# Modeling change

can you help me to solve this question?

A Killer Virus:

You have volunteered for the Peace Corps and have been sent to Rwanda to help in humanitarian aid. You meet with the World Health Organization (WHO) and find out about a new killer virus, Hanta. If just one copy of the virus enters the human body, it can start reproducing very rapidly. In fact, the virus doubles its numbers in 1 hour. The human immune system can be quite effective, but this virus hides in normal cells. As a result, the human immune response does not begin until the virus has 1 million copies floating within the body. One of the first actions of the immune system is to raise the body temperature, which in turn lowers the virus replication rate to 150% per hour. The fever and then flu-like symptoms are usually the first indication of the illness. Some people with the virus assume that they have only a flu or a bad cold. This assumption leads to deadly consequences because the immune response alone is not enough to combat this deadly virus. At maximum reaction, the immune systems alone can kill only 200,000 copies of the virus per hour.

Model this initial phase of the illness (before antibiotics) for a volunteer infected with 1 copy of the virus.

How long will it take for the immune response to begin?

You need to decide between a discrete-time (one data point per hour) or a continuous-time model.

In the discrete model, at each step first update the number of virus copies according to the growth model for the last hour, then subtract anything killed by the immune system.

In the continuous model, describe the rate of change as a differential equation where the birth rate of the virus is proportional to the current number of virus copies and the killing rate is initially zero and later a constant of 200,000 per hour. Solve the resulting differential equation separately for the two killing regimes.

Two pedantic remarks:

1. Aren't antibiotics ineffective against viruses?
2. You need the two killing regimes for a complete model, but the question about how long it takes for the immune response to begin can be solved using only the first regime.
• thanks. what is the model for a discrete-time (one data point per hour) before antibiotics effect and after that?(the immune system and the antibiotics together kill 500,000,000 copies of the virus per hour) Mar 31, 2016 at 12:29
• That is additional data, you should add that to the question instead of hiding it in a comment. The model without killing is a pure exponential. The model with killing is a shifted exponential, i.e., there exists a positive constant $K$ such that $N-K$ is exponential (with $N$ the time-dependent number of virus copies). I have not worked out the value of $K$ for the discrete model, but in the continuous model it is the hourly killing rate ($500\times10^6$) divided by the natural logarithm of the growth rate, in this case $\ln(1.5)$ Mar 31, 2016 at 12:34
• if change model is pure exponential (before antibiotic) the term "the human immune response does not begin until the virus has 1 million copies floating within the body" didn't consider. what can I do? Mar 31, 2016 at 14:05