The Curve: The basics of a Pandemic

Covid-19 has made many phrases like ‘flattening the curve’ become part of News broadcasts, political announcements, and even everyday conversations. But, what is the curve? Why is there a curve to flatten? This post explores, the ‘curve’ and finds how herd immunity is the sole factor that produces the ‘bell curve’ resembling shape.

• Creating Your Own Curve
• Immunity: Why Not Simply Exponential Growth?
• Spread Factor (R number or R0) & Effective Spread Factor.
• Herd Immunity
• Individual Immunity & Infections
• Managed Curves vs Unmanaged Curves
  • Adjusting Effective Spread Rate: ‘Flattening the curve‘.
  • Lockdowns
  • A Lockdown Peak
  • An Early Lockdown Peak
• Lifecycle: A More Accurate Curve/Model
• Edit, More basics: What is a case? Infection? Immunity? What do vaccines do?
• Summary: The shape of the curve is a direct result of continually rising herd immunity. The curve is only accurate for an outbreak shaped by herd immunity.

Creating Your Own Curve

You can create the classic curve using a spreadsheet program. Two constants: Population and Spread are required. Two rows: ‘cases’ and ‘cumulative cases’ are required. The cases row will take the initial number of cases, and multiply by spread and by ‘fraction-of-population-already infected’. In other words, formulae after the first cell in the ‘cases’ column is:

= Previous-Cell * $Spread * (1 - Previous-Cell-In-Cumulative-Row/Population)

The cumulative cases divided by the population give the ‘fraction of the population already infected’. Subtracting this from 1 gives the ‘fractions not already infected’

This multiplication by the ‘fraction not already infected’, creates an ‘effective spread ratio’. So each cell is the cases is being multiplied by the effective spread ratio.

The cumulative row will add the previous cumulative and the ‘cases’ number for that column.

= previous-cell + this-cell-from-cases-row

Graph an area graph, and you have your curve.

It is that simple.

The curve produced is an unmanaged curve, or what will happen in the event of an outbreak without any management by government or society.

Mathematically, the basic pattern of the function would be:

y = e^{-(x-e)^2}

But add the population and r₀ requires the a better ability to display formulas than I have currently.

No Immunity: No Curve

The Outbreak Curve

The standard outbreak curve is the result of exponential growth of the number of cases, with a continually decaying exponent.

The nature of exponential graphs is:

• exponential growth when the exponent is greater than 1.0,
• a flatline if the exponents is 1.0
• exponential decay if the exponents is less than 1.0.

This results in the curve starting as exponential growth which moves to linear growth and then flattens as the exponent approaches a value of 1.0, reaches a peak when the exponent reaches 1.0, than then enters exponential decay as then exponent becomes less than 1.0.

Alternatives to the Standard Outbreak Curve.

What happens without immunity?

A simple exponential graph have a formulae:

f(x) = C^x

Where C is a constant number. For the outbreak curve, in place of a constant number ‘C’, the variable ‘effective spread rate’ is used. This effective spread rate continues to decrease throughout the outbreak, as the number of people who, having already been infected, are now immune increases.

Exponential Growth: Alternate Curve 1

The decreasing exponent in the standard curve arises as the level of ‘herd immunity’ rises with every case. This produces the continual reduction in ‘effective spread rate’ which creates the decaying exponent.

‘The curve’ only becomes the correct projection of an outbreak if either, no one, or an insignificant number of people, will be reinfected.

In other words this whole ‘curve’ only applies to diseases where people once infected, then become immune, and then always, or almost always, remain immune throughout the outbreak. So, effectively, no reinfections.

Eventually there are no more infections, even though the ‘cumulative infections’ row normally will not get to reach the value of the total population. Some people will be spared. The outbreak ends not because there are no more people to infect, but because the number of ‘now immune’ people is sufficient to kill the outbreak.

Take out the ” – Previous-Cell-In-Cumulative-Row/Population” and the graph as shown in ‘creating your own curve‘ above, and the graph would become exponential growth. Once the peak is reached, the level would remain at the peak forever. Everyone always infected.

The standard curve shape, results from he assumption each person is only effectively infected once. If people who recover are soon again vulnerable to reinfection, then exponential growth is result. No longer is there any herd immunity, and therefore no curve that results in the end of the outbreak.

