4/9/2019

I met with Prof Kruse this past Friday, and we talked about how the SIR model works and the derivation of the differential equations. The following are questions I have answered from Duke University's SIR epidemic model example guide.

Our next steps are working more on practicing the model, and fitting a mathematically defined function to fit a real data set.

 

Under the assumptions we have made, how do you think  s(t)  should vary with time? How should  r(t)  vary with time? How should  i(t)  vary with time?

Before the spread of an infection, the amount susceptible to the infection within a population will naturally be close enough to the population, that the difference is negligible. So we can use the constant N to equal our population. If we are modelling as a fraction, we may divide throughout the equation by our population constant, N.

The interaction between the infected and susceptible population will cause a loss in susceptibles, and a gain in infected, with respect to time. This logistic increase will inflect, then begin to level out as it approaches equilibrium, whatever it may be (likely 0, but it differs.)

The interaction between the infected and the recovered will be negligible. However, the infected will naturally recover based on a constant equal to the life of the disease, so the recovered will logistically increase over time before leveling out.

The infected will be modeled by the difference between the susceptible population and recovered population. As you will see in the next question, the summation must equal 1 at a given time.

Explain why, at each time  t,  s(t) + i(t) + r(t) = 1.

each function of time is a fraction of the whole population. Assuming we are in a closed system (no fatalities, immigration, individuals permanently leaving, etc.), then we can reasonably infer that the summation of the fractions will equal the whole.

 

Algebraically, we can multiply the equation throughout by N to regain our original equation 

S(t) + I(t) + R(t) = N

The Susceptible Equation. Explain carefully how each component of the differential equation

(1)

follows from the text preceding this step. In particular,

• Why is the factor of  I(t)  present?

we are supposing that the constant "b" is the number of contacts an individual has per day that is sufficient to spreading the infection. If we are assuming the mixing of the population is homogeneous, then the number of contacts mixed in with the susceptible population (as a fraction) is represented as s(t). Thus, the average number of newly infected generated by an infected individual is bs(t). However, at a given time, there will be a multiple of infected individuals, which can be represented by bs(t)I(t).

• Where did the negative sign come from?

We begin with a susceptible population. Therefore, it is reasonable to say that the rate of change of susceptible individuals will DECREASE over time. This is due to the interaction in the model between the infected and susceptible population.

Now explain how this equation leads to the following differential equation for  s(t).

b = number of contacts, per individual, representative of infection spread s(t) = number of contacts mixed with susceptible population at a given time i(t) = fraction of infected at a given time negative (-) = susceptibility is a rate of change that decreases as time approaches infinity The Recovered Equation. Explain how the corresponding differential equation for  r(t), (3) follows from one of the assumptions preceding Step 4. given that the constant "k" is a fixed fraction of the population that will recover (based on the recovery rate of the infection,) and i(t) is the fraction of infected at a given time, then we can assume that ki(t) will provide an accurate approximation of the population that has recovered at that given time. This is because of the negligible interaction between the infected and recovered population, that is recovered can not get infected again in this model. The