}}_t{u}_k^0.$$

Passing to the limit in H gives

$$X\left(t\right)=\underset{n\to \infty }{lim}{\displaystyle \sum _{k=1}^n}{a}_k{\mathcal{K}}_t{u}_k^0.$$

This proves the claim. □

Corollary 1 

(Uniform boundedness on compact time intervals). Assume, in addition, that for every $T>0$ there exists $C_T>0$ such that

$${∥{\mathcal{K}}_t∥}_{\mathcal{L}\left(H\right)}\le C_T,\forall t\in [0,T].$$

Then, for every $T>0$,

$$\underset{t\in [0,T]}{sup}{∥X\left(t\right)∥}_H\le C_T{∥X_0∥}_H.$$

Moreover,

$$\underset{t\in [0,T]}{sup}{∥X_n\left(t\right)∥}_H\le C_T{∥X_n^0∥}_H,\forall n\ge 1.$$

In particular, if $\left(X_n^0\right)$ is bounded in H, then $\left(X_n\left(t\right)\right)$ is uniformly bounded on every compact time interval.

Proof. 

For every $t\in [0,T]$,

$${∥X\left(t\right)∥}_H={∥{\mathcal{K}}_t{X}_0∥}_H\le {∥{\mathcal{K}}_t∥}_{\mathcal{L}\left(H\right)}{∥X_0∥}_H\le C_T{∥X_0∥}_H.$$

Taking the supremum over $t\in [0,T]$ gives the first estimate. The second follows in the same way by replacing $X_0$ with $X_n^0$. □

Remark 1 

(Natural metric induced by the aggregated dynamics). The Hilbert structure of H induces the canonical metric

$$d_H(f,g):={∥f-g∥}_H,f,g\in H.$$

Therefore, once the aggregated evolution $X\left(t\right)={\mathcal{K}}_t{X}_0$ is well defined, the family ${\left(X\left(t\right)\right)}_{t\ge 0}$ admits a natural metric description. More generally, if $W:H\to H$ is bounded, self-adjoint and positive definite, one may define

$$d_W(f,g):=\sqrt{{\langle W(f-g),f-g\rangle }_H},$$

which is equivalent to $d_H$ whenever W is coercive. Such a metric may later be used to quantify structural features of the aggregated dynamics.

4. Convergence of the Aggregated Initial Data

In order to define the aggregated dynamics rigorously, we first need to ensure that the initial partial sums

$$X_n^0={\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0$$

converge in the Hilbert space H.

Theorem 2 

(Convergence for orthonormal generating modes). Let H be a real Hilbert space, and let ${\left\{u_k^0\right\}}_{k\ge 1}\subset H$ be an orthonormal family. Let ${\left(a_k\right)}_{k\ge 1}\in {\ell }^2$, and define

$$X_n^0:={\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0.$$

Then there exists $X_0\in H$ such that

$$X_n^0\to X_0\mathrm{in}H,$$

as $n\to \infty$. Moreover,

$${∥X_n^0∥}_H^2={\displaystyle \sum _{k=1}^n}|a_k{|}^2\le {\displaystyle \sum _{k=1}^{\infty }}{\left|a_k\right|}^2,\forall n\ge 1.$$

In particular, the sequence ${\left(X_n^0\right)}_{n\ge 1}$ is uniformly bounded in H.

Proof. 

Let $m>n$. Using orthonormality, we obtain

$${∥X_m^0-X_n^0∥}_H^2={∥{\displaystyle \sum _{k=n+1}^m}{a}_k{u}_k^0∥}_H^2={\displaystyle \sum _{k=n+1}^m}{\left|a_k\right|}^2.$$

Since $\left(a_k\right)\in {\ell }^2$, the series ${\sum }_{k=1}^{\infty }{\left|a_k\right|}^2$ converges, and therefore

$${\displaystyle \sum _{k=n+1}^m}{\left|a_k\right|}^2\to 0\mathrm{as}n,m\to \infty .$$

Hence $\left(X_n^0\right)$ is a Cauchy sequence in H. Since H is complete, there exists $X_0\in H$ such that

$$X_n^0\to X_0\mathrm{in}H.$$

The identity

$${∥X_n^0∥}_H^2={\displaystyle \sum _{k=1}^n}{\left|a_k\right|}^2$$

follows directly from orthonormality. The uniform boundedness is immediate. □

Remark 2. 

