So, before I provide you a time series alternative, let me discourage you from using time series analysis for this problem. I would point out that having five data points is equivalent to having four time series data points. To use time series, you will need to throw out twenty percent of your data. If you add in the idea of degrees of freedom, it gets worse. If you think about the internal dependence structure of the data, you have even less free variability. It isn't that you cannot do it, but tossing out twenty percent of your data on a small set is costly.
If you did decide you needed a time series model and if you thought you understood the data generation process so that you could build a model of how it is working in nature and if you can find material research on the time series of similar disease progressions, then you could construct a Bayesian time series, but almost all of your information will have to come from the other long time series. That is a huge limitation if this disease does not actually behave like the others.
If you used a Bayesian model, your data would be conditioned on the prior distribution based on the estimators of the other diseases. This would increase the value of your free information. It is not something I would do unless my boss came to me and began the conversation "I know you like working here, and we love having you here, I read your opposition but...."
A moving average is sort of okay, but do you have a feel for how time is impacting the spread of the disease? If you do, then that is information. You could make time an explicit variable, with the proviso that time isn't a random variable and it is standing in for a latent variable that you are not observing. You always need to keep that in mind. Another option is to use a simple average. When you have five data points your best choice is to incorporate any relevant exterior information that you can. You should also honestly and aggressively look at any censoring issues.
Is your data only five years because the problem used to be considered to be too small to track and has suddenly reached the eyes of policymakers? Are the people in the series at time one also in the series at time zero? Is this disease confused for other diseases? Is the data in other diseases inflated by data that should be in your series?
If you begin to think about all of the issues, the simple sample average could look pretty good unless you can think in terms of physical world models. In that case, Bayesian or maximum likelihood models with time substituting for a missing latent variable may be a good proxy. The best choice is to go back to the ecology of the disease and similar diseases and see if you can fit a physical model of the illness.
No model with five data points can be expected to have high precision. However, if you are doing predictive work, consider using Frequentist predictive intervals or Bayesian predictive distributions. They will account for both the uncertainty about how the sample was generated and the future yet to be observed uncertainty. Your Frequentist intervals will be wider than your confidence intervals, which do not consider future uncertainty. Your Bayesian predictive distribution will be more diffuse than the density from plugging in point estimates to a distribution because it will fully account for the uncertainty in both the past and the future.