In the case of disease courses that extend over longer periods of time, it can also make sense to take demographic developments within the population into account. These demographic elements can be integrated into the systems in the Tables 3.1 and 3.2. We consider this using the example of theSIR model, see Figure 3.1.
S I R
β
NSI γI
µBN
µDS µDI µDR
Figure 3.1: Flow chart of the SIR model with demography (3.1).
If we assume, for example, that the population has a birth rate ofµBand a natural death rate ofµD and that all newborns are born susceptible, the S compartment increases by µBN newborns and decreases byµDS naturally deceased in each time step. Analogously, the other compartments decrease by the respective number of deceased µDE, µDI etc.
which leads to the ODE system dS
dt =µBN −µDS− β
NSI , S0≥0, (3.1a)
dI dt = β
NSI−(γ+µD)I , I0≥0, (3.1b)
dR
dt =γI−µDR , R0≥0. (3.1c)
The derivation of the death rate also corresponds to those for γ, κ etc.. For example, if one assumes an average life expectancy of 70 years, one can chooseµD = 70·3601 . The birth rate can usually be read from statistics. For the total populationN at timet we assume thatN(t) =S(t) +I(t) +R(t) holds true. This leads to
dN dt = dS
dt +dI dt +dR
dt = (µB−µD)N ,
which means that forN an ODE has to be solved with initial valueN0 :=S0+I0+R0. In this case, this can be done byN(t) =N0e(µB−µD)(t−t0).
For simplicity, however, we assume that birth and death rates are the same and can therefore be replaced byµ:=µB=µD. Thus, we receive
dS
dt =µ(N −S)− β
NSI , S0 ≥0, (3.2a)
dI dt = β
NSI−(γ+µ)I , I0 ≥0, (3.2b)
dR
dt =γI−µR , R0 ≥0. (3.2c)
Since in (3.2) we have dNdt = 0, it follows thatN is constant withN =S0+I0+R0. If we substituteR=N −S−I the system is reduced to a two-dimensional system
dS
dt =µ(N−S)− β
NSI , S0 ≥0,
dI dt = β
NSI−(γ+µ)I , I0 ≥0,
which must be solved numerically. However, a further analysis allows to extract more precise properties of this system. For this purpose we rescale both sides of the equations and divide them byN and obtain for ˆx1 := NS and ˆx2 := NI
dˆx1
dt =µ(1−xˆ1)−βxˆ1xˆ2, xˆ10= ˆx1(t0)≥0, dˆx2
dt = (βxˆ1−(γ+µ)) ˆx2, xˆ20= ˆx2(t0)≥0.
It should be noted that the new variables ˆx1 and ˆx2 no longer have units. The next step is to do the same for the time variable by introducingτ := (γ+µ)t. This variable has no unit and leads to, e.g. x1(τ) := ˆx1
τ γ+µ
= ˆx1(t) with dˆdτx1 = γ+µ1 dˆdtx1 and finally to the dimensionless ODE system
dx1
dτ =ρ(1−x1)− R0x1x2, x10=x1(τ0)≥0, (3.3a) dx2
dτ = (R0x1−1)x2, x20=x2(τ0)≥0, (3.3b) including the substitutionsρ:= γ+µµ and R0:= γ+µβ .
For further analysis we need the following definition:
Definition 3.2.1. (Equilibrium points)
Consider a given autonomous IVP with continuousg:Rm−→Rm dx
dt =g(x(t)), x(t0) =x0. (3.4)
Then aconstant solution x∗ : [t0,+∞)−→Rm of (3.4) satisfying g(x∗) = 0
is called equilibrium or singular point. An equilibrium x∗ is called locally asymptotically stable, if a neighbourhood X ⊂Rm of x∗ exists such that
t→+∞lim x(t) =x∗
holds true for all solutionsx: [t0,+∞)−→Rm of (3.4) withx(t0)∈ X. So in our case we have to solve the equations
0 =ρ(1−x∗1)− R0x∗1x∗2, 0 = (R0x∗1−1)x∗2
and find the so–calleddisease-free equilibrium x∗DE = (1,0) and the endemic equilibrium x∗EE =
1 R0, ρ
1−R1
0
. The former occurs when the entire population is susceptible and no infected persons are present. In the second case, a fixed proportion of the population is always infected with the disease, i.e. endemic.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x2
Figure 3.2: Direction fields of the dimensionless ODE system (3.3) with ρ := 0.5 and initial conditions x(1)0 := (0.1,0.7) (blue/solid), x(2)0 := (0.9,0.7) (red/dashed) and x(3)0 := (0,0.1) (magenta/dotted). On the left we have R0 = 1.3 and on the right R0= 0.8.
The direction field in Figure 3.2 shows that the trajectories of the system withR0 >1 tend towards the endemic equilibrium x∗EE. On the other hand, in the case R0 <1 the solutions tend to the disease free equilibriumx∗DE. These observations are no coincidence.
The following theorem helps to investigate the stability of the present equilibria in more detail.
Theorem 3.2.2. (Local stability of equilibria)
Consider a given autonomous IVP with continuousg:Rm−→Rm dx
dt =g(x(t)), x(t0) =x0,
and equilibrium pointx∗. If the eigenvalues of the Jacobian matrix Jg(x∗) =
∂gi(x∗)
∂xj
i,j=1,...,m
have onlynegative real parts, then the equilibrium pointx∗ islocally asymptotically stable.
The explanation for this theorem is based onlinearizing the ODE system for solutions sufficiently close to the equilibrium.
In our example the following applies to the Jacobian matrix Jg(x) =
−ρ− R0x2 −R0x1 R0x2 R0x1−1
. Inserting the disease-free equilibrium (1,0) into Jg leads to
Jg(x∗) =
−ρ −R0 0 R0−1
.
In this case the eigenvalues can be read directly from the diagonal and we get z1 =−ρ <0,
z2 =R0−1<0, if and only ifR0<1. (3.5) Analogously, the local stability of the endemic equilibrium is examined and the following result is obtained:
IfR0<1, there exists a unique disease-free equilibriumx∗DE = (1,0) which is locally asymptotically stable.
If R0>1, there exist a disease-free equilibrium x∗DE = (1,0) which is unstable, and an endemic equilibrium x∗EE =
1 R0, ρ
1−R1
0
which is locally asymptotically stable.
It can even be shown that
If R0<1, the disease-free equilibriumx∗DE = (1,0) is globally stable.
If R0 >1 andx20>0, the endemic equilibriumx∗EE = 1
R0, ρ
1−R1
0
isglobally stable.
Global stability here means that the initial values do not have to be chosen near the equilibrium, but only have to meet certain conditions, e.g. x20>0 in the second case.
Mathematically, the meaning of
R0= β
γ+µ (3.6)
becomes clear at this point since this value represents a threshold for the stability of the two equilibria. Biologically, this value corresponds to the so–called Basic Reproduction Number. This is the average number of new infections that an infected individual causes during the course of its disease in an otherwise susceptible population. It therefore seems logical that the disease ”dies out” if an individual infects less than one other person with the disease. In our derivation, we have already used a method to determine this Basic Reproduction Number using the Jacobian matrix, also called theJacobian approach. This is done by inserting the disease–free equilibrium in Jg and checking for which threshold this equilibrium becomes stable, see equation (3.5). Since this approach does not always work, there are also alternative ways, such as theNext–Generation approach.