# Modeling of viral zoonotic infectious diseases from wildlife to humans

The COVID-19 pandemic has made painfully clear the importance of studying zoonotic diseases, infectious diseases that are passed from animals to humans. Today, Linda Allen, a mathematical biologist at Texas Tech University, gave an AMS-MAA Invited Address on how mathematical modeling helps us understand emerging zoonoses, particularly viruses.

Before a virus passes from a natural animal host to humans, it generally passes through an intermediate animal host. For example, SARS originated in horseshoe bats, jumped from them to civets, and jumped from civets to humans. Seasonality in ecological interactions can affect a virus’s spread within the intermediate host population as well as its transmission to humans. In her talk, Allen focused on how seasonality impacts the probability and timing of a spillover—the moment when an intermediate animal host first infects a human with the disease.

Mathematicians harness differential equations and Markov chain methods to model the spread of infectious diseases. In the context of COVID-19, most of us have heard about susceptible-infected-recovered (SIR) models, which are characterized by a transmission rate $\beta$ and a recovery rate $\gamma$. Allen applied an SIR model to a spillover event by including both an animal-animal transmission rate $\beta_{aa}$ and an animal-human transmission rate $\beta_{ah}$. She modeled seasonality by considering both transmission rates (as well as the animal recovery rate) to be continuous periodic functions.

Allen explained how an SIR model can be used to study the spillover of an infectious disease from animals to humans.

If seasonal effects are strong, the time of year when the virus infects the intermediate animal host has a major influence on the probability of a spillover into humans. Allen presented the results of computations and simulations with different relationships between the transmission rates and animal recovery rate.

Unsurprisingly, if the seasonal peaks in the two transmission rates align, the probability of a spillover is highest near those peaks. If the peaks don’t align, the resulting trends in spillover probability are not as intuitive. Plus, the timing of the maximum spillover probability can depend on the number of initially infected intermediate hosts.

Allen gave an example of the model applied to H5N1 (a type of avian influenza) in domestic poultry and humans. Data from the World Organization for Animal Health show sharp annual spikes in outbreaks among domestic poultry. The results of Allen’s approximate calculations as well as Markov chain simulations clearly showed the resulting seasonality in spillover probability.

The probability of a spillover shows seasonal variation. In the left column, the animal-animal transmission rate (dashed red) and animal-human transmission rate (black) are plotted as functions of time. In the other two columns, the time of the first animal infection is on the x-axis. The middle column shows the probability of a spillover if one animal host becomes infected initially. The right column shows the probability of a spillover if five animal hosts become infected initially.

Researchers estimate that over 60% of human infectious diseases are zoonotic. Of those, around 75% are emerging or reemerging, so understanding the dynamics of spillovers is crucial if we hope to prevent future pandemics. Worryingly, the frequency of zoonoses is rising due to deforestation, climate change, globalization, and other factors.

Allen closed her talk with some thoughts on how we can promote public health. She mentioned three interconnected tasks: collaboration between mathematicians and experts in diverse fields like ecology, geography, and epidemiology; cooperation within and between agencies at the local, national, and international levels; and education of the public about the sources and prevention of zoonoses. Given the ongoing pandemic, now is the time to build momentum in all three of these areas. Maybe next time a dangerous zoonotic disease emerges, humanity will be better-prepared.

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