Here is an attempt to describe the epidemic diffusion of Covid-19 and its effect on the population of a State. The evolution through time is described by a discrete time micro-founded model. The model is calibrated on Italian data, available thanks to the Dipartimento Protezione Civile, and available at this GitHub repository. After the calibration, it is possible to study the characteristics of the parameters on the evolution of the disease, forecast its spread and time to peak, and evaluate the impact of restraining measures put in place by the Government.
Another feature of the model that may be of interest is the possibility to infer some key indicators of the phenomena, like r0 (which is a measure of the capacity of an infected individual to infect other people), time to recover, death probability, etc.
For those interested in looking at the code, please visit this GitHub repo (I’ll update it in the next days, so if you still find it empty, please come back in a little while).
DISCLAIMER: This is a preliminary attempt of a home-made exercise, so please consider it as NON-SCIENTIFIC. I’ve no medical background, and I’m not a researcher, just a person who likes numbers and coding 🙂 I’ll update my findings in this blog as new data comes in, and as I discover and fix bugs / implement new features.
DISCLAIMER 2: This is by no means a post about estimating number of people infected or dead, it is just a math/logical/coding exercise. The high number of infected / dead may be due to the fact that the calibration takes into consideration what happened in the first days of the infection, which is typically characterised by high contagious rate.
The model describes the evolution in time of a society in which an initial number of people affected by the virus is introduced. At each point in time, every person in the society is in one of the following categories:
- : unknown asymptomatic, people that will not suffer from the severe consequences of the virus and will continue their normal life. They will recover from the virus after periods, becoming .
- : unknown seriously affected, people that are currently not showing symptoms but will be hit by the serious consequences of the virus after periods, becoming either with probability or with probability .
- : known seriuos affected in intensive care, they may die during each period of the hospitalization with probability , or, if they don’t die, they will recover after periods becoming .
- : known serious affected but not in intensive care, they also may die during each period with probaility , or, alternatively, they will recover if they survive after periods becoming .
- : people that have not been affected by the virus and so can be affected by an infected person. Once infected, the person can become an unknown serious affected () with probability , or an unknown asymptomatic () with probability
The equations that describe the evolution of each variable in time are desribed below. We wrote the equation with the following convention, to make it more readible: the prefix is used to descrive people that are new to the category refers to, while is used in the same fashion for those that exited the category in which they where before.
where is the number of people been infected by either or during period and become new asymptomatic infected at time , and are those that recovered during/after time
Finally, and can be seen as a way to calculate what in medical theory is called . They represent the number of people that will be infected in a period of time by an unknown seriously affected person and by an unknown asymptomatic, respectively. and are not constant parameters, as they vary due to environmental conditions: the higher are the chances of contacts between individuals, for instance, the higher they will be. From a mathematical perspective, they represents an important closure condition: we’ll make them depened on the population not yet affected, and this will make the model not exploding as time goes.
where is the total population at time 0. In other words, you can see and as , where is the starting time when the first infected person is introduced in the society.
Model characteristics and parameter sensitivity
The model is highly dependent on exogenous parameters: here we describe how the most important variable evolve in time given changes in the exogenous parameters.
The rest of the sensitivity graphs are shown at the end of the page, for convenience reasons.
In order to calibrate the model, we proceed with an extensive grid search on all parameters and initial conditions. The procedure we developed consists on three different grid searches, that will be described below, each of which follows this path:
- create a grid of parameters
- run the model for every combination of parameters in the grid
- compare the results of the model to actual data coming from the Dipartimento della Protezione Civile Italiana, and calculate an error measure (more on this later)
- select an optimal model for each error calculated: we calculate errors on the number of total infected people since the beginning of the infection , number of people currently infected , total number of deaths , total number of known recovered peolpe and the average of all the above
The calibration steps mentioned above are the following:
- Initial grid search: create a grid of all parameters and find optimal models
- Parameter fine tuning: starting from the optimal parameters from step 1, create a grid with +- a percentage increase of the optimal parameters and find optimal models
- Window Search: we divide the periods in windows of fixed length (7 periods), and for each of them we allow the and parameter to change (like in step 2), so to allow for changing conditions in the infection rate. This will try to find changes in exogenous conditions affecting the infection rate, like social restriction measures taken by the government.
Here are the results of a calibration performed on Italian data at national level, updated as of March 28th. Differences between the model and the window optimized one will be also presented.
DISCLAIMER: this is a quite preliminary version of the model, so its results are still under investigation. In the next days/weeks I’ll be working on it and more robust analysis will be shared. For this reasons, please consider the
following analysis only a pure academic exercise. For the same reasons, I’m not investing time in describing the results,
but I’m only showing them as they are, also to get feedbacks that are always very very wellcome.
|Peak Var||Model||Opt Window|
|Total Infected (Igc_cum)||9.85M @ Sep 08||8.72M @ Sep 08|
|Daily Infected (Igc)||2.76M @ May 28||2.14M @ Jun 03|
|Daily Infected Int. Care (Igci)||175K @ May 28||136K @ Jun 02|
|Daily Deaths (M)||26K @ May 29||20K @ Jun 04|
|Total Deaths (M_cum)||1.27M @ Aug 04||1.12M @ Aug 20|
Finally, here is a comparison between the model, optimal window model and actual data for key variables.