📂Dynamics

Inter-Species Transmission Model: Disease Spread among Three Populations

Overview

Species Barrier refers to the phenomenon where it’s difficult for a pathogen to transmit from its original host to another species. The event where a disease crosses this barrier is called Cross-species Transmission, for which we introduce the Host-vector-host model as a mathematical representation.

Model ¹

$$\begin{align*} \\ {{d I_{1}} \over {d t}} =& \beta_{12} I_{2} S_{1} - \gamma I_{1} \\ {{d I_{2}} \over {d t}} =& \left( \beta_{21} I_{1} + \beta_{23} I_{3} \right) S_{2} - \gamma I_{2} \\ {{d I_{3}} \over {d t}} =& \beta_{32} I_{2} S_{3} - \gamma I_{3} \end{align*}$$ Note that for $k=1,2,3$, it applies that $S_{k} + I_{k} = 1$.

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• The differential equation for $S_{k}$ is intentionally omitted, although it should be explicitly stated under normal circumstances.

Variables

• $S_{k}(t)$: Represents the ratio of the $k$th group that can be susceptible to the disease at time $t$.
• $I_{k}(t)$: Represents the ratio of the $k$th group that can be infectious at time $t$. In the context of information spread, it’s also common to use the first letter of Informed.

Parameters

• $\beta_{ij}>0$: The infection rate of the $j$ group by the $i$ group.
• $\gamma_{k}>0$: The recovery rate of the $k$ group.

Explanation

While it primarily uses the SIS model, that’s not the main concern; the key aspect is that the total population has been divided into two groups, $k=1,2,3$. When considering the index $1,3$ as the Host, the two groups do not directly transmit the disease to each other but share the pathogen through the Vector, indexed by $2$.

Even into 2021, there remains controversy over COVID-19, which is believed to have started in bats, passed through pangolins, and then transmitted to humans². If $I_{1}$ had been the bat carrying this virus, then $I_{2}$ would be the pangolins, and the initial $I_{3}$ would have been the Chinese in Wuhan who indulged in them as medicines or delicacies.

While not strictly about cross-species transmission, setting compartments, not just by state but by multiple species levels, opens up more possibilities. Even within the same species, assuming differences in transmission rates based on spatial characteristics (such as by country), or assuming different contact amounts due to differences in age/life patterns, is worth considering.