Mathematics of influenza epidemics
Introduction
Coverage of the COVID-19 pandemic provides, almost real time, updates on the numbers, statistics, forecasts and heated debates on the best policies to be put in place.
The main difficulty, when it comes to disease propagation, is that the mathematical models — even a simplistic one like the SIS model — are usually described by nonlinear differential equations and depend on parameters that are relatively difficult to measure accurately.
Nonlinear equations are difficult to work with. Most of them do not have an analytical solution and can only be solved with numerical methods; on top of that nonlinear differential equations are tightly associated with chaotic systems.
Chaotic systems
Chaotic systems are such that the final state —unstable, oscillatory or steady — may vary greatly with minute changes in the initial conditions.
In such systems there are usually multiple paths that it can go — and these can be predicted and calculated — but it’s not possible to predict which one the system will “choose”.
The SIS model
The SIS model is used for diseases that do not provide long term immunity, like influenza. Influenza is a highly mutable virus and any immunity acquired by a population to a certain strand of the virus doe snot guarantee immunity against a new strand.
In this case, a susceptible person can become infected while an infected person can become susceptible over time.
The SIS model for disease propagation is derived as follows:
Let S be the subset of the total population susceptible to infection.
Let I be the subset of the total population infected and with severe symptoms (requiring hospitalization).
Let N be the total population.
Let pi be the probability of one person being infected in a certain time interval.
Let ps the probability being becoming susceptible in a certain time interval.
Based on this assumptions we can write:
The probability of one person to be infected at a given time is proportional to the number of possible interactions between a susceptible person and an infected person divided by all possible contacts between 2 people.
The proportionality constant α represents the fraction of the interactions that actually occur within a certain time interval.
On the other hand, the probability of one person infected becoming susceptible at a given time is:
The proportionality constant γ represents the fraction of the infected that become susceptible within a certain time interval.
Therefore, the total possible number of people that can be infected in a certain time interval is:
And the total number of people that can become susceptible in a certain time interval is:
To calculate the total number of people infected over time we have to integrate the difference between the number of people that get infected and number of people that become susceptible.
Consequently, over time, the total number of people infected is:
And from Equation 1:
But, also from Equation 1:
Substituting Equation 8 in Equation 6 we get:
The solution for this nonlinear differential equation is the logistic function.
This differential equation represents the speed of infection. Let’s take a closer look at it.
Equation analysis
If we solve Equation 9 for the case where the time derivative is 0 (system equilibrium), we get to values for the number of people infected I:
The second solution is the one relevant to our analysis.
- If α > γ, then the disease becomes endemic, where the factor (1-γ/α) determines the fraction of the total population that will be permanently infected
- If α < γ, then the speed of infection is negative and the number of infected eventually becomes 0 and the virus disappears.
Conclusion
How can these equations help us?
This is a system that is inherently chaotic; if γ and α are very close to each other the slightest difference can affect the outcome.
The key lies on the values of α and γ. These numbers are determined by several factors: age, gender, health preconditions, treatment availability, vaccine availability, frequency of contact, etc.
So the problem boils down to answering these questions:
- What are the natural values of α and γ for different groups classified by age, gender, health preconditions, etc.? In other words, what are these values if the virus propagates without interference?
- For these numbers, what is the expected number of infected people in each group when the virus becomes endemic?
- What is the percentage of the infected population that can develop serious symptoms and require hospitalization?
- How deadly is the disease in each group?
- What resources are available to change or control the natural values of α and γ?
By answering these questions, policies can then be adopted to control α, γ or both.
