# COVID-19/Mathematical Modelling

The following learning module focuses on the introduction of mathematical modelling.

## If you don't test, you don't have reported COVID-19 cases

In epidemiology it is important to have a basic understanding of reported numbers of COVID-19 infected people. To explain that we create two extreme fictional settings:
* Country A does not test at all and therefore reports 0 cases of infected people,
* Country B tests the total population and reports 0.5 million cases of infected people.
Complete testing of the total population is not possible due to time constraints and the availability of resources for testing. So in real life we have to deal mathematically with the fact, that the number of tested people is lower than the number of the total population in a country. Furthermore the tests are applied as a limited resource mostly on people
* that are exposed to confirmed COVID-19 positive citizens or
* stayed in an area with a large number of COVID-19 positive citzens or
* working in health care with COVID-19 positive patients and a possible infection must be clarified

## Subtopics

* (Exponential Growth) - no counterforce for the epidemiological spread of a viral disease (e.g. Vaccination or the immune system is not able to produce a immunological response by production of anti-bodies, assumption of an unlimited number of people - discuss application for COVID-19)
* (Logistical Growth) - the population develops a counterforce during the exposure to a new virus (e.g. by the immune system response
* (Compartment Models) - integrate different immunological status of patients (e.g. Susceptible, Infected and Recovered ) into the epidemiological modelling. The number of recovered/immune citizens are an indicator of the counter force against the epidemiological spread.
* (Spatial Risk Maps) - analysis of spatial patterns of risk and response
* (Estimation Cases) - estimate the number of true case
* (Connectivity) - Connectivity between people and locations have an impact on spreading of communicable diseases.
* (Disease Modelling Time span - DiMoT) describes mathematically the development of the disease over time.

## Learning Tasks

* (Comparison exponential growth and logistical growth) Compare exponential growth and logistical growth and discuss which model is more appropriate for the epidemiological spread of the disease (see Geogebra-Applet by Athur Lee)..
* (Disease Modelling Period - DMP) Try to draw in the Cartesian coordinate system for a curve that determines the probability of infection as an integral over the probability density function!
* (Capacity of logistic Growth) Try to identify the capacity of epidemiological spread of the disease COVID-19. Keep in mind that the capacity will the number of the population that will be infected during the considered time span. If the capacity is lower than the total population (e.g. 60% of total population the number of people that were infected by a virus (and recovered) is converging to $M$ (i.e. $\lim \limits _{x\to +\infty }f(x)=M$. If the capacity $M$ is larger than the total population, then the aggregate number of infected people over time show more or less an exponential behaviour until the epidemiological spread of the disease reaches the total number of population.
* (Parameter Logistic Growth Function) Analyze the impact of the parameters on the graph of logistic growth function by using the Geogebra Applet for Logistical Growth in Epidemiology, wie die Parameter $x_{o},M,k$ auf den Verlauf der Funktion bestimmen.
* (Social Connectivity) What is the impact of reduction of social contacts on the epidemiological spread? Discuss the role of Risk Literacy in the population for having a significant impact on the epidemiological spread of COVID-19? What are the parameters in the Geogebra-Applet about Logistical Growth in Epidemiology, that would be altered if social physical social contacts are reduced and people stay at home?
* (Compartment Models) Analyze compartmental models in epidemiology like the SIR model (Succeptible, Infected, Recovered) and compare the approach with logistical growth approach!
* (Spatial Risk Maps) The number of infected people differ from region to region and from country to country. The spatial difference in the epidemiological spread can be represented by risk maps that alter in time! Explain how the Compartment Models can be extended to spatially heterogenous distribution of the epidemiological fraction of succeptible, infected and recovered people? (see COVID-19/Time Series and COVID-19 Data)
* (Zoonosis) Analyze the epidemiological term of a Zoonosis and then extend the basic compartment models to a zoonosis. Is the notion of zoonosis applicable on COVID-19?

## Adjustments due to the delay in the tests

At the initial phase of contagion, the number of infections seems to follow an exponential growth. When the time between an infection and the moment when that infection is reported is significant, the information of cases of infection becomes quickly outdated.
Let us see an example. A person gets infected and does not know neither suspect it. After 5 days of incubation, this patient feels ill and requests an infection test. After 2 days, the test is ready and the result is officially reported.
Let us assume that the number of reported cases is growing at a daily rate of 22% and the reported cases is 402 when the person gets infected. After 7 days, the number of reported cases is 1617, which in turn includes the mentioned patient. However, this person was effectively infected 7 days before. In fact, when only 402 cases were reported, there were already 1617 true cases.
We would like to approximate how many true cases are there. Let us assume that:
* the time between a patient gets infected and the case is reported is always the same
* people do not significantly change the growth of infected cases
The variables are:
* $k$ is the number of days between a patient gets infected and the case is reported
* $r_{i}$ is the reported cases for day $i$
* $t_{i}$ is an approximation of true cases for day $i$
| day | reported cases | daily growth | true cases | approx. of true cases |
| 0 | $r_{0}$ | | $r_{k}$ | |
| 1 | $r_{1}$ | ${\frac {r_{1}}{r_{0}}}$ | $r_{k+1}$ | |
| $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | |
| $k$ | $r_{k}$ | ${\frac {r_{k}}{r_{k-1}}}$ | $r_{2k}$ | $t_{k}$ |
| $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | |
| $n-k$ | $r_{n-k}$ | ${\frac {r_{n-k}}{r_{n-k-1}}}$ | $r_{n}$ | $t_{n-k}$ |
| $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | |
| $n$ | $r_{n}$ | ${\frac {r_{n}}{r_{n-1}}}$ | $r_{n+k}$ | $t_{n}$ |
We would like to find a formula for $t_{n}$ that approximates $r_{n+k}$.
One possibility is using the previous $k$ growth rates. In this case:
$t_{n}=r_{n}\cdot {\frac {r_{n}}{r_{n-1}}}\cdot {\frac {r_{n-1}}{r_{n-2}}}\cdot \ldots \cdot {\frac {r_{n-k+2}}{r_{n-k+1}}}\cdot {\frac {r_{n-k+1}}{r_{n-k}}}$
Hence, $t_{n}={\frac {r_{n}^{2}}{r_{n-k}}}$.

### Learning Task

* Analyze the mathematical model behind these mathematical terms mentioned above!
* What are the basic assumptions for the model?
* How could you improve the model?
* What are methods to validate the mathematical approach in general?
* How could you improve the mathematical approach above and what are the benefits and potential drawback?