flowchart LR Susceptible --> Infected --> Recovered
Overarching questions in Consumer–resource dynamics
Consider a system with two species (\(R_1\) and \(R_2\)) that compete with one another for resources, but that are also predated on by the same predator (\(C\)).
Write a model that captures the interactions in this system.
For competing species \(R_1\) and \(R_2\):
\[\frac{dR_1}{dt} = r_1R_1(1-\alpha_{11}R_1 - \alpha_{12}R_2)\] \[\frac{dR_2}{dt} = r_2R_2(1-\alpha_{22}R_2 - \alpha_{21}R_1)\]
Modify these to account for consumer effects:
\[\frac{dR_1}{dt} = r_1R_1(1-\alpha_{11}R_1 - \alpha_{12}R_2)-a_1R_1C\] \[\frac{dR_2}{dt} = r_2R_2(1-\alpha_{22}R_2 - \alpha_{21}R_1)-a_2R_2C\]
The consumer equation also needs to reflect both sources of food:
\[\frac{dC}{dt} = e_1 a_1 R_1 C + e_2 a_2 R_2 C - mC\]
Figure from Bell and Fortier-Dubois 2017, Proc B
Such simplification enables predictions, e.g. what should we expect to happen if one of the herbivores is eliminated?

Of course, there’s some limit to how complicated a system we can reasonably model
One producer, an herbivore, and a carnivore
e.g. Kelp, sea urchins, and otters
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Assumptions:
Let’s start with some history
In 1926-27, Volterra and Lotka developed the model of consumer–resource dynamics




Assumptions
flowchart LR Susceptible --> Infected --> Recovered
Assumptions
To model this system we need three equations: \(\frac{dS}{dt}\), \(\frac{dI}{dt}\), \(\frac{dR}{dt}\)
\[\frac{dS}{dt} = -\beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
\[\frac{dS}{dt} = -\beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
flowchart LR Susceptible --> Infected --> Recovered Recovered --> Susceptible
flowchart LR Susceptible --> Infected --> Recovered Susceptible --> Vaccinated
flowchart LR Susceptible --> Infected --> Recovered Infected --> Dead
Some disease move through populations slowly enough that vital rates (birth and daeth rates) become important
Some diseases exist in which there are individuals who are carriers but not infected. These carriers don’t get sick but can get others sick.
Some diseases have an ‘Incubation’ period after infection (i.e. infected individual doesn’t immediately show signs of infection)
Stage/age structure: some diseases cause more effects in older or younger patients – combining matrix model approaches with SIR approach
Consider a simple model of disease dynamics (“SIR”)
flowchart LR Susceptible --> Infected --> Recovered
\(S = \frac{\text{total susceptible}}{\text{total number of individuals}}\),
\(I = \frac{\text{total infected}}{\text{total number of individuals}}\)
\(R = \frac{\text{total recovered}}{\text{total number of individuals}}\)
flowchart LR Susceptible --> Infected --> Recovered
\[\frac{dS}{dt} = -\beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
In infectious disease modeling, an important question is:
Under what conditions would a new disease spread through a population?
To know whether an infection will spread, we need to know whether the following is true:
\[\frac{dI}{dt} = \beta SI - \gamma I \gt 0\]
What needs to be true for
\[\frac{dI}{dt} = \beta SI - \gamma I \gt 0\]
\[\beta SI - \gamma I \gt 0\]
\[\beta SI \gt \gamma I\]
\[\beta S \gt \gamma \]
\[\frac{\beta}{\gamma} > 1 ~~~\text{for}~~~ \frac{dI}{dt} > 0\]
\[\frac{\beta}{\gamma} > 1 ~~~\text{for}~~~ \frac{dI}{dt} > 0\]
On the other hand, for the disease to not spread,
\[\frac{\beta}{\gamma} < 1 ~~~\text{for}~~~ \frac{dI}{dt} < 0\]
We give this number a special name:
\[\overbrace{R_0}^{\substack{\text{basic} \\ \text{reproductive} \\ \text{rate}}} = \frac{\beta}{\gamma}\]
\[R_0 = \frac{\beta}{\gamma}\]
\(R_0 > 1\) implies an infection can spread; \(R_0<1\) implies an infection cannot spread
\(R_0\) can change over time (e.g. because of change virulence of the disease or change in behavior), so we can focus on \(R_{0,t}\) (\(R_0\) at a given time \(t\))
When is \(R_{0, t}\) big?
What can we do to reduce \(R_{0, t}\)?
\(R_{0,t}\) is used by public health agencies to monitor trends:
https://www.cdc.gov/cfa-modeling-and-forecasting/rt-estimates/index.html
\(R_0\) for important infections, from the Center for Evidence Based Medicine, Oxford Univ.
\[\frac{dS}{dt} = -\beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\beta SI > \gamma I\]
\[\beta S > \gamma\]
\(\beta > \gamma\), which is the same as \(\frac{\beta}{\gamma} > 1\)
\[\frac{\beta}{\gamma} > 1\]
We give the ratio \(\frac{\beta}{\gamma}\) a special name: \(R_0\) (“R-naught”) or basic reproductive rate of the disease
If \(R_0>1\), the disease is predicted to spread after it first enters the system; otherwise, it cannot spread.
Recall that the assumption behind \(R_0\)’s equation is that we are thinking about the disease immediately after it enters the population, so \(S \approx 1\)
So in real life, disease ecologists track \(R_t\), or disease reproductive rate at a given time.
\[R_0 = \frac{\beta}{\gamma}\]
Reducing \(R_0\) requires either reducing \(\beta\) or increasing \(\gamma\)
\(\beta\) is the transmission probability if an interaction does happen – can be hard to modify
\(\gamma\) is the recovery rate of infected individuals – can be hard to lower for brand new infections
Over time, these numbers can change, e.g. due to evolution of the infectious disease itself or due to better recovery
But at the time of disease introduction, these can be hard
\[\frac{dS}{dt} = -\beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
Fewer interactions requires measures like quarntining, or moving individuals out of the \(S\)uscptible category – which is possible through vaccination.
Other resources for learning about applications of SIR models:
https://media.hhmi.org/biointeractive/click/modeling-disease-spread/advanced-simulate.html
Modeling Epidemics With Compartmental Models, overview by JAMA (Journal of the American Medical Association)
Video series by US CDC: https://www.youtube.com/watch?v=1QLgXzyXOH0