Computing human to human Avian influenza $\mathcal{R}_0$ via transmission chains and parameter estimation

The transmission of avian influenza between humans is extremely rare, and it mostly affects individuals who are in contact with infected family member. Although this scenario is uncommon, there have been multiple outbreaks that occur in small infection clusters in Asia with relatively lowtransmissibility, and thus are too weak to cause an epidemic. Still, subcritical transmission from stut-tering chain data is vital for determining whether avian influenza is close to the threshold of ℜ₀ > 1.In this article, we will explore two methods of estimating ℜ₀ using transmission chains and parameterestimation through data fitting. We found that ℜ₀ = 0.2205 when calculating the ℜ₀ using the maxi-mum likelihood method. When we computed the reproduction number for human to human transmis-sion through differential equations and fitted the model to data from the cumulative cases, cumulativedeaths, and cumulative secondary cases, we estimated ℜ₀ = 0.1768. To avoid violating the assumptionof the least square method, we fitted the model to incidence data to obtain ℜ₀ = 0.1520. We tested thestructural and practical identifiability of the model, and concluded that the model is identifiable undercertain assumptions. We further use two more methods to estimate ℜ₀ : by the ℜ₀ definition whichgives an overestimate of 0.28 and by Ferguson approach which yields ℜ₀ = 0.1586. We conclude that ℜ₀ for human to human transmission was about 0.2.