Stopping the Curve: Alternate Curve 2

Even in the absence of decaying effective growth rate as a result of ever increasing immunity, a outbreak can be brought to an end by reducing effective growth rate to below 1.0 by other means, typically by lockdowns and social distancing. The downward curve has two important characteristics:

• Quite rapid reduction in cases
• A long tail before all cases are eradicated

Not that this no immunity in this model. The natural ‘spread rate’ of outbreak must be above 1.0, or there would have been no initial outbreak. With no immunity, if this natural ‘spread rate’ is restored (such as by ending lockdowns or relaxing distancing measures), the outbreak will simply restart.

Spread Factor (R0 or R number)& Effective Spread Factor.

(see this page for useful background)

Spread Factor (R0 or R Number)

‘Spread factor‘ is the number to feed into models in order to calculate how an outbreak will progress. The spread factor represents the number of individuals each infected person, in their normal environment, will in turn infect during their illness. For example, if each person infected will in turn infect 3 other people, the spread factor would be 3.

Spread factor is specific to a virus and an environment. Modelling a virus such as Covid-19 will usually settle on single averaged out spread factor for the society wide typical environment, but with most viruses, the closer the people are to each other, and the higher number the contacts, the higher the spread. The spread factor will actually vary from society to society and for different areas within that society. Normal everyday life an inner city area, could have a higher spread factor for a given virus, than will be typical for the same virus in a rural setting.

It seems quite clear that Covid-19 has a different spread pattern on a cruise ship, and within communities such as aged care facilities, prisons etc.. Further, people specific events such as at a wedding event may have a higher spread factor than normal everyday life.

Further, as will be seen in ‘end games’, and ‘flattening the curve’, society may bring in rules such as ‘social distancing’, reducing contact between people from the society normal, to produce an engineered reduction in spread factor.

People isolated in hospital should in theory have a spread factor of zero, provided all protective measures are effective.

Effective Spread Factor.

‘Effective spread factor‘ is the product of ‘spread factor‘ multiplied by ‘fraction of the population not yet infected‘. For example, with a ‘spread factor or 3’, a person will, statistically, infect 3 other people. However if 1/3 of the population has already been infected and is now immune, the ‘fraction not yet infected‘ would be the remaining 2/3, so the effective spread factor would be 2 x 2/3 = 2.

For any given environment, ‘spread factor’ is constant, but effective spread factor will continue decrease throughout the outbreak, as more people become immune as a consequence of having already been infected.

Herd Immunity & Herd Immunity Level

Herd Immunity Level.

One use of the term ‘herd immunity’ is ‘herd immunity level’ as the term in the phrase ‘reach herd immunity‘. The herd immunity level is simply the percentage of the population that is immune to the disease. Some people may be naturally immune. Some may be immune as a result of being vaccinated. But with a new disease, the main way to become immune is to have recovered from the disease. Once infected, it becomes a race between the immune system to get ready and build an immune response, and the disease spreading in the body. The disease is defeated if the immune response happens in time, eventually killing the virus within that host, and leaving the host with an immune system able to defeat the virus should it return.

If the herd immunity level is at 25%, then 1/4 of all people are immune. This herd immunity level reduces the effective spread ratio. A virus with a spread factor of 2 within a community with herd immunity of As more cases occur, and more people have already overcome the infection, the herd immunity level will continue to rise.

Herd Immunity (Immunity to give Spread of 1.0 or less).

The term ‘herd immunity’ without the extra word ‘level’, is used to describe the level herd immunity (percentage of people immune) required to prevent an outbreak from growing. Consider a virus where each generation infects two new people.

Note: Herd immunity does not immediately end an ongoing outbreak. Herd immunity is the point where outbreak stops growing. The flat part at the top of the curve. At the point of reach herd immunity an existing outbreak continues at the same level, which will make even more people immune. With the above example, the extra people mean now more than half are immune, so statistically case 1,000 cases will now lead to less than 1,000 cases, and numbers will decrease. As people are people are being infected, raising immunity levels even further, resulting in faster and faster decline in case numbers. During the entire right hand half of an infection curve, herd immunity is present and the cause of the drop in case numbers.

In mathematical terms for the model, the peak of the curve occurs because there is a value when ever shrinking ‘ratio not yet infected‘ multiplied by ‘spread factor‘ had dropped to being be equal to one. That is, effective spread factor = 1. When the spread factor is 1, then the next ‘number of cases’ will be equal to the previous ‘number of cases’, so the graph will at this point be a horizontal line. Of course there are still cases, so the percentage of immune people is still rising and as a consequence the effective spread factor will drop below 1.0 almost immediately, and the curve will now trend downward.