The previous theorem shows that, when the generating modes form an orthonormal family, square-summability of the coefficients is sufficient to guarantee both existence and boundedness of the aggregated initial observable.

Theorem 3 

(Convergence for Riesz generating modes). Let H be a real Hilbert space, and let ${\left\{u_k^0\right\}}_{k\ge 1}\subset H$ be a Riesz sequence. That is, assume there exist constants $A,B>0$ such that, for every finite scalar sequence $\left(c_k\right)$,

$$A{\displaystyle \sum _{k=1}^N}|c_k{|}^2\le {∥{\displaystyle \sum _{k=1}^N}{c}_k{u}_k^0∥}_H^2\le B{\displaystyle \sum _{k=1}^N}{\left|c_k\right|}^2.$$

Let ${\left(a_k\right)}_{k\ge 1}\in {\ell }^2$, and define

$$X_n^0:={\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0.$$

Then there exists $X_0\in H$ such that

$$X_n^0\to X_0\mathrm{in}H,$$

as $n\to \infty$. Moreover,

$${∥X_n^0∥}_H^2\le B{\displaystyle \sum _{k=1}^n}|a_k{|}^2\le B{\displaystyle \sum _{k=1}^{\infty }}{\left|a_k\right|}^2,\forall n\ge 1.$$

Thus $\left(X_n^0\right)$ is uniformly bounded in H.

Proof. 

Let $m>n$. By the upper Riesz bound,

$${∥X_m^0-X_n^0∥}_H^2={∥{\displaystyle \sum _{k=n+1}^m}{a}_k{u}_k^0∥}_H^2\le B{\displaystyle \sum _{k=n+1}^m}{\left|a_k\right|}^2.$$

Since $\left(a_k\right)\in {\ell }^2$, the right-hand side tends to zero as $n,m\to \infty$. Hence $\left(X_n^0\right)$ is Cauchy in H, and therefore converges to some $X_0\in H$.

The uniform bound follows directly from the upper Riesz inequality:

$${∥X_n^0∥}_H^2\le B{\displaystyle \sum _{k=1}^n}|a_k{|}^2\le B{\displaystyle \sum _{k=1}^{\infty }}{\left|a_k\right|}^2.$$

□

Corollary 2 

(Well-posed aggregated Koopman dynamics). Under the assumptions of either of the previous theorems, the initial aggregated observable

$$X_0=\underset{n\to \infty }{lim}{\displaystyle \sum _{k=1}^n}{a}_k{u}_k^0$$

is well defined in H. Therefore, the aggregated evolution

$$X\left(t\right)={\mathcal{K}}_t{X}_0$$

is well posed for every $t\ge 0$, and the partial evolutions

$$X_n\left(t\right)={\mathcal{K}}_t{X}_n^0$$

converge to $X\left(t\right)$ in H.

Proof. 

The existence of $X_0$ follows from the previous theorems. The conclusion then follows from Theorem 3.1 on the Koopman representation of the aggregated dynamics. □

5. Inherited Invariants of the Aggregated Dynamics

The Koopman framework also provides a natural way to identify quantities preserved by the aggregated dynamics.

Definition 1 

(Invariant linear observable). Let H be a real Hilbert space and let ${\left({\mathcal{K}}_t\right)}_{t\ge 0}$ be a strongly continuous Koopman semigroup on H. A continuous linear functional

$$\mathsf{\Lambda }:H\to \mathbb{R}$$

is said to be invariant under the Koopman evolution if

$$\mathsf{\Lambda }\left({\mathcal{K}}_{t}g\right)=\mathsf{\Lambda }\left(g\right),\forall g\in H,\forall t\ge 0.$$

Proposition 1 

(Inherited invariant for the aggregated dynamics). Let $X_0\in H$ and let

$$X\left(t\right):={\mathcal{K}}_t{X}_0,t\ge 0,$$

be the aggregated Koopman evolution. Assume that $\mathsf{\Lambda }:H\to \mathbb{R}$ is a continuous linear functional invariant under ${\left({\mathcal{K}}_t\right)}_{t\ge 0}$. Then

$$\mathsf{\Lambda }\left(X\left(t\right)\right)=\mathsf{\Lambda }\left(X_0\right),\forall t\ge 0.$$

In particular, Λ defines a conserved quantity for the aggregated dynamics.

Proof. 

