Competition Between Contagious Viruses in Populations
Analyzing how two viruses spread reveals their impact on public health.
― 6 min read
Table of Contents
In this article, we look into how two competing viruses spread in a given area. Imagine a situation where one virus is more contagious but spreads quickly, while the other is less contagious but has a longer time before people show symptoms. This difference means that even if the second virus spreads slower, it might still reach more people because those infected can travel or interact with others before they show symptoms.
We want to answer an important question: which virus poses a greater threat? This is not just a hypothetical question. During outbreaks like COVID-19, we have noticed that new variants behave differently than the original. Understanding these differences can help us decide how best to respond to outbreaks.
Our goal is to create a math model that represents how these two viruses compete to spread through a population. Each virus can not only spread at different rates but can also have different strengths when it comes to spreading over longer distances. This paper marks the beginning of exploring models that consider these long-distance effects in viral competition.
Basics of Virus Spread
When we talk about viruses spreading, we often refer to how quickly they can infect others. This spread can happen in two ways: locally and over longer distances. Local spread happens when people come into close contact with each other, while long-distance spread can occur because an asymptomatic person travels or interacts with others far away.
In our model, we treat the spread of a virus like water moving through porous material. There might be pathways that are difficult for the virus to travel, just like water faces resistance when it moves through porous rocks or soil. We use graphs to represent these pathways, where each connection between points (or vertices) represents a way the virus can spread.
Concept of First-Passage Percolation
First-passage percolation (FPP) is a way to understand how things move through a network. In our case, we are interested in how viruses spread through a population. Each connection between individuals has a "weight," which indicates how difficult it is for the virus to spread along that connection. Lower weights mean it is easier for the virus to spread, while higher weights indicate more resistance.
When we consider two types of viruses, we look at how they compete for the same connections in the network. We can determine if both types can coexist and infect a positive proportion of the population or if one type dominates and infects almost everyone.
Setting Up the Model
To simplify our study, we consider a finite area, which we represent as a grid called a torus. In this grid, connections represent the possible ways the viruses can spread between individuals. By including not only nearest neighbors but also connections that are farther apart, we account for the long-range interactions between the viruses.
In our scenario, each virus starts from a different source, and we examine how they spread over time, based on their respective weights and rates of infection. Some paths may be quicker for one virus than another, allowing it to reach people more efficiently.
Key Concepts
Coexistence and Non-Coexistence
We define coexistence as the scenario where both viruses manage to infect a significant portion of the population. Non-coexistence occurs when one virus infects almost everyone while the other barely spreads.
Parameters Affecting Spread
The spread of each virus can depend on various parameters, such as:
- Infection Rates: How quickly each type can infect others.
- Transmission Times: The time it takes for a virus to spread from one individual to another.
- Distance weights: A measure of how difficult it is for the virus to travel between individuals, which can vary based on the nature of the connection (e.g., direct contact vs. remote).
Long-Range Effects
Long-range effects are significant because they show how the virus can spread even without direct contact. If one virus has longer distances it can effectively spread through, it may reach more people, even if it has a lower transmission rate. Our model captures these long-range dynamics.
Analyzing Outcomes
When analyzing how many people each virus will infect, we look for patterns based on the parameters we define. We can determine if both viruses can coexist and reach significant proportions or if one will outcompete the other.
Direct Comparison
One way to compare the viruses is by examining their transmission characteristics. If both viruses have the same long-range parameters but different infection rates, we find that they can coexist if and only if their overall rates match. On the other hand, if the long-range parameters differ, one type may dominate even if its rate is much lower.
Examples
Equal Long-Range Parameters
Imagine both viruses share the same long-range effects. In this case, if the infection rates are different, we might observe that one virus spreads to more individuals. If one virus has a much higher rate, it might infect most of the area while the other barely reaches any people.
Unequal Long-Range Parameters
If one virus is better at traveling longer distances, it might still infect a large portion of the population, even if its infection rate is lower. This behavior shows how the strengths of the viruses in long-distance transmission can outweigh pure infection rates.
Implications for Public Health
Understanding how competing viruses affect one another can offer crucial insights for public health decisions. It can help determine which virus should be prioritized in control measures or vaccination efforts. Our model can inform strategies for managing outbreaks and controlling the spread of new variants.
Conclusion
In summary, the paper discusses how competing viruses can spread in a population, particularly when long-range effects are taken into account. The model highlights important parameters that affect coexistence and competition between different types of viruses. This understanding provides valuable knowledge for responding to future outbreaks and managing public health efforts more effectively.
By analyzing this competition closely, we can develop better strategies to control the spread and impact of viral infections on communities. The research opens up opportunities for further exploration and insights into how complex interactions in viral spread play out in real-world scenarios.
Title: Long-range competition on the torus
Abstract: We study a competition between two growth models with long-range correlations on the torus $\mathbb T_n^d$ of size $n$ in dimension $d$. We append the edge set of the torus $\mathbb T_n^d$ by including all non-nearest-neighbour edges, and from two source vertices, two first-passage percolation (FPP) processes start flowing on $\mathbb T_n^d$ and compete to cover the sites. The FPP processes we consider are long-range first-passage percolation processes, as studied by Chaterjee and Dey. Here, we have two types, say Type-$1$ and Type-$2$, and the Type-$i$ transmission time of an edge $e$ equals $\lambda_i^{-1} \|e\|^{\alpha_i}E_e$ for $i\in\{1,2\}$, where $(E_e)_e$ is a family of i.i.d.\ rate-one exponential random variables, $\lambda_1,\lambda_2>0$ are the global rate parameters, and $\alpha_1,\alpha_2\geq 0$ are the long-range parameters. In particular, we consider the instantaneous percolation regime, where $\alpha_1,\alpha_2\in[0,d)$, and we allow all parameters to depend on $n$. We study \emph{coexistence}, the event that both types reach a positive proportion of the graph, and identify precisely when coexistence occurs. In the case of absence of coexistence, we outline several phase transitions in the size of the losing type, depending on the relation between the rates of both types. One of the consequences of our results is that for constant intensity competition, i.e.\ when the long-range parameters of the two processes are the same, while their rates differ by a constant multiplicative factor, coexistence of the two processes at the scale of the torus volume happen if and only if their global rates are equal. On the other hand, when the long-range parameters differ, it is possible for one of the types, e.g.\ Type-$2$, to reach a significant number of vertices, even when its global rate parameter $\lambda_2$ is much smaller than $\lambda_1$.
Authors: Bas Lodewijks, Neeladri Maitra
Last Update: 2024-03-08 00:00:00
Language: English
Source URL: https://arxiv.org/abs/2403.05536
Source PDF: https://arxiv.org/pdf/2403.05536
Licence: https://creativecommons.org/licenses/by/4.0/
Changes: This summary was created with assistance from AI and may have inaccuracies. For accurate information, please refer to the original source documents linked here.
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