If the spread factor is three (3), then it would require two thirds (2/3) of the population to be have been infected, creating a ‘ratio not yet infected’ of 1/3, to reach the point of ‘herd immunity’. If the spread ratio is 2, then only half need be infected to reach herd immunity . The higher the spread rate of a virus, the higher the of population that must be immune to reach ‘erd immunity’. With measles, the most contagious virus so far, 90% of people would normally need to be infected to reach this peak or ‘herd immunity’.

Again, herd immunity is reached when sufficient are immune that the effective spread factor is reduced to 1 or less.

Note that as ‘herd immunity is already reached at the top of the curve, during the whole second half of the curve the ‘immunity percentage’ continues to increase from total number of cases, is continuing to increase beyond having achieved herd immunity.

On the positive side, this means the result of an unmanaged outbreak will normally be even more immunity than required for ‘herd immunity’: a safety buffer. However, this also means that more people that necessary to reach ‘herd immunity’ have been exposed to the infection.

Herd Immunity and New outbreaks.

Since herd immunity is when in each infected person only infects at most one new person. With herd immunity, there will be no new outbreaks, as the number of cases will be unable to grow. An outbreak requires each infected person to infect more than one person, resulting in a growth of cases.

Individual Infections & Immunity.

To Continue to Exist, Viruses Must Continue to Infect New Hosts

Viruses cannot reproduce without a host. Viral infections require a previously uninfected hosts to not just spread, but survive. Viral infections infect cells within the host. Infect too many cells in the host and the host dies, leaving the virus without a host. Trigger an immune system response in the host, and the host immune system will in time destroy all copies of the virus, and the virus dies. Even if the virus has a way to remain within a host, for the life of that host, eventually the host will die. For every case, infecting a single host has a limited future. The virus must keep finding new hosts or die out.

Infections: Infecting New Hosts.

To infect a new host, the virus has to reach cells inside the new host. For a human host, entry is normally through the eyes, nose or mouth. Some viruses remain airborne and can be breathed in. Others can be in droplets that can be in the air for a short time in the breath, sneezes or coughs from an infected person. Viruses can also survive for periods of time on surfaces. If the virus is on the surface of food, the food may later be eaten. The other path is a person touches the surface, picking up the virus, which then becomes transferred to eyes, nose or mouth when a person touches their face, or touches food they are about to eat.

Individual Immunity

Inside a persons body, an infection follows a path very much like the outbreak curve. In place of decreasing infections rates, the decline the virus infecting cells comes from antibodies creates by the immune system of the host. It takes about two weeks for an antibody response. Provided the host survives sufficiently healthy to produce an antibody response, that response will then start destroying the virus, and even after the virus is eliminated, the presence of the antibody in the system of the host will prevent reinfection.

How long the antibodies remain active in the host and remain effective varies from virus to virus. For the common cold and influenze, the antibodies only remain effective for around 6 months. Once the antibody response starts but the individual immunity that brings the curve back down is a very different mechanism. A virus reproduces in each infected cell, so the more infected cells the more reproduction, producing exponential growth. There are two separate immune responses, the first createing quickly reproduces, infecting every increase number of cells, until the person’s immune system is fully activated and then starts reducing the number of infected cells. From the peak of the infection within the individual, the person’s immune system is able to continually reduce the number of infected cells. This ability to reduce the number of infected cells will also quickly end any new infections. With a new infection, the immune system is ready immediately, so the infected cell count will never reach a high enough level for serious illness.

This ability of the immune system to respond to a virus without delay is know as ‘immunity’.

ould does not develop the ability to defeat the virus, that person would never get better. In the persons’ own body, the virus would continue to multiply exponentially, and all cells in the host body would become infected. There can be a virus where ‘hosts’ can survive with every cell infected, but as infected cells normally do not function fully, once too many cells are infected, the host would die. A virus which kills the host too quickly will not succeed in infecting new hosts, and will die with the original host. Since the virus relies on the host surviving for the virus to stay alive, the most successful viruses trigger an immune response in the host that allows the host to stay alive, and gives the virus a far longer time to infect new hosts.

All of this means, the most effective way for a virus to spread, is if ‘host’ immune systems can reduce the virus within the host. In short, for a virus to create an outbreak, the normal characteristics will mean anyone who recovers, will have a period of immunity.

Vaccines.

