# Modelling the course of viral illness

The aim of this essay is to model the course of a viral illness and its treatment.

When viral particles of a certain virus enter the human body, they replicate rapidly. In about four hours the number of viral particles has doubled. The immune system does not respond until there are about 1 million viral particles in the body.

The first response of the immune system is fever and the rise in temperature lowers the rate at which the viral particles replicate to 160% every four hours. The immune system tough can only eliminate about 50 000 viral particles per hour. If the number of viral particles, however, reaches, the person dies.

For a person infected with 10 000 viral particles we could determine how long it will take for the immune system response to begin. As we have an ordered set of numbers and a common ratio we could consider this to be a geometric sequence. The form of a geometric sequence is So , where is the initial term of the sequence and is the common ratio.

In this case  where is the number of viral particles approximately,  is the initial phase of the illness respectively 10 000 viral particles, is the growth of rate of 200% every four hours and,  is the time in hours where fever begins.

To determine how long it will take for the body’s immune system response to begin we can substitute with the values into the formula:

Hence it will take around hours for the immune system response to begin.

In order to determine how long it will be before the patient dies if the infection is left untreated, we have to consider that after the fever begins at , will be 160% every ...