{"id":445,"date":"2020-04-27T17:07:24","date_gmt":"2020-04-27T15:07:24","guid":{"rendered":"https:\/\/houzardc.wordpress.com\/?page_id=445"},"modified":"2020-06-01T17:52:35","modified_gmt":"2020-06-01T15:52:35","slug":"prediction-of-the-epidemic-final-size","status":"publish","type":"page","link":"https:\/\/houzardc.fr\/?page_id=445","title":{"rendered":"Prediction of the epidemic final size"},"content":{"rendered":"\n

This article was originally published on Apr, 28. The graphs and values are updated regularly at <\/em>the end of this article.<\/em> <\/a><\/p>\n\n\n\n

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Abstract<\/strong>: The purpose of this article is to predict the final size of the COVID-19 epidemic based on a logistic regression model by fitting the data of France, the US, Italy and Spain. <\/p>\n<\/div><\/div>\n\n\n\n

In this article, it will be tried to predict the final size of the COVID-19 epidemic per country based on a logistic regression.<\/p>\n\n\n\n

It is divided in 3 parts:<\/p>\n\n\n\n

  1. Logistic regression<\/strong>: the properties of this function will be explored<\/li>
  2. Fitting of the model<\/strong>: it will be described how the best fit was chosen<\/li>
  3. Results<\/strong>: the model will be applied to 4 countries: France, the US, Italy and Spain<\/li>
  4. Conclusion<\/strong><\/li>
  5. Updated prediction<\/strong><\/li><\/ol>\n\n\n\n

    The codes that were used for these graphs are available on this GitHub repository<\/a>. The code used for the visualisation of the RMSE (\u00a72.1) is 32 – Visualisation of RMSE.py<\/a>. The one used for the prediction is 31 – Mod\u00e9lisation v2.0.py<\/a> (\u00a72.2 and \u00a73).<\/p>\n\n\n\n

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    1. Logistic regression<\/h2>\n<\/div><\/div>\n\n\n\n

    1.1. Models to predict an epidemic<\/strong><\/h3>\n\n\n\n

    An epidemic can be described by many models, more or less complex. The most usual models are the compartmental one<\/a>. They are called this way because they divide a population into categories. For example, the most basic compartmental model is the SIR: Suspected (i.e. not yet infected), Infectious and Recovered. It can give interesting insight on an epidemic like the number of people one can infect on average. <\/p>\n\n\n\n

    On this model, the infectious curve is essentially a logistic regression if the constants of the model don’t change. Hence, for simplicity, we will consider here that the number of cases follows a logistic regression. Such an application \"c\" can be expressed as a function of time with 3 parameters: \"\alpha, \beta, \gamma\"<\/p>\n\n\n\n

    \"\displaystyle<\/p>\n\n\n\n

    1.2. Properties of<\/strong> a logistic regression <\/h3>\n\n\n\n

    The function \"c\" defined above is clearly \"C^\infty\", hence, we can determine the first few derivates.<\/p>\n\n\n\n

    From these, nice properties can be found. They are summed up in on this graph:<\/p>\n\n\n\n

    \"\"<\/a>
    Fig. 1.1 – Representation of a logistic regression, its first and second derivate<\/figcaption><\/figure>\n\n\n\n

    1.2.1. Maximum of the function<\/h4>\n\n\n\n

    Finding the maximum of this function presents an obvious interest in our problem: it is the final estimated size of the epidemic. For a logistic regression, we have:<\/p>\n\n\n\n

    \"\displaystyle<\/p>\n\n\n\n

    1.2.2. Inflexion point<\/h4>\n\n\n\n

    The function clearly presents an inflexion point. Let’s call it \"P_I\". We have:<\/p>\n\n\n\n

    \"P_I=(-\frac{\beta}{\gamma}, \frac{\alpha}{2})\"<\/p>\n\n\n\n

    By definition, the inflexion on the function \"c\" is also a maximum (or a minimum) of the derivate \"\frac{dc}{dt}\". Let’s call this maximum \"P_{I'}\". We also have:<\/p>\n\n\n\n

    \"P_{I'}=(-\frac{\beta}{\gamma}, \frac{\alpha \gamma}{4})\"<\/p>\n\n\n\n

    1.2.3. Growth speed<\/h4>\n\n\n\n

    Let’s focus on the interval \"\left]-\infty, -\frac{\beta}{\gamma} \right]\". <\/p>\n\n\n\n

    By looking at the 2nd derivate \"\frac{d^2c}{dt^2}\", we can see that the function increases more and more rapidly up until a point we shall name \"t_1\". The 2nd derivate then decreases. Before that point, the growth is essentially exponential. <\/p>\n\n\n\n

    By solving \"\frac{d^3c}{dt^3} (t_1)=0\", we get :<\/p>\n\n\n\n

    \"\displaystyle <\/p>\n\n\n\n

    Similarly, on the interval \"\left[-\frac{\beta}{\gamma}, \infty \right[\", the growth decreases slower and slower, until \"t_2\", when it starts decreasing faster. The solution to equation \"\frac{d^3c}{dt^3} (t_2)=0\" on this interval gives us:<\/p>\n\n\n\n

    \"\displaystyle<\/p>\n\n\n\n

    The difference between these 2 points \"t_2\" and \"t_1\" gives us a first characteristic time \"\tau\". It expresses the time it takes to go from an exponential increase to an exponential decrease:<\/p>\n\n\n\n

    \"\displaystyle<\/p>\n\n\n\n

    1.2.4. Total duration of the epidemic<\/h4>\n\n\n\n

    It can be interesting to quantify the time \"t_x\" at which the function has reached the proportion \"x\" of its final value i.e. at which it reaches the value \"x \alpha\". <\/p>\n\n\n\n

    By solving a simple equation, \"c(t_x)=x\alpha\", we find that:<\/p>\n\n\n\n

    \"\displaystyle<\/p>\n\n\n\n

    Now we can imagine a new characteristic time that would be defined as the required time to go from the proportion \"(1-x)\" to the proportion \"x\". Let\u2019s call it \"\Delta t_x\". <\/p>\n\n\n\n

    For example, \"\Delta t_{99}\" would be the time it took to go from 1% to 99% of the final value \"\alpha\". <\/p>\n\n\n\n

    It can be expressed as a function of \"\gamma\" or \"\tau\":<\/p>\n\n\n\n

    \"\displaystyle <\/p>\n\n\n\n

    These values will be used to characterise the epidemic afterwards.<\/p>\n\n\n\n

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    2. Fitting of the model<\/h2>\n<\/div><\/div>\n\n\n\n

    Now, let’s see how we can fit that logistic function to our data.<\/p>\n\n\n\n

    For the rest of this paragraph<\/span>:<\/p>\n\n\n\n