It may seem that one way to end an outbreak is to infect every one at once. If everyone had the virus at once, there would be no future for the infection, as almost everyone would be immune, and an outbreak would end with one cycle of infection (normally within a month). However so many people would require medical treatment if everyone had a regular infection at the same time that medical systems would be so overloaded that almost no one would be able to receive any medical treatment. that infecting everyone at once would mean medical assistance would not be feasible.

That is, unless you could infect everyone at once with a modified form of the virus that has been ‘crippled’ in order to prevent serious illness, but still produce an immune response. This is commonly known as a vaccine. Most vaccines , but not all, are a ‘crippled’ version of the original virus. It is also technically possible to have a vaccine which provides protection other than via antibodies, or triggers the production of appropriate antibodies without being a derivative of the original virus.

Multiple Infections.

Note also, that even before recovery and having immunity, a person who is already infected will significantly be vulnerable to being reinfected. This makes those currently infected able to be treated as immune. The number of virus cells to start an infection is very small, and with exponential growth in number of infected cells, after just a few days there so many copies of the virus within an infected person as to completely outnumber any extra virus cells from a new additional infection. There are two caveats:

• new additional infections in the first day, or perhaps even two days, will have a similar effect to a very large initial infection, resulting in a doubling or more of the number of virus infected cells for every day of the outbreak, robbing the immune system of important days to respond
• a new infection could be in some way different to the original infection, being a genetic variation that requires a second immune response, it should normally have no significant impact on the level of infection.

Other than those two caveats, being again infected whilst already suffering from Covid-19 poses no additional risk of any significance.

Managed Curves vs Unmanaged curves.

Adjusting Spread Rate: Flattening the curve.

Effective spread rate is adjusted by adding and removing a variety of measures to reduce the spread rate. Adding such measures is often called ‘Flattening the curve’, but that term is used in many different ways as described in a separate post. This section is about how to reduce spread rate.

While ‘herd immunity’ (above) changes how many people may be infected by reducing the number of contacts who will become infected, adjustments to the spread rate itself work by changing the number contacts. Reduce the personal contacts can have a similar effect to contacts being immune. In place of some contacts being immune, people can be removed from being contacted. ‘Social distancing’ is designed to ensure people remain sufficient distant from each other that direct infection is not possible. Since keeping sufficient distance from all people is at best impractical, a unit of people who share an environment can be considered a bubble. If all the people in each bubble remain sufficiently distant from all other people, there will be no spread between bubbles. People should not be part of multiple bubbles, which requires either isolation at work or at home! If such measures are successful, the other problem to avoid is infection from surfaces. All these steps change the environment and as a result change the number of contacts people will have in a given environment. Reduce contacts and you will reduce infections. Effectively, change the environment to reduce the virus ‘spread rate‘.

Lockdowns

At set of measures to adjust spread rates (as described above) are often referred to as a ‘lockdown’. Typically there will be rules such as:

• distance people who are not in the same household group or ‘bubble’ to keep at set distance apart (e.g. 1.5 meters or 5 feet)
• all but essential businesses to be closed
• work from home where possible
• school from home where possible
• schools closed

Lockdown level 1 will normally combine the least restrictive set of measures, level 2 a more restrictive set of measures etc.

Grouping measures into a ‘lockdown’ or lockdown level, provides convenient communication of what set of measures will apply during which periods.

A Lockdown Peak

The measures within a ‘lockdown’ will, if successful, reduce spread rate. A reduced spread rate means not only a flattened curve, but also a lower herd immunity level required to cause the curve to peak. This means that a peak of cases observed under lockdown only means there is herd immunity under that lockdown. Consider the graph to the left. St Louis saw a ‘lockdown peak’ in early November and then lifted their lockdown. True herd immunity was only reached in December following the surge in cases that followed lifting the lockdown.

The lesson is, do not assume a lockdown peak is the end of the outbreak. Cases can only be assumed to continue to fall following a peak that occurs outside of lockdown.

An Early Lockdown Peak

Sometimes a lockdown is so effective that cases will start within a single infection cycle. Under lockdown, the spread rate is reduced to less than 1.0 and cases will drop. Such a lockdown will produce an ‘early lockdown peak’. That is a peak driven by the lockdown alone, and not driven by rising herd immunity. The may be very low herd immunity at this time. The graph to the left was created from a model where the ‘early lockdown peak’ triggers lifting the lockdown completely, and then the curve resumes in full force.

