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8 Solving Ordinary Differential Equations
Let the mathematical function S(t) count how many individuals, at time t, that
have the possibility to get infected. Here, t may count hours or days, for instance.
These individuals make up a category called susceptibles, labeled as S. Another
category, I, consists of the individuals that are infected. Let I (t) count how many
there are in category I at time t. An individual having recovered from the disease is
assumed to gain immunity. There is also a small possibility that an infected will die.
In either case, the individual is moved from the I category to a category we call the
removed category, labeled with R. We let R(t) count the number of individuals in
the R category at time t. Those who enter the R category, cannot leave this category.
To summarize, the spreading of this disease is essentially the dynamics of moving
individuals from the S to the I and then to the R category:
We can use mathematics to more precisely describe the exchange between the
categories. The fundamental idea is to describe the changes that take place during a
small time interval, denoted by Δt.
Our disease model is often referred to as a compartment model, where quantities
are shuffled between compartments (here a synonym for categories) according to
some rules. The rules express changes in a small time interval Δt, and from these
changes we can let Δt go to zero and obtain derivatives. The resulting equations then
go from difference equations (with finite Δt) to differential equations (Δt → 0).
We introduce a uniform mesh in time, t n = nΔt, n = 0, . . . , N t , and seek S
at the mesh points. The numerical approximation to S at time t n is denoted by S n .
Similarly, we seek the unknown values of I (t) and R(t) at the mesh points and
introduce a similar notation I n and R n for the approximations to the exact values
I (t n ) and R(t n ).
In the time interval Δt we know that some people will be infected, so S will
decrease. We shall soon argue by mathematics that there will be βΔtSI new
infected individuals in this time interval, where β is a parameter reflecting how easy
people get infected during a time interval of unit length. If the loss in S is βΔtSI ,
we have that the change in S is
S
n+1
− S
n
= −βΔtS
n I
n .
(8.9)
Dividing by Δt and letting Δt → 0, makes the left-hand side approach S (t n ) such
that we obtain a differential equation
S
= −βSI .
(8.10)
The reasoning in going from the difference equation (8.9) to the differential
equation (8.10) follows exactly the steps explained in Sect. 8.2.1.
Before proceeding with how I and R develops in time, let us explain the formula
βΔtSI . We have S susceptibles and I infected people. These can make up SI pairs.
Now, suppose that during a time interval T we measure that m actual pairwise
meetings do occur among n theoretically possible pairings of people from the S
8 Solving Ordinary Differential Equations
Let the mathematical function S(t) count how many individuals, at time t, that
have the possibility to get infected. Here, t may count hours or days, for instance.
These individuals make up a category called susceptibles, labeled as S. Another
category, I, consists of the individuals that are infected. Let I (t) count how many
there are in category I at time t. An individual having recovered from the disease is
assumed to gain immunity. There is also a small possibility that an infected will die.
In either case, the individual is moved from the I category to a category we call the
removed category, labeled with R. We let R(t) count the number of individuals in
the R category at time t. Those who enter the R category, cannot leave this category.
To summarize, the spreading of this disease is essentially the dynamics of moving
individuals from the S to the I and then to the R category:
We can use mathematics to more precisely describe the exchange between the
categories. The fundamental idea is to describe the changes that take place during a
small time interval, denoted by Δt.
Our disease model is often referred to as a compartment model, where quantities
are shuffled between compartments (here a synonym for categories) according to
some rules. The rules express changes in a small time interval Δt, and from these
changes we can let Δt go to zero and obtain derivatives. The resulting equations then
go from difference equations (with finite Δt) to differential equations (Δt → 0).
We introduce a uniform mesh in time, t n = nΔt, n = 0, . . . , N t , and seek S
at the mesh points. The numerical approximation to S at time t n is denoted by S n .
Similarly, we seek the unknown values of I (t) and R(t) at the mesh points and
introduce a similar notation I n and R n for the approximations to the exact values
I (t n ) and R(t n ).
In the time interval Δt we know that some people will be infected, so S will
decrease. We shall soon argue by mathematics that there will be βΔtSI new
infected individuals in this time interval, where β is a parameter reflecting how easy
people get infected during a time interval of unit length. If the loss in S is βΔtSI ,
we have that the change in S is
S
n+1
− S
n
= −βΔtS
n I
n .
(8.9)
Dividing by Δt and letting Δt → 0, makes the left-hand side approach S (t n ) such
that we obtain a differential equation
S
= −βSI .
(8.10)
The reasoning in going from the difference equation (8.9) to the differential
equation (8.10) follows exactly the steps explained in Sect. 8.2.1.
Before proceeding with how I and R develops in time, let us explain the formula
βΔtSI . We have S susceptibles and I infected people. These can make up SI pairs.
Now, suppose that during a time interval T we measure that m actual pairwise
meetings do occur among n theoretically possible pairings of people from the S