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ODE Analysis of Pathogen Fitness and R0 ——updating markdown in progress!

Jess Valiarovski 2025-01-16

Background

Many environments within ecological communities are hotbeds for evolving resistant and dangerous pathogens. It has been observed that host population traits influence the outcome of an emerging disease outbreak. For example, in the avian flu, there are species of migratory birds that are asymptomatic carriers who are tolerant, but spread it to other species that are highly vulnerable, causing rapid population decline and ecological instability.

The two primary traits that characterize different strains on an epidemiological scale are transmission and virulence. Our focus will be on micro parasites as macro parasites exhibit survival and reproduction outside the host. All pathogenic strains have a level of virulence which is generalized here defined by infection mortality rate. Virulence is a phenomenon that arises as a consequence of a pathogen multiplying, spreading within a host, and transmitting to new host contacts. In the case of avian flu, a higher virulent strain is prevalent because pathogen population is maintained as a reservoir in a tolerant host species, but is a detriment to other species.

Research Question

The research question I explored in Allison Shaw’s lab with Martha Torstenson and Allison Shaw at the University of Minnesota Department of Ecology, Evolution, and Behavior is whether the host adaptations tolerance and resistance have an effect on the optimal pathogen virulence and how effective is the virulence strategy in maximizing infection spread. By using a theoretical model, the systemic dynamics and interplay of disease transmission become apparent. These insights can be used to inform ecological host population strategy effectiveness in minimizing pathogen harm.

This model is inspired by the work in the below citation:

Torstenson, M., & Shaw, A. K. (2024). Pathogen evolution following spillover from a resident to a migrant host population depends on interactions between host pace of life and tolerance to infection. Journal of Animal Ecology, 93, 475–487. https://doi.org/10.1111/1365-2656.14075

NSF Grant summary: https://allisonkshaw.weebly.com/migration-and-parasites.html

Model Overview

My theoretical model is based on a common mathematical model used the in the epidemiology field called a susceptible infected ordinary differential equation (SI ODE) model which depicts how biological processes such as transmission, infection recovery, and mortality, affect a host population that is divided into two subgroups: susceptible (S) or currently infected (I). There are near infinite amount of ecological and evolutionary strategy combinations that host species use to combat off disease. The two notable strategies I will model are host tolerance to the infection (by reducing infection-related mortality rate) and host resistance (reduce the rate that an individual becomes infected upon exposure to a pathogen). In my model, host and pathogen traits mutually contribute to the transmission and mortality processes which shift population abundances over time. My model assumes that virulence increases transmission rate, where more virulent pathogens have the potential to be evolutionary favored long term, which is not the case in the classic SI epidemiology mathematical models. I have encoded a transmission mortality trade off in my model, so it is not immediately evident what virulence phenotype is most competitive to exist in a host population with different defensive adaptations.

Realized Transmission and Mortality Rates

To capture the host and pathogen contributions to disease transmission and mortality, we denote a new rate called realized transmission rate $\beta(\alpha_z)$ and realized mortality rate $\mu_I$:

$$ \beta(\alpha_z) = \alpha_z \beta_z \cdot (1 - m_R) $$

Realized transmission is a function of virulence $\alpha_z$ and scalar term $m_R$ as the susceptible host’s resistance to new infection.

$$ \mu_I = (\sqrt{\mu_S} + (1 - m_T) \left(\frac{\beta(\alpha_z)}{\beta_{\text{max}}}\right))^2 $$

Realized mortality rate of infected hosts by a pathogen with virulence strategy $z$. Host tolerance $m_T$ decreases mortality rate by infection. Mortality is parabolic, where increasing transmission is traded off with higher infection-mortality rates.

ODE Model

The ODE model functions to predict host population abundances of 11 cohorts (susceptibles and 10 infected classes by virulence alpha) over time until stability is reached.

Rate of Change of Susceptible Individuals

$$ \frac{dS}{dt} = fN - \mu_S S + \gamma I - \sum_{z=0}^{10} \frac{\beta(\alpha_z) SI}{N^q} $$

Rate of Change of Infected Individuals

$$ \frac{dI_z}{dt} = \frac{\beta(\alpha_z) SI}{N^q} - \mu_I I - \gamma I $$

Average Pathogen Strategy

$$ V(X) = \frac{\sum_{z=0}^{10} \alpha_z I_z}{\sum_{z=0}^{10} I_z} $$

The average pathogen strategy is calculated as the sum of individuals infected by each pathogen class and the alpha value associated with it, divided by the total infected individuals in the population.