Here is a graph from New York data modelling. The scenario presented is quite similar to the model from the curve page. The same risk applies: the lower the level of herd immunity at the lockdown peak, the worse the outcome if lockdowns are lifted completely.

Consider the situation a hospital. Everyone wears masks and other hazmat level equipment to hopefully ensure that despite medical staff being in close proximity to an infected patient, there is no spread of infections. Patients are kept isolated from each other, or at least they should be if a hospital is not going to create more outbreaks. If all goes well, an infected person infects no others. If a lockdown has the same effectiveness, cases will end immediately, but there is still a completely vulnerable society to protect.

Lifecycle: ‘Growth Rates’ For A Refined Curve.

Generation Factor vs Growth Rates.

The outbreak is a population of cases. Consider the population of a human society, where all parents have 3 children, and average age of a parent when a child is born is age 25 years. Exponential growth just like with the virus. Every step transforms 2 parents to 3 children. Start with 100 people, and a graph would have values of 100, 150, 225, 337.5, 506.25, 759.375 etc, at 25 year intervals.

But what about between the 25 years? Aside from the problem of there being 337.5 people at one point in time, is the delay of 25 years before there is another value. Generations will not really distinct in this way, unless the couples from the 337.5 people (168.75 couples?) all having triplets at the same point in time, the numbers will not all jump by this fixed multiple every 25 years. There will be a continuous growth, with population growth every year, such that if on pattern, any two years 25 years apart will be in the 1.5x ration. This would mean the step between each year would be in the ration of 1.5 to the power of 1/25th. So annual exponential growth at the rate of approximately 1.10635.

Calculating Daily Growth Rate.

Knowing the rate of growth, requires knowing not just the spread factor, but also the length of time for a spread cycle to complete. So for example, for a virus with a spread rate of 3 and a cycle average of 10 days, the daily growth rate would be 3^(1/10) = 1.116. Note that for single day growth rates, the numbers will typically be very close to 1.0. Just above 1.0 for an outbreak, and just below 1.0 when case levels are dropping. Exponential growth or decay will result in steep curves even for daily growth rates between 1.2 and 0.8. Using these daily growth rates will produce data where the intervals are 1 day, in place of ‘1 infection cycle’. However, just as children are not born to 1 year olds born last year, infections are not all a result of people infected yesterday. Average of cases for the past n days (where n are the days of the infection cycle generating the new infections), and applying growth for mean of the number of days sampled, will produce a more accurate model.

Continual Model Improvement.

Moving from simple model as in ‘creating your own curve‘ above, to the use of daily growth rates, improves the model. Calculating based on average case number from across infectious days is a further improvement. But improvements do no stop there.

Each country has its own underlying ‘spread rate’, and typically has taken steps such as shutdowns and social distancing to introduce changes to the underlying spread rate. A model can introduce different values for spread rate under different sets of lockdown rules.

Of course, accurately predicting spread rates for any given set of lockdown rules is almost impossible without data from what has happened previously.

Using Available Data

Data on number of cases confirmed per day has been collected in many countries. This data has been used to calculate daily growth rates for different countries. This enables a model to set spread rates that result in matching daily growth rates. Such reverse calculation of spread rates, using infected population numbers to extract the data from effective spread rates, and provided the ability to predict how the outbreak will behave as measures are introduced or removed at different levels of herd immunity.

The ‘creating a curve’ describes ‘unmanaged curves’, of what would result without any intervention. ‘more accurate curve’ only Without intervention the ‘effective spread rate’ continually declines during the outbreak as the percentage of people immune increases.

In a managed curve, rather than allow ‘effective spread rate’ to be set by rising immunity alone, takes action to change spread rates, such as social distancing, which introduces

Summary: The Curve Shape is a result of herd immunity

‘The curve’ predicts the outbreak of a typical virus. Typical viruses have only a limited time in each host, with hosts either dying or overcoming the virus and being left with a period of immunity.

An outbreak starts with a virus with when then virus is in an environment where it has a spread factor greater than 1.0, resulting in exponential growth of the number of cases. The level of immunity within the population increases with the number of infections, until the ‘effective spread rate’ drops to 1, which will cause the outbreak to peak and then begin to decline.

Herd immunity will eventually result in the curve declining and reaching zero at end of the outbreak. It is the continual increase of herd immunity reducing the effective spread rate from the a level greater than 1.0, to below 1.0 that results in the ‘curve’.