Question

(5) A new virus has emerged in the Southwest United States for which no one has immunity (so re-infection is possible). The Centers for Disease Control has tasked you with modeling the outbreak assuming a population of healthy people (H) and sick people (S). Some of the sick people (S) will recover while others will perish (D). This can be modeled by the following elementary reactions:

Getting sick from virus exposure (rate constant k1): H -> S

Getting sick from contact with a sick person (rate constant k2): H + S -> 2S

Recovering from sickness (rate constant k3): S -> H

Dying from the infection (rate constant k4): S -> D

(a) Develop an equation that describes the death rate of the population from the virus.

(b) Plot the number/concentration of health people as a function of time.

(c) Determine the concentration of health people for which humanity wil not be able to recover.

Answer 1

The rate of population growth at any given time can be written:

dN/dt = rN

Where:

• r is the rate of natural increase and is usually expressed as a percentage (birth rate - death rate)
• t a stated interval of time, and
• N is the number of individuals in the population at a given instant.

The algebraic solution to this differential equation is

N = N_oe^rt

where:

• N₀ is the starting population
• N is the population after
• a certain time, t , has elapsed,
• r is the rate of natural increase expressed as a percentage (birth rate - death rate) and
• e is the constant 2.71828... (the base of natural logarithms).

Essential to understanding the mathematics of population growth is the concept of doubling time. Doubling time is the time it takes for population to double and it is related to the rate of growth. When the population doubles, N = 2N₀. Thus the equation becomes

ln 2/r = t or

0.69/r = t; where r is the rate and t is the doubling time.