Symbol Definitions

Symbol Meaning Numerical Range
$S$ Host variable: Susceptible population count Any integer
$I_z$ Host variable: Infected population by $\alpha$ Any integer
$t$ Time in days $t \geq 0$
$N$ Total number of individuals in the population Varies
$q$ Host population density effect on transmission $0 \leq q \leq 1$
$\mu_S$ Host parameter: Natural mortality rate —
$\alpha_z$ Virulence factor $0.1 \leq \alpha \leq 1$ in steps of 0.1
$z$ Pathogen strategy location in parameter space Integers $0$ to $10$
$f$ Per capita fecundity rate —
$m_T$ Host adaptive tolerance to infection mortality $m_T \leq 1$
$m_R$ Host adaptive resistance to new infections $m_R \leq 1$
$\mu_I$ Realized mortality rate induced by $\alpha$ Quadratic
$\beta$ Base pathogen transmission rate Individual $\cdot$ time$^{-1}$
$\beta_{\text{max}}$ Maximum instantaneous transmission rate —
$\gamma$ Recovery rate $\gamma > 0$
$V(X)$ Mean pathogen strategy Evolves

ODE Analysis

Figures 1-3 in the github repository are parameter sweep simulations of the equations above.

The $R_0$ equation divides the favorable pathogen rates (that allow it to proliferate) by the unfavorable/trade-off rates, giving you a standarized pathogen fitness value.

If $R_0$ is greater than 1 ($R_0$>1) then the strain will proliferate and induce an epidemic state in the host population; if $R_0$ is one ($R_0$=1), then it’s at steady state; if $R_0$ is less than 1 then the strain will eventually go extinct

$$ R_0 = \frac{\beta(\alpha)}{\mu_I + \gamma} \to R_0 = \frac{\alpha_Z \beta^(1-m_R) S}{\sqrt{\mu_S + (1-m_T)(\frac{\alpha_Z \beta^(1-m_R)}{\beta_{\text{max}}})^2} + \gamma} $$

Where: - $\beta(\alpha)$: Transmission rate as a function of $\alpha$

  • $\mu_I$: Infected mortality rate

  • $\gamma$: Recovery rate

ODE analysis Goals

  1. Investigate the conditions under which $R_0 > 1$, $R_0 = 0$, or $R_0 < 1$.
  2. Determine the optimal virulence $\alpha$ value by finding the derivative of $R_0$ and locating the global maximum where $\frac{d}{d\alpha} = 0$.

Methodology

Partial Derivative of R0 with Respect to Alpha

We independently vary host parameters (e.g., tolerance and resistance) to examine the derivative of alpha (represented as A below) and host trait constants (represented as H below) to determine ## Partial Derivative of $R_0$ with Respect to $\alpha$

We independently vary host parameters tolerance T and resistance R to examine the derivative of $\alpha$ (represented as $A$) and host trait constants R and T.

  1. When Tolerance is Zero ($m_r = R$):
library(Deriv)

# Define R0 for tolerance
R0_tolerance <- expression(
  alpha * 0.4 * (1 - R) / 
  (sqrt(0.1 + alpha * 0.4 * (1 - R) / 0.4)^2 + 0.01)
)

# Compute the partial derivative
partial_tolerance <- Deriv(R0_tolerance, "alpha")
print(partial_tolerance)
## expression({
##     .e1 <- 1 - R
##     .e2 <- alpha * .e1
##     .e3 <- 0.11 + .e2
##     (0.4 - 0.4 * (.e2/.e3)) * .e1/.e3
## })

$$ \left( 0.4 - 0.4 \cdot \frac{A \cdot (1 - R)}{0.11 + A \cdot (1 - R)} \right) \cdot \frac{1 - R}{0.11 + A \cdot (1 - R)} $$ 2. When Resistance is Zero ($m_t = T$):

# Define R0 for resistance
R0_resistance <- expression(
  alpha * 0.4 / 
  (sqrt(0.1 + (1 - T) * alpha * 0.4 / 0.4)^2 + 0.01)
)

# Compute the partial derivative
partial_resistance <- Deriv(R0_resistance, "alpha")
print(partial_resistance)
## expression({
##     .e2 <- alpha * (1 - T)
##     .e3 <- 0.11 + .e2
##     (0.4 - 0.4 * (.e2/.e3))/.e3
## })

$$ \left( 0.4 - 0.4 \cdot \frac{\alpha \ \cdot (1 - T)}{0.11 + \alpha \ \cdot (1 - T)} \right) \cdot \frac{1 - T}{0.11 + \alpha \ \cdot (1 - T)} $$ They’re both the same result! The derivatives of resistance and tolerance are identical equations because the host traits in the model are constants, but the basic reproductive $R_0$ equations when virulence level is varied will be different because those same constants affect the trajectory curve differently when virulence level alpha is varied.