Convergences for a Virus-like Evolving Population driven by Mutually-exciting Hawkes Processes

Imagine a bustling, chaotic city where new citizens are constantly being born, and old ones are constantly dying. But this isn't a normal city; it's a virus-like population evolving in real-time.

This paper by Rahul Roy, Dharmaraja Selvamuthu, and Paola Tardelli tries to predict how this city grows, shrinks, and changes over time. They use a fancy mathematical tool called Hawkes Processes, but let's translate that into everyday language.

The Core Idea: The "Contagion" of Events

In a normal city (modeled by a standard Poisson process), births and deaths happen randomly and independently, like raindrops hitting a roof. One drop doesn't make the next one fall faster.

But in this virus city, events trigger more events.

The "Hawkes" Effect: Think of a viral TikTok trend. When one person posts a video, it excites others to post their own. The more posts there are, the more likely another post is to happen soon.
Births are Mutually Exciting: If a new mutant virus is born, it makes it more likely that another mutant (or a regular virus) will be born soon. They are "hype-maning" each other.
Deaths are Self-Exciting: If one virus dies, it somehow makes it slightly more likely that another one will die soon after (perhaps due to a spreading weakness or a sudden immune system attack).

The Characters in Our Story

The authors track three main things:

The Mutants ( $N_1$ ): New, random viruses with unique "fitness" levels (how good they are at surviving). Their fitness is like a random number between 0 and 1.
The Non-Mutants ( $N_2$ ): Copies of existing viruses. If a virus with "fitness 0.8" exists, a new one is likely to be born with that same fitness. They copy the winners.
The Deaths ( $N_3$ ): When a death happens, the system is ruthless. It always kills the weakest virus (the one with the lowest fitness number closest to 0).

The Big Problem: The "Memory" Trap

Usually, to predict the future of a system, you just need to know where it is right now (this is called the Markov Property). But because these viruses "remember" their past (every past birth makes future births more likely), the system has a long memory. It's like trying to predict the weather without knowing the temperature, humidity, or wind speed from the last hour.

The Breakthrough:
The authors realized that if you track two things together—the number of viruses AND their current "excitement level" (intensity)—you can turn this complex, memory-heavy system into a simple, predictable one.

Analogy: Instead of trying to remember every single tweet ever posted, you just track the current "trending score." If you know the score and the number of tweets, you can predict the next one.

The Main Discovery: The "Critical Fitness" Line

The paper's most exciting finding is about a Phase Transition. Imagine a line drawn on a graph called the Critical Fitness Level ( $f_c$ ). This line acts like a border between two different worlds:

Scenario A: The Death Trap (Death Rate > Birth Rate)
If the viruses die faster than they are born, the population crashes. The city empties out. The population keeps hitting zero and staying there. It's a failed ecosystem.

Scenario B: The Explosion (Birth Rate > Death Rate)
If the viruses are born faster than they die, the population grows forever. But where do they live?

The Magic Line ( $f_c$ ): The authors found that the population doesn't just grow randomly. It concentrates above a specific fitness level.
The Analogy: Imagine a crowded party where the weakest people are kicked out immediately. If the party is growing fast enough, eventually, everyone left in the room will have a "fitness" higher than a certain threshold.
- If the threshold is 0.5, eventually, you won't find anyone with a fitness below 0.5. The whole population clusters in the "high fitness" zone (between 0.5 and 1.0).
- If the birth rate is super high, the population clusters near 1.0 (the absolute best fitness).

Why Does This Matter?

This isn't just about math; it's about understanding evolution.

It explains how a virus population might suddenly "evolve" to become much stronger.
It shows that there is a tipping point. If the environment (death rate) gets too harsh, the population dies out. If the environment is just right, the population explodes, but it forces itself to become "fitter" to survive, leaving the weak behind.

Summary in One Sentence

The authors built a mathematical model showing that when a virus population grows in a "contagious" way (where events trigger more events), there is a critical point where the population either dies out or explodes, and if it explodes, it naturally sorts itself so that only the strongest, fittest viruses survive.

Here is a detailed technical summary of the paper "Convergences for a Virus-like Evolving Population driven by Mutually-exciting Hawkes Processes."

1. Problem Statement and Motivation

The paper addresses the modeling of a virus-like evolving population subject to birth and death processes with varying mutation rates. Traditional models often assume Poisson processes for births and deaths, implying constant intensities and independent events. However, in real-world biological and epidemiological scenarios (e.g., viral spread, financial contagion), events are often self-exciting (a birth increases the likelihood of future births) or mutually-exciting (births of one type stimulate births of another).

The authors aim to:

Develop a continuous-time stochastic model where births and deaths are driven by Hawkes processes (point processes with memory).
Overcome the non-Markovian nature of general Hawkes processes to derive rigorous convergence results.
Identify conditions for a phase transition in the population dynamics, specifically regarding the distribution of fitness levels (survival of the fittest).

2. Methodology and Model Construction

The Stochastic Model

The population is modeled as a queueing system with three interacting counting processes:

$N_1(t)$ : Births of mutants. Each new mutant is assigned a fitness level $f \in [0, 1]$ chosen uniformly at random.
$N_2(t)$ : Births of non-mutants (existing subspecies). The fitness level is chosen from existing fitness levels in the population with probability proportional to the current count of individuals at that level.
$N_3(t)$ : Deaths. When a death occurs, the individual with the lowest fitness (closest to 0) is eliminated.

