SEIR model

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In mathematical epidemiology, the SEIR model is an approach used to describe the spread of infectious diseases . The description is closer to the real course than that of the SIR model , since it takes into account that an individual is not immediately infectious himself after being infected. In contrast to an individual-based model, the description is macroscopic; H. the population is considered as a whole.

System of equations

Let the population of N individuals be divided into the four compartments S , E , I , R , so that

or.

Any individual can do the procedure

Susceptible (S) → Exposed (E) → Infectious (I) → Recovered (R)

run through. The spread of the disease is described by

It is a non-linear system of ordinary differential equations . Various forms are in use in the literature, for example there is also a modified form of the equations with a term instead . This form is used in the SIR model article . It is also often not on normalized, but it applies to the general population.

size unit Explanation
S ( t ) 1 Percentage of those at risk susceptible . Not infected yet.
E ( t ) 1 Proportion of those exposed, engl. exposed . Infected, but not yet infectious.
I ( t ) 1 Proportion of infectious people infectious .
R ( t ) 1 Proportion of recovered, engl. recovered or resistant . Or. deceased or quarantined after symptoms .
t d Time in days time .
Indv Indv Number of individuals
β 1 / (Indv * d) Transmission rate. The reciprocal is the mean time between contacts.
γ 1 / d Recovery rate. The reciprocal is the mean infectious time.
a 1 / d The reciprocal is the mean latency.

The mean latency is the average time an individual spends in group E of those exposed; this must be differentiated from the mean incubation time , because the onset of infectivity does not have to coincide with the onset of symptoms.

The term contact rate is also common for the transmission rate. A more precise consideration breaks this down into a product , with the transmission probability and the actual contact rate being.

Relationship to the base reproduction number

A relationship to the base reproduction number can be established. So that the disease does not spread further, must and be. Substituting these conditions into the differential equations leads to and , and thus . This corresponds to the case of the basic reproduction number and it results with :

If one uses normalized numbers of people or uses modified SEIR equations (as in the article SIR model , but the definition of in the formula for differs from the one used here), the result is:

.

The parameter that describes the transition to the intermediate stage in which the person is infected but not yet contagious does not play a role in the base reproduction number as long as one neglects the possibility of death in group E. If this is not done, you will get a formula for as given below in the section Including demographic dynamics .

You can also define a time-dependent net reproduction number or with .

Maximum possible proportion of infected people

The basic reproduction number essentially determines the maximum possible proportion of infected people in a given population. If we denote the total proportion of all infected people with , so it follows from the above system of differential equations

,

or simply with the basic reproduction number

By rearranging this equation , an integration by separating the variables provides

for all t , where is the natural logarithm . In particular, it applies

or equivalent

The auxiliary function has a maximum at . The maximum possible proportion of infected people is therefore reached, i.e. depends only on the base reproduction number and the initial values ​​of and :

For an emerging disease as an epidemic by an unknown virus applies and , that is the maximum possible proportion of those infected in the population depends in this case, as follows from the basic reproduction number from:

This equation corresponds to the equation for the SIR model specified under SIR model # Maximum number of infected people .

Percentage of people recovered at the end of the epidemic

In connection with the base reproduction number, there is also the proportion of the population as a whole that is infected, assuming that the epidemic would go through without any quarantine. Using the differential equations one finds

For the initial value is therefore

At is now , and therefore . This gives the equation

Algebraic transformation leads to the equation

With and is the last equation that with the Lambertian W function to be changed, d. H. what back replaces and converted after

results.

For the practical calculation of the part of the W function relevant here, one considers the quadratic Taylor polynomial of the function at the point −1 and from this determines the zero point to which the fixed point iteration is applied once again. The result is

A good approximation of the W function is now obtained as

where is the nth iteration of Newton's method

is. For all practical needs is perfectly adequate.

Exponential initial phase

At the beginning of the epidemic, the spread of the disease is exponential to a good approximation. With the approach comes the relationship . Herewith applies and . As a result

If you derive this equation on both sides, use it and then divide by , you get

or.

Since at the beginning is a very good approximation , it can be set what a relationship between the parameters and the growth constant is obtained.

An alternative systematic approach considers the same , which simplifies the differential equations to a linear system, which is called

can be described. All further considerations follow from the theory of linear differential equations with constant coefficients and their eigenvalue theory . The growth constant is an eigenvalue of the system matrix , the equation already found can be obtained from

As with any exponential growth process, the growth constant is equivalent to the more illustrative multiplication time

especially at the doubling time .