By definition of the aggregated dynamics and the invariance of $\mathsf{\Lambda }$,

$$\mathsf{\Lambda }\left(X\left(t\right)\right)=\mathsf{\Lambda }\left({\mathcal{K}}_t{X}_0\right)=\mathsf{\Lambda }\left(X_0\right),\forall t\ge 0.$$

This proves the claim. □

Definition 2 

(Koopman eigenobservable). A nonzero observable $g\in H$ is called a Koopman eigenobservable associated with $\lambda \in \mathbb{C}$ if

$${\mathcal{K}}_{t}g=e^{\lambda t}g,\forall t\ge 0.$$

Proposition 2 

(Modal evolution of Koopman eigenobservables). Let $g\in H$ be a Koopman eigenobservable associated with $\lambda \in \mathbb{C}$. Then its evolution under the aggregated dynamics is given by

$${\mathcal{K}}_{t}g=e^{\lambda t}g.$$

In particular:

• if $\lambda =0$, then g is invariant;
• if $\Re \left(\lambda \right)<0$, then the mode decays exponentially;
• if $\Re \left(\lambda \right)=0$, then the mode is neutrally stable;
• if $\Re \left(\lambda \right)>0$, then the mode grows exponentially.

Proof. 

The statement follows directly from the definition of eigenobservable. If ${\mathcal{K}}_{t}g=e^{\lambda t}g$, then

$${∥{\mathcal{K}}_{t}g∥}_H=\left|e^{\lambda t}\right|{∥g∥}_H=e^{\Re \left(\lambda \right)t}{∥g∥}_H,$$

from which the four cases follow immediately. □

Theorem 4 

(Spectral representation of the aggregated dynamics). Assume that the initial aggregated observable admits the expansion

$$X_0={\displaystyle \sum _{k=1}^{\infty }}{a}_k{g}_k$$

in H, where each $g_k\in H$ is a Koopman eigenobservable associated with ${\lambda }_k\in \mathbb{C}$, and the series converges in H. Then, for every $t\ge 0$, the aggregated dynamics is given by

$$X\left(t\right)={\mathcal{K}}_t{X}_0={\displaystyle \sum _{k=1}^{\infty }}{a}_k{e}^{{\lambda }_{k}t}{g}_k,$$

with convergence in H whenever the evolved series remains summable.

Proof. 

Using linearity of ${\mathcal{K}}_t$ and the eigenobservable property,

$${\mathcal{K}}_t{X}_0={\mathcal{K}}_t\left({\displaystyle \sum _{k=1}^{\infty }}{a}_k{g}_k\right)={\displaystyle \sum _{k=1}^{\infty }}{a}_k{\mathcal{K}}_t{g}_k={\displaystyle \sum _{k=1}^{\infty }}{a}_k{e}^{{\lambda }_{k}t}{g}_k,$$

where the interchange of operator and series is justified by convergence in H. □

6. Example: A Social Risk Metric

The metric framework developed above allows the definition of application-specific quantities derived from the geometry of the aggregated dynamics.

Definition 3 

(Social risk). Let H be the Hilbert space of aggregated observables and let $W:H\to H$ be a bounded, self-adjoint and positive operator. The social risk associated with a state $X\in H$ is defined as

$$R\left(X\right):=\sqrt{{\langle WX,X\rangle }_H}.$$

This quantity represents the magnitude of the aggregated social state under the metric induced by W.

Remark 3. 

If the aggregated observable admits the expansion

$$X={\displaystyle \sum _{k=1}^{\infty }}{a}_k{u}_k,$$

and W is diagonal in the generating basis $\left\{u_k\right\}$ with weights $w_k>0$, then

$$R{\left(X\right)}^2={\displaystyle \sum _{k=1}^{\infty }}{w}_k{a}_k^2.$$

Each coefficient $w_k$ measures the contribution of the corresponding interaction mode to the overall risk.

Proposition 3 

(Metric robustness of social risk). Let $W_1$ and $W_2$ be coercive operators on H. Then the corresponding risk metrics

$$R_1\left(X\right)=\sqrt{{\langle W_{1}X,X\rangle }_H},R_2\left(X\right)=\sqrt{{\langle W_{2}X,X\rangle }_H}$$

are equivalent. In particular, they generate the same stability topology for the aggregated dynamics.

