Growth models, otherwise sometimes known also as “population models”, became very prominent in 2020, as the Corona virus epidemic was raging all over the world. Population models want to estimate the future situation, of a population, given as fragments: how big part of the initial population will be in certain state (compartment) at given date.
The models have their roots in classical differential equations. As mathematical modeling of epidemics are essentially about biological growth (and transmission of illness causing pathogen), the natural choice was to take a differential equation as basis.
This blog Post first explains the background in differential equations, and then goes to show how SIR model uses differential equations.
Why are differential equations used?
Differential equations describe the quantity of something, in terms of the “steepness of change”. The equations contain differentials – the slope of another, external thing. Differential equation thus predicts the amount of the target variable using quantities of change in other variables. The other variables, free parameters, guide a differential equation. Solving a differential equation is about iterative looping through the whole timeline, always applying the change to the calculation of target variable.
For example, dropping a rock can be presented as differential equation: the rock is pulled by gravity – gravity is the external thing, that doesn’t change – but gravity changes the rock’s position. The rock goes toward the surface of Earth, until it hits the surface, and stays in place.
Differential and an integral – they make a pair
Differentials are the operations which help you going from target function to change. Example:
dv/dt = d f(t) / dt
Substitute the falling body equation into f(t). Then take the derivative from all terms in the f(t) equation, according to the rules of derivation. You can learn the rules from any algebra tutorial – such as MIT 18.01 Single Variable calculus (video).
Integrals work vice versa: going from change equation -> to value of a target function. A round of integrating practically makes your function more complex, and “peels” derivatives, replacing them by higher power variables.
If you have the rate of change function known, you can find out the values of the target function as a function of time, by taking the integral of the derivative function.
Differentiation and integration are complementary operations.
One minor note: calculating the integral needs also a initial quantity, C, value chosen. The reason why we need to also tell the integral what C was, is that the integral function cannot “know” how much of something had already been accumulated at the time of calculation. C is called the ‘free parameter’ of an integral function.
Drop the ball! Newtonian example
Let’s take a simple example: In physics, we know that gravity is a force. All objects react to gravity. Gravity is a conservative, two-way force: dropping a rock, in actuality the rock also pulls Earth toward itself, but since Earth is way more massive than the rock, it seems as if Earth only pulls the rock towards the ground.
In everyday practical situations, it’s a planet that pulls all objects near it, towards the center of the planet. Space travel is different: after the initial escape from Earth’s gravity, space flights need to take into account other things (gravity assist).
The magnitude of gravity (called ‘g’, a number) doesn’t change much, since it’s the force created by mass of Earth. Near earth g = 9,81 m/s2. Thus the magnitude of gravity in practical sense is quite constant; the objects we are talking about on surface of Earth, move a maximum height of about 30-40 kilometers. During that transition, the value of Earth’s gravity force changes very little. In practise, we use g=9,81 for calculations.
We could use a table to mechanically calculate the very first seconds:
But wait.. the simple gravity model doesn’t look credible?!
This model of Gravity is one of the cornerstones of Newtonian physics. However in real world, this model alone would imply that a rock thrown from a high altitude would have a huge speed at its impact to ground!! That’s not true. By observation, we know that objects reach a limited speed, called ‘terminal velocity‘.
Adjustment of rock-throw equation with two foes
The reason is simple: gravity is not the only force in the rock-throw: since the rock is coming through Earth’s atmosphere, we’re talking about matter – a fluid. So in reality there’s also 2 counteracting forces to gravity:
Equilibrium of forces: acceleration hits zero
The terminal velocity (for falling rock) is reached when the forces (in vertical direction) are equal to each other. The net sum of forces acting on the object is zero => acceleration is zero, and thus velocity (which was a “function of acceleration”) stays the same. All good.
Resistance is created by the object pushing away a mass of air. The object, as it is ‘sweeping’ directly downwards, pushes away to the sides all the air molecules in front of the falling path. This is the same kind of thing that happens also in any other direction:
- a car driving forward on a road pushes away air molecules, as the car goes
- there’s actually no difference between a falling object pushing air and a car pushing away air
- the direction (up-down, or left-right) doesn’t “matter”
- ..but whereas the “dropping object” keeps moving towards center of Earth, a car stops if enough time passes, and the engine doesn’t provide forward force to push the car going
- thus car needs energy from fuel, whereas a dropping object gets “fuel” from the gravity force of Earth
Back to SIR model – with new understanding
So, using the variables and binding the compartments together we can:
- define a time-dependent function for each individual compartment’s value (amount of population likely to be in S, I or R at time t)
- make the compartments work logically in conjunction (“connecting” the compartments)
- keep the logic watertight – by requiring that the sum of population be constant.
Why SIR and SEIR are useful modeling methods for epidemics?
The beauty and usefulness of SIR, SEIR and other variants of the model are:
- by adjusting the free parameters early on in an epidemic, one can estimate the whole epidemic wave’s shape
- different levels of infections can be estimated
- epidemiologist, government and general public can be kept aware of different scenarios
- as there are new facts learned from the field, the model parameters can be adjusted accordingly and new forecasts made instantly (this is why testing for the infection is also important; naturally the proper healthcare of an individual is put first)
- the date when the epidemic starts to level off (slow growth in end stage of an epidemic) can be estimated
SIR model can be used to estimate the peak level of infected individuals. That is the maximum fraction of a population that will eventually get the disease. This peak level may not always be 100%. Not even close. There are a few things that make the peak rate lower than 100% of population:
- virus is too aggressive, self-limiting its spread by causing too quick deaths of the “I” – infected persons
- a vaccination comes out, that once given to, immunizes “S” people so that they skip “I” stage, and go to “R” (being safe from infection)
- virus mutates to a less potent form in the population, affecting the parameters
4 parameters in SIR model
These parameters are explained better in the next part of this blog series, where a computer-based R language simulation is shown.
‘S’ compartment – susceptible (disease-free)
Epidemiologic models are a set of functions, that draw the curves of various “populations” during an epidemic. Typically in times of no epidemia, the population is considered healthy (regarding a particular pathogen). Pathogens that become epidemics, also may not have existed for long time in human-transmissible form. The Corona virus epidemic of 2019 is a prime example: until the nCov virus jumped initially from animals to human, somewhere in late 2019, it wasn’t kind of a threat to humans. It was a threat which apparently wasn’t registered as a dire threat to humankind. Sometimes pathogens can be widely distributed in animal kingdom, but are of no danger to humans – and vice versa.
However, in reality, unless vaccinated, the population indeed is in “S” state – susceptible to getting infected. This is normal business, a rather academic definition indeed. It basically means that normal people, since we have not yet invented a global, universal vaccination against all possibly harmful viruses, is susceptible to new pathogens.
I – Infected
Next compartment, “I” means the pathogen has invaded a person. Viruses are present in sufficient amounts in the body, that they will typically soon start showing signs of illness.
SIR and others are also called compartmentalized models: the populations are the compartments. People still stay the same, essentially, but they get labeled (and counted for in statistical models) differently according to their factual status of having or not having the illness (or, as in case of “R”, having had and gone through into Resolved population).
R – Resolved (cured or died)
The R population is often considered not capable of infecting S population; thus once resolved, person is both immune to reinfection and does not infect others. This, however, in medicine is a case-by-case thing, again dependent on the real biological and systemic properties of the virus.