Sample calculation

The following is an example calculation for a parameter assignment as it was estimated for the COVID-19 pandemic 2020 in Germany. The spread is with respect to a base reproduction number of 2.4, which corresponds to the omission of essential quarantine. The Euler method is sufficient for the numerical solution of the initial value problem .

from numpy import array as vector

# Explizites Euler-Verfahren
def euler_method(f,t0,x0,t1,h):
    t = t0; x = x0
    a = [[t,x]]
    for k in range(0,1+int((t1-t0)/h)):
        t = t0 + k*h
        x = x + h*f(t,x)
        a.append([t,x])
    return a

def SEIR_model(beta,gamma,a):
    def f(t,x):
        S,E,I,R = x
        return vector([
            -beta*S*I,
            beta*S*I - a*E,
            a*E - gamma*I,
            gamma*I
        ])
    return f

def SEIR_simulation(beta,gamma,a,E0,I0,days,step=0.1):
    x0 = vector([1.0-E0-I0,E0,I0,0.0])
    return euler_method(SEIR_model(beta,gamma,a),0,x0,days,step)

def diagram(simulation):
    import matplotlib.pyplot as plot
    plot.style.use('fivethirtyeight')
    figure,axes = plot.subplots()
    figure.subplots_adjust(bottom = 0.15)
    axes.grid(linestyle = ':', linewidth = 2.0, color = "#808080")
    t,x = zip(*simulation())
    S,E,I,R = zip(*x)
    axes.plot(t,S, color = "#0000cc")
    axes.plot(t,E, color = "#ffb000", linestyle = '--')
    axes.plot(t,I, color = "#a00060")
    axes.plot(t,R, color = "#008000", linestyle = '--')
    plot.show()

def simulation1():
    N = 83200000 # Einwohnerzahl von Deutschland 2019/2020
    R0 = 2.4; gamma = 1/3.0
    return SEIR_simulation(
        beta = R0*gamma, gamma = gamma, a = 1/5.5,
        E0 = 40000.0/N, I0 = 10000.0/N, days = 140)

diagram(simulation1)
Dynamics of an infectious disease, according to the basic SEIR model

The four parts, each depending on the time in days.
S in blue , E in yellow dashed , I in magenta , R in green dashed .

It can be seen from this example that the epidemic, due to the heating up by the infectious, continues after the critical immunization threshold has been reached

continues. Overall, 88% of the population would contract the disease. However, the epidemic could be stopped by a one-month strict quarantine at the latest after reaching the critical immunization threshold.

Inclusion of demographic dynamics

State diagram

Assuming a constant death rate and a coincident birth rate, the extended model was made

formulated. Compared to the simple SEIR model, this model also describes a long-term endemic course in which an oscillation of the susceptibles can occur until it gets through

has settled in a defined equilibrium.

For the basic reproduction number one finds the relationship here when considering the equilibrium

Time-dependent transmission rate

Processes such as behavior changes, quarantine and seasonality cause a change in the transmission rate. These circumstances are taken into account in the modeling of the transmission rate as a time-dependent function , with the rest of the model being retained identically. The simplest approaches to seasonality take the transmission rate e.g. B. as a sine wave, with mountain in the colder and valley in the warmer months.

It should be noted that the differential equation system with the explicit time dependency is no longer an autonomous one and is therefore no longer a direct dynamic system. However, one can artificially gain an autonomous one from the system by adding the equation .

Web links

  • Model calculator for the course of an epidemic according to the SEIR model with changeable input variables

See also

literature

Individual evidence

  1. a b Pauline van den Driessche, Reproduction numbers of infectious disease models , Infectious Disease Modeling, Volume 2, August 2017, pp. 288-303
  2. Odo Diekmann, Hans Heesterbeek, Tom Britton, Mathematical tools for understanding infectious disease dynamics, Princeton UP, 2013, p 35
  3. Institute for Disease Modeling: SEIR Model (accessed April 12, 2020)
  4. C. Hubbs (2020): Social Distancing to Slow the Coronavirus (accessed April 12, 2020)
  5. Junling Ma: Estimating epidemic exponential growth rate and basic reproduction number . In: Infectious Disease Modeling, Volume 5, 2020, pp. 129–141, KeAi Publishing (December 17, 2019).
  6. ^ Statement by the German Society for Epidemiology (DGEpi) on the spread of the new coronavirus (SARS-CoV-2). (PDF) German Society for Epidemiology , March 18, 2020, accessed on March 26, 2020 . The version of March 19 on which the sample calculation is based can no longer be found. Here the version from the previous day as an alternative. The current version see here .
  7. Gerardo Chowell, Cécile Viboud, Lone Simonsen, Seyed M. Moghadas: Characterizing the reproduction number of epidemics with early subexponential growth dynamics . In: JR Soc. Interface 13: 20160659 (August 17, 2016). DOI: 10.1098 / rsif.2016.0659.
  8. M. Keeling, P. Rohani: Modeling Infectious Diseases in Humans and Animals . Section 5.2. (P. 159): Modeling forcing in childhood infectious diseases: Measles .