Proposition 4 

(Risk-metric sufficient condition for exponential stability). Let $\mathcal{H}$ be a real Hilbert space and let $W:\mathcal{H}\to \mathcal{H}$ be a bounded, self-adjoint, positive definite operator. Define the risk inner product and norm by

$${\langle x,y\rangle }_R:={\langle Wx,y\rangle }_{\mathcal{H}},{∥x∥}_R:=\sqrt{{\langle x,x\rangle }_R},$$

and the social-risk functional

$$\mathcal{R}\left(x\right):={\displaystyle \frac{1}{2}}{∥x∥}_R^2.$$

Consider the infection dynamics near the disease-free equilibrium (DFE)

$$\dot{I}\left(t\right)=\left(\beta \left(t\right)-(\gamma +\mu )\right)I\left(t\right),\gamma ,\mu >0,$$

(1)

where the transmission rate depends on the social state through

$$\beta \left(t\right)=\mathsf{\Phi }\left(\mathcal{R}\left(x\right(t\left)\right)\right),$$

with $\mathsf{\Phi }:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ continuous and non-decreasing. Assume that there exist constants $\rho >0$ and $\delta >0$ such that, for all $t\in [0,T]$,

$${∥x\left(t\right)∥}_R\le \rho ⟹\mathsf{\Phi }\left({\displaystyle \frac{1}{2}}{\rho }^2\right)\le (\gamma +\mu )-\delta .$$

(2)

Then the DFE $I\equiv 0$ is exponentially stable on $[0,T]$, and moreover

$$\left|I\left(t\right)\right|\le \left|I\left(0\right)\right|e^{-\delta t},\forall t\in [0,T].$$

Proof. 

Equation (1) is a scalar linear ODE and admits the explicit representation

$$I\left(t\right)=I\left(0\right)exp\left({\int }_0^t\left(\beta \left(\tau \right)-(\gamma +\mu )\right)d\tau \right).$$

Assume ${∥x\left(\tau \right)∥}_R\le \rho$ for all $\tau \in [0,T]$. Since $\mathsf{\Phi }$ is non-decreasing and

$$\mathcal{R}\left(x\left(\tau \right)\right)={\displaystyle \frac{1}{2}}{∥x\left(\tau \right)∥}_R^2\le {\displaystyle \frac{1}{2}}{\rho }^2,$$

we obtain

$$\beta \left(\tau \right)=\mathsf{\Phi }\left(\mathcal{R}\left(x\right(\tau \left)\right)\right)\le \mathsf{\Phi }\left({\displaystyle \frac{1}{2}}{\rho }^2\right).$$

By condition (2) it follows that

$$\beta \left(\tau \right)-(\gamma +\mu )\le -\delta ,\forall \tau \in [0,T].$$

Therefore,

$${\int }_0^t\left(\beta \left(\tau \right)-(\gamma +\mu )\right)d\tau \le -\delta t,$$

and substituting into the explicit formula yields

$$\left|I\left(t\right)\right|\le \left|I\left(0\right)\right|e^{-\delta t},\forall t\in [0,T].$$

Hence $I\equiv 0$ is exponentially stable on $[0,T]$ with decay rate at least $\delta$. □

Corollary 3 

(Risk-metric control of the effective reproduction number). Under the assumptions of the previous proposition, define the effective reproduction number

$$R_t:={\displaystyle \frac{\beta \left(t\right)}{\gamma +\mu }}.$$

If there exists $\rho >0$ such that

$${∥x\left(t\right)∥}_R\le \rho \forall t\in [0,T],$$

and

$$\mathsf{\Phi }\left({\displaystyle \frac{1}{2}}{\rho }^2\right)<\gamma +\mu ,$$

then

$$R_t<1\mathrm{for}\mathrm{all}t\in [0,T],$$

and consequently the disease-free equilibrium is exponentially stable.

Proof. 

From the hypothesis,

$$\beta \left(t\right)\le \mathsf{\Phi }\left({\displaystyle \frac{1}{2}}{\rho }^2\right)<\gamma +\mu .$$

Dividing by $(\gamma +\mu )>0$ yields $R_t<1$ for all $t\in [0,T]$. Exponential stability follows from the previous proposition. □

Remark 4 

(Geometric interpretation of the stability region). Define the closed ball in the risk-metric space

$$B_R(0,\rho ):=\{x\in \mathcal{H}:{∥x∥}_R\le \rho \}.$$

If ρ satisfies

$$\mathsf{\Phi }\left({\displaystyle \frac{1}{2}}{\rho }^2\right)<\gamma +\mu ,$$

then $B_R(0,\rho )$ is a sufficient stability region for the epidemic dynamics. That is, any social trajectory $x\left(t\right)$ remaining inside $B_R(0,\rho )$ ensures $R_t<1$ and exponential decay of infections.

