Swine Flu

I've been trying to understand the current outbreak of Swine Flu.
In humans, common symptoms of the disease are chills, fever, pharyngitis, muscle pains, severe headache, coughing and weakness.
In more serious cases, influenza causes pneumonia, which can be fatal, particularly for the young and the elderly.
Influenza is transmitted from infected mammals through the air by coughs or sneezes, creating aerosols containing the virus, and from infected birds through their droppings.
It can also be transmitted by saliva, nasal secretions, faeces and blood.
Infections also occur through contact with these body fluids or with contaminated surfaces.

In April 2009, a dangerous variant of the H1N1 flu strain, combining human, pig, and bird flu elements, dubbed the "swine flu," emerged.
Symptoms show up in one to three days after being infected with the virus ... and infected people are ill for two to four days.
What follows may (or may not!) be correct.

Here's a picture of an influenza virus:

Its surface is studded with hemagglutinen.
That's a protein whose name derives from its ability to cause blood to "agglutinate" (or clot).

There are many different types of influenza viruses. ... indeed, there are currently 16 known types: H1 to H16.
H1, H2 and H3 are human influenza viruses. Others don't usually attack humans -- but they can mutate.
In fact, a vaccine formulated for one year may be ineffective in the following year, since the influenza virus evolves rapidly
For example, H5, although normally a bird virus, may mutate and infect human cells. This strain of avian virus is called H5N1.
Although H5 doesn't have an appropriate hemagglutinen to attack human cells, it can attack pigs.
Pigs (with genetic makeup very similar to humans) are susceptible to viruses that attack birds as well as viruses that attack humans.
If a pig should harbour both types of viruses, they may exchange genetic information generating a new virus capable of attacking humans.

By virtue of the molecular structure of the hemagglutinen coating, the virus is able to attach to the surface structure of a normal cell.
In fact, hemagglutinen seeks out particular sugar molecules on the cell surface.
The surface membrane of the normal cell then envelops the virus, attempting to digest it.
Alas, the inner contents of the hemagglutinen protein contain hydrophobic peptides.
When they are exposed, the "normal" cell is compromised.
The inner contents of the virus (viral RNA) is then free to pour out into the "normal" cell's cytoplasm ... and the cell is no longer "normal".

In addition to hemagglutinen, there's also neuraminidase protein on the surface of the virus.
A particular combination of Hemagglutinen and Neuraminidase identifies the type of virus: H3N2 or H1N3 ... or H1N1.

A "typical" influenza virus

The hemagglutinen coating

Vaccines are killed (or deactivated) viruses, sometimes grown (in an outdated ritual!) in fertilized chicken eggs.
When these weakened /killed viruses are injected into a human, their own immune system generates disease-fighting protein antibodies (without the symptoms of the actual infection).
These antibodies then circulate in the blood stream, ready to attack subsequent occurrences of the virus.
This is why a child who has had chickenpox will only rarely develop the disease again: antibodies are present.

The best fit exponential approximation A exp(b x)
selecting A and b so as to minimize:
Σ yj ( log yj - log(A) - b xj)2
See mathworld

The best fit logistic approximation to data,
selecting K, A and b to get a minimum least squares fit to:
y = K / (1 + A e-b x)

Data obtained from
World Health Organization

The logistic approximation was obtained by brute force, using this Excel spreadsheet:

Click on the spreadsheet to download.

See also Logistic Growth.htm

In the spreadsheet described above, I've added another criterion for "best parameters".
The errors are weighted so that recent case numbers are weighted more heavily than ancient ones.

The Logistic model is pretty simple. There's also the SIR Model.