The total population size is $N(t) = N_1(t) + N_2(t) - N_3(t)$ .

Hawkes Process Dynamics

The intensities of these processes are defined as follows:

Mutually-Exciting Births ( $N_1, N_2$ ):
$\lambda_i(t) = \lambda_i^0 + \sum_{j=1,2} \int_0^t \phi_{ji}(t-u) dN_j(u), \quad i=1,2$
Here, $\phi_{ji}$ represents the excitation kernel where an event in process $j$ influences the intensity of process $i$ .
Self-Exciting Death ( $N_3$ ):
$\lambda_3(t) = \lambda_3^0 \mathbb{1}_{N(t-)>0} + \int_0^t \psi(t-u) dN_3(u)$
The death process is self-exciting but only active if the population is non-empty.

The Markovian Challenge

General Hawkes processes are non-Markovian because their intensity depends on the entire history of events. To analyze convergence and stability, the authors seek conditions under which the system can be treated as a Markov process.

Key Technical Approach:

Shot Noise Representation: The authors introduce shot noise processes $\xi_i(t)$ to represent the history-dependent part of the intensities.
Kernel Restriction: They derive necessary and sufficient conditions for the memory kernels ( $\phi_{ji}$ and $\psi$ ) to allow the Markov property. They prove that the kernels must be exponential:
$\phi_{ji}(t) = \delta_{ji} + \alpha_{ji}e^{-\beta_i t}, \quad \psi(t) = \delta_3 + \alpha_3 e^{-\beta_3 t}$
Under these conditions (specifically with $\delta=0$ to avoid explosion), the joint process of population counts and intensities $(N_1, \lambda_1, N_2, \lambda_2, N_3, \lambda_3)$ becomes a Markov process.
Infinitesimal Generator: The authors construct the infinitesimal generator $G$ for this Markov process, enabling the use of martingale problem techniques for stability analysis.

3. Key Contributions

Markov Property Characterization: The paper provides the necessary and sufficient conditions for a system of mutually-exciting and self-exciting Hawkes processes to possess the Markov property. This is achieved by restricting the excitation kernels to exponential forms.
Stability Analysis: The authors derive explicit conditions for the stability of the population (i.e., preventing infinite growth or ensuring the population does not go extinct). They establish that stability requires the expected death intensity to exceed the expected birth intensity in the long run.
Phase Transition at Critical Fitness: The most significant contribution is the identification of a phase transition at a critical fitness level $f_c$ . The behavior of the population changes drastically depending on whether the fitness level $f$ is below or above this threshold.

4. Main Results

Expected Values and Stability

The paper derives closed-form expressions for the expected intensities $E[\lambda_i(t)]$ and expected population sizes $E[N_i(t)]$ .

Stability Condition: The system is stable if the long-term mean death rate exceeds the mean birth rate:
$\lim_{t \to \infty} E[\lambda_3(t)] > \lim_{t \to \infty} (E[\lambda_1(t)] + E[\lambda_2(t)])$
If this holds, the population size $N(t)$ returns to zero infinitely often. If the birth rate dominates, $N(t) \to \infty$ .

Phase Transition (Theorem 8.1)

Assuming the system grows ( $N(t) \to \infty$ ), the authors define a critical fitness level:
$f_c = \lim_{t \to \infty} \frac{E[\lambda_3(t)]}{E[\lambda_1(t)]}$
The behavior of the population distribution is determined by $f_c$ :

Case $f_c \leq 1$ (High Death Rate / Low Birth Rate):
- The population concentrates in the fitness interval $[f_c, 1]$ .
- Individuals with fitness $f < f_c$ die out almost surely.
- The proportion of the population with fitness $> f_c$ converges to 1.
Case $f_c \geq 1$ (Very High Birth Rate):
- The population concentrates near the maximum fitness $f=1$ .
- Even individuals with relatively low fitness survive because the birth rate is so high that the "survival of the fittest" mechanism is overwhelmed by the sheer volume of new births.

Empirical Distribution (Corollary 8.4)

If $f_c < 1$ , the empirical distribution of fitness levels $F_t(f)$ converges uniformly almost surely to:
$\frac{\max(f - f_c, 0)}{1 - f_c}$
This indicates a sharp cutoff: no individuals exist below $f_c$ , and the distribution is uniform above it.

5. Significance and Implications

Theoretical Advancement: The paper bridges the gap between discrete-time evolutionary models (like those by Roy and Tanemura) and continuous-time stochastic processes. It successfully extends these models to include contagion effects (Hawkes processes), which are crucial for modeling viral spread and financial markets but mathematically difficult due to non-Markovianity.
Biological Insight: The results provide a rigorous mathematical basis for "survival of the fittest" in a dynamic, contagious environment. It quantifies how the "contagion" of births (Hawkes excitation) shifts the critical fitness threshold required for a species to survive.
Methodological Rigor: By proving that exponential kernels are the only kernels allowing the Markov property in this coupled system, the paper sets a standard for future modeling of interacting point processes. It demonstrates how to handle the "enlargement of filtration" problem inherent in systems with history-dependent intensities.

In summary, this work provides a robust framework for analyzing evolving populations where events trigger future events, revealing that the interplay between birth/death rates and excitation kernels creates a distinct critical fitness threshold that dictates the long-term survival and distribution of the population.