Most Deaths Are Multiplicative
Disclaimer: I am not an epidemiologist and I consider epidemiology a very complex and specialized field. This brief article is not about the advantages or disadvantages of any policies to combat covid-19 spread and reduce fatalities and their impact on economic growth. Any such analysis is naive and premature at this point and results will probably become clearer after several years.
The covid-19 pandemic has ignited debates between those who propose strict policies to reduce fatalities and those who claim that measures to deal with other more common causes of deaths have not been considered with comparable urgency while the impact on economy is disproportionally severe. The latter group brings up examples of deaths from common diseases, accidents and substance abuse, to name a few causes. However, the former groups insists that these causes of death are not as serious as a virus pandemic because they are not multiplicative.
Specifically, if there is a fatal accident in New York or someone dies in Los Angeles from cancer, there is no effect on people in Miami whereas if someone was affected by the virus and traveled to New York and Miami could affected people there who may subsequently die besides affecting close encounters in his/her city. This sounds like a strong argument in favor of extreme isolation measures to limit rate of infection and deaths. This may have to be done irrespective of the debate anyway; debates often irrelevant when lives are at stake.
However, it is not correct to claim that deaths from pandemic are multiplicative and deaths from diseases or accidents are not. The common sense of most people who are not mathematically inclined is correct and those who insist deaths from pandemics are more severe than deaths from other causes are possibly wrong. The truth can be found in mathematics but before that below are the leading causes of death from CDC, the Centers for Disease Control and Prevention. [1]
The above 10 leading causes resulted in more than 2 million deaths in 2017. The argument of some people is that even 200,000 deaths from a pandemic does not justify putting the economy on hold and causing unemployment to rise to 15% in a few weeks while many corporations are driven to bankruptcy. The most frequent counter-argument made is that, as mentioned already, the 10 leading causes of death are not multiplicative, whereas virus deaths are. Is this correct?
According to mathematics, this is not correct but this has nothing to do with whether strict measures must be taken because this goes beyond simplistic arguments and debates. In fact, all deaths are multiplicative in the longer-term because they affect the birth and death rates although at specific moments, or instantaneously, deaths are additive or what is called in population dynamics, reinjection (positive or negative.)
Below is a discrete stochastic equation that describes the change of the number of individuals n(t) within a population during a time interval Δt [2]
n(t+Δt) = n(t)[1+α(t)] +f(t) (1)
For those in finance, you will notice immediately how the above equation relates to multiplicative equity growth. If f(t) = 0, then this is related, for example, to how the stock price changes from one day to the other when a(t) is the daily return.
In population dynamics, a(t) and f(t) are random variables with some distribution. More importantly, a(t) is the difference between the birth and the death rate, and f(t) is the “reinjection rate”. Reinjection rate is a random variable that depends on many factors, immigration is one of them.
Therefore, it may be seen that the contribution to the population of a(t) is proportional to the population itself and this is a multiplicative process.
Deaths due to pandemics have both multiplicative and additive affects but so do deaths from common causes, as it is seen from equation (1). This is trivial since all deaths of people of reproductive age affect a(t) whether they occur due to pandemic or common diseases and accidents.
Moreover, deaths from pandemics are additive in short-term, multiplicative only in longer-term. In fact, if a pandemic, as for example covid-19, impacts primarily older people, as the data show for now at least, then the impact on a(t), and as a result on the expectation of n(t) at some future time t+T, may be comparable to impact from deaths by accidents, alcohol, substance abuse and genetic disorders combined among younger people. However, this is just speculation as the dynamics are too complex and detailed simulations must be performed to find out. Equation (1) may appear “simple” but in reality it is very complex.
Specifically, for f(t) other than 0, the probability distribution P(n,t) is a power law of the form
where the exponent γ is determined by another complex equation that is beyond the scope of this article.
Therefore, all deaths are multiplicative in the longer-term because they all affect population dynamics and growth. Commons sense is vindicated that not only deaths from pandemics are multiplicative. Also it is a fact that some deaths are considered “normal” and some others “not normal” although they are all deaths. Pandemics have a very low probability to wipe out large parts of a population as history has shown. Unfortunately, other causes of deaths that can be minimized are not treated with the same vigor, especially those that affect young people who are in reproductive age.
Also see: https://mikeharrisny.medium.com/the-hidden-multiplicative-nature-of-additive-processes-eb610178f2a2
References
[1] Centers for Disease Control and Prevention
[2] Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology, and Genetics, Bernd Blasius, Jurgen Kurths, Lewi Stone, World Scientific, 2007
Also see: Pandemic Trilemma.
