Gonorrhea ranks high among reportable communicable diseases in the United States. Public health officials estimate that more than 2,500,000 Americans contract the disease every year. This disease is spread by sexual contact and if untreated can result in blindness, sterility, arthritis, heart failure, and possibly death. Gonorrhea has a very short incubation time (3-7 days) and does not confer immunity to those individuals who have recovered from the disease. It often causes itching and burning for males, particularly during urination, while it is often asymptomatic in females. Thus, males tend to seek treatment more often than females.
We will assume a sexually active heterosexual population with c1 females and c2 males. If the number of infective females is given by x and the number of infective males is given by y, then a mathematical model that describes this disease is given by the following system of differential equations:
y' = - a2y + b2 (c2 - y)x,
where x' = dx/dt and y' = dy/dt. The cure rates for infective females and males are proportional to the infective populations with proportionality constants a1 and a2, respectively. New infective females are added to the population at a rate proportional to the number of infective males and susceptible females, b1 (c1 - x)y. (A similar term adds infectives to the male population.)
There is not an exact solution to this differential equation, so you need to use an Euler's method to approximate the solution for this system of differential equations. The Euler's formula for this system is given by
yn+1 = yn + h(- a2yn + b2 (c2 - yn)xn).
Notice that this is simply a discrete dynamical system, so you can simulate this system very much like other discrete dynamical systems.
a. Assume a sexually active female population of c1 = 1000 and a sexually active male population of c2 = 1000. Begin the initial infected populations with x0 = 20 females and y0 = 20 males. Since males seek treatment more frequently for this disease, we assume the treatment rates of a1 = 0.33 for females and a2 = 0.53 for males. Take the rate of transmission (or infective rate) for each population to be similar, say b1 = 0.00052 and b2 = 0.00047. Let h = 1 and simulate the disease for 100 months (or 100 iterations). Make a table of the number of infected females and males with entries at every 10th month. Also, graph the populations of infected females and males on a single graph (being sure to label which population is which). Find all equilibria for this model with these parameters. Discuss what is happening to this disease from your graph, including a reason why one population has a higher number of infected individuals.
b. Because AIDS is such a problem in today's society, the sexually active population is taking more precautions. This may result in a lowering of the infective rates. Let b1 = 0.00042 and b2 = 0.00037. Let x0 and y0 be the last population of infectives from your simulation in Part a. Now simulate the model for 50 months. Graph your solution and discuss how this affects the model. Find all equilibria for this model with these parameters.
c. Alternately, people fearing the more dangerous disease AIDS may seek medical treatment earlier. This would affect the treatment rate. Let a1 = 0.43 for females and a2 = 0.63. (Use the same b1 and b2 from Part a.) Again let x0 and y0 be the last population of infectives from your simulation in Part a. and simulate the model for 50 months. Graph your solution and discuss how this affects the model. Find all equilibria for this model with these parameters.
d. Discuss the biological significance of your simulations and what it predicts about the disease. What are the strengths of this model for discussing public policy toward sexually transmitted diseases? What are the weaknesses that you see in the model?