🔣 🍴 💡 Epidemiological model Covid-19 🤷🏼 ☦️ 🤶🏻

Recently, there have been quite different models for the development of the epidemic, including on Habré. This topic has not bypassed me either. I would hardly have written here, but given what I managed to find, the importance of the discovered dependencies and their impact on our lives, I cannot but share the find. There will be many formulas, graphs and little text. Basic information and graphics for Germany, where I live.

So, the epidemiological model in the first approximation is described by the growth formula of the infected.

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N_{t} = N_{0} 2^{t / T_{d}}

$N_t = N_02^{t/T_d}$

Where

T_{d}

$T_d$ - the doubling time of infected in our case in days,

t

$t$ number of days

N_{0}

$N_0$ number of infected at a certain point in time and

N_{t}

$N_t$ - the number of cases through

t

$t$ days. If we divide both parts of the formula by the total population of the region, we get the same formula, but in parts of the population

P

$P$ .

P_{t} = P_{0} 2^{t / T_{d}} (1)

$P_t = P_02^{t/T_d} (1)$

The problem with this formula is that the formula does not take into account the limited population and

P

$P$ will soon be more than 1. This does not happen in real life.

There is an epidemiological factor determining to what level the number of cases can increase. It is calculated based on thebase number of reproductions.

R_{0}

$R_0$ . This number shows us how many people approximately one infected person infects, is constant and specific for each particular region depending on the population density and other characteristics of the region. It can be determined only at the beginning of the epidemic, when there are no limiting factors. The formula itself looks like this:

P_{s a t} = 1 - 1 / R_{0}

$P_{sat} = 1 - 1/{R_0}$

There is also an effective reproduction rate

R_{t}

$R_t$ , which also lets us know how many people the patient infects. Unlike the base number, the effective one is constantly changing. You can determine this value using the formula above and knowing the number of infected at a certain point in time:

R_{t} = R_{0} (1 - P_{t}) (2)

$R_t = R_0(1 - P_t) (2)$

If we take the simplified SEIR ^[1] model of the epidemic, we can find additional factors describing the characteristics of the epidemic, such as growth rate

G_{r}

$G_r$ or time of infectivity of the patient

D

$D$ . The following formulas show the relationship between the quantities.

G_{r} = \frac{R_{t} - 1}{D}

$G_r = \frac{R_t - 1}{D}$

T_{d} = \frac{l n (2)}{G_{r}}

$T_d = \frac{ln(2)}{G_r}$

Using the above formulas, we can derive the following dependence

T_{d} = \frac{D l n (2)}{R_{t} - 1}

$T_d = \frac{D ln(2)}{R_t - 1}$

and substituting it in (1) we get:

P_{t} = P_{0} 2^{\frac{t (R_{t} - 1)}{D l n (2)}}

$P_t = P_0 2^\frac{t (R_t - 1)}{D ln(2)}$

or after simplification

P_{t} = P_{0} e^{\frac{t (R_{t} - 1)}{D}}

$P_t = P_0 e^\frac{t (R_t - 1)}{D}$

If we need to determine the value the next day, then

t = 1

$t = 1$

P_{1} = P_{0} e^{\frac{R_{t} - 1}{D}}

$P_1 = P_0 e^\frac{R_t - 1}{D}$

The effective number of reproductions for a specific time

c u r r e n t

$current$ can be calculated from formula (2) and then knowing only the basic number of reproductions

R_{0}

$R_0$ and the number of infected at the moment

P_{c u r r e n t}

$P_{current}$ we can easily calculate the percentage of people infected the next day.

P_{n e x t} = P_{c u r r e n t} e^{\frac{R_{0} (1 - P_{c u r r e n t}) - 1}{D}}

$P_{next} = P_{current} e^\frac{R_0 (1 - P_{current}) - 1}{D}$

There is only one parameter in this formula

, which can be calculated from the doubling time at the beginning of the epidemic. Taking for example

R_{0}

$R_0$

P_{c u r r e n t} = 0, 0001

$P_{current} = 0,0001$ and taking n steps, we will get an epidemiological state in n days. Time, the shape of the curve, the saturation value, the number of cases at a certain point in time and other parameters jump out of the formula "like a devil from a snuff box."

What about quarantine and other factors affecting the course of the epidemic?

Each of the measures taken corrects the base number of reproductions by a certain factor (factor)

μ \in [0; 1]

$μ ∈ [0;1]$ in the following way:

P_{n e x t} = P_{c u r r e n t} e^{\frac{μ R_{0} (1 - P_{c u r r e n t}) - 1}{D}}

$P_{next} = P_{current} e^\frac{μ R_0 (1 - P_{current}) - 1}{D}$

For simplicity, you can even define a “limiting” base reproduction number:

R_{0}^{'} = μ R_{0}

$R'_0 = μ R_0$

Further, at some time points, you can simply replace one reproduction number with another, thus adjusting the spread of the epidemic. The epidemic continues to spread with new conditions for a certain period of time, until the moment of a new change. Change points or intervention points are determined by external factors, such as quarantine, closure of schools, or the need to wear masks. The time and extent of exposure to these factors, as a rule, cannot be known in advance. However, if the value of the correction number change is known

μ

$μ$ , which determines the effectiveness of quarantine, you can find out how its cancellation will affect at a certain point in time in the future. This provides good predictive opportunities for this model. This is also one way to test its validity in practice.

As the time of infectivity of the patient

D

$D$ for Covid-19, a value of 10 was taken ^[2] .

There are no other parameters in the model, as well as additional degrees of freedom.

What about verification?

Charts based on data from Germany.

There were only 3 intervention points indicated in the following table:

Which led to the following results.

The points of intervention and comparison of model data with actual values extracted from public data are visible on the graph of changes in the effective number of reproductions.

The coincidence of the data and the quality of the model can be checked on the regression charts:

The model and calculations for Germany are posted on GitHub . There are not only these data, but also studies on deaths.

Update:
Additional checks made. Each country has its own correction factor.

Russia:

Italy:

USA:

Update 2:
Added a version of “simple” on GitHub in which everything superfluous is removed, you can insert the values of other countries, change the points of intervention and the correction factor. There is a high probability that this same correction factor is the ratio of those diagnosed to infected. But it needs to be checked. Further development and completion of the epidemic will confirm or refute this hypothesis.

In the graphs, the values of the detected cases are in% and these values are not corrected for the Ratio value (Correction Factor). Dividing by this number we get the actual percentage of detected cases and the prognosis of infection. In the simple version, this correction has been made.

References
[1] JM Heffernan et al. Perspectives on the basic reproductive ratio. doi.org/10.1098/rsif.2005.0042 JR Soc. Interface 2005 2, 281–293 (2005)

[2] Xi He, Eric HY Lau et al. Temporal dynamics in viral shedding and transmissibility of COVID-19. www.nature.com/articles/s41591-020-0869-5 Nature (2020)

Epidemiological model Covid-19

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