Proposition 5 

(Convex stability region induced by the social-risk metric). Let $\mathcal{H}$ be a real Hilbert space endowed with the risk norm ${∥·∥}_R$ and define $\mathcal{R}\left(x\right)={\displaystyle \frac{1}{2}}{∥x∥}_R^2$. Let $\mathsf{\Phi }:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be non-decreasing and set

$$\beta \left(x\right):=\mathsf{\Phi }\left(\mathcal{R}\right(x\left)\right).$$

Fix $\gamma ,\mu >0$ and define the stability region

$$\mathcal{S}:=\left\{x\in \mathcal{H}:\beta \left(x\right)<\gamma +\mu \right\}=\left\{x\in \mathcal{H}:\mathsf{\Phi }\left(\frac{1}{2}{∥x∥}_R^2\right)<\gamma +\mu \right\}.$$

Define the threshold radius

$${\rho }_{★}:=sup\left\{\rho \ge 0:\mathsf{\Phi }\left(\frac{1}{2}{\rho }^2\right)<\gamma +\mu \right\}\in (0,\infty ].$$

Then

$$\mathcal{S}=\{x\in \mathcal{H}:{∥x∥}_R<{\rho }_{★}\}.$$

In particular, $\mathcal{S}$ is convex, balanced, and radially symmetric with respect to ${∥·∥}_R$.

Proof. 

Let $x\in \mathcal{H}$ and set $r:={∥x∥}_R$. Since $\mathsf{\Phi }$ is non-decreasing, the condition

$$x\in \mathcal{S}⟺\mathsf{\Phi }\left(\frac{1}{2}{r}^2\right)<\gamma +\mu$$

depends on x only through the scalar radius r. By definition of ${\rho }_{★}$, we have

$$r<{\rho }_{★}⟹\mathsf{\Phi }\left(\frac{1}{2}{r}^2\right)<\gamma +\mu ,$$

hence $\{x:{∥x∥}_R<{\rho }_{★}\}\subset \mathcal{S}$.

Conversely, if $r>{\rho }_{★}$, then by the definition of supremum there exists $\rho$ with ${\rho }_{★}<\rho <r$ such that

$$\mathsf{\Phi }\left(\frac{1}{2}{\rho }^2\right)\ge \gamma +\mu .$$

Since $\mathsf{\Phi }$ is non-decreasing and $r>\rho$, it follows that

$$\mathsf{\Phi }\left(\frac{1}{2}{r}^2\right)\ge \mathsf{\Phi }\left(\frac{1}{2}{\rho }^2\right)\ge \gamma +\mu ,$$

so $x\notin \mathcal{S}$. Therefore, $\mathcal{S}\subset \{x:{∥x∥}_R\le {\rho }_{★}\}$ and combining both inclusions yields $\mathcal{S}=\{x:{∥x∥}_R<{\rho }_{★}\}$.

Finally, open balls of a norm are convex, balanced, and radially symmetric, hence so is $\mathcal{S}$. □

Remark 5 

(Bounded (sigmoidal) transmission maps and global stability regions). Assume that Φ is bounded above, i.e.,

$$0\le \mathsf{\Phi }\left(r\right)\le {\beta }_{max}\forall r\ge 0,$$

for some ${\beta }_{max}>0$ (this includes sigmoidal maps). Then two qualitatively different regimes arise:

(i) Globally safe regime.If ${\beta }_{max}<\gamma +\mu$, then

$$\mathsf{\Phi }\left(\frac{1}{2}{∥x∥}_R^2\right)\le {\beta }_{max}<\gamma +\mu \forall x\in \mathcal{H},$$

hence the stability region satisfies $\mathcal{S}=\mathcal{H}$; in other words, the epidemic decay condition is met for any admissible social state.

(ii) Threshold regime.If ${\beta }_{max}\ge \gamma +\mu$, then the stability region is nontrivial and there exists a critical radius ${\rho }_{★}\in (0,\infty ]$ such that

$$\mathcal{S}=\{x\in \mathcal{H}:{∥x∥}_R<{\rho }_{★}\}.$$

If, in addition, Φ is continuous and strictly increasing on $[0,\infty )$, then ${\rho }_{★}\in (0,\infty )$ is uniquely characterized by the threshold equation

$$\mathsf{\Phi }\left(\frac{1}{2}{\rho }_{★}^2\right)=\gamma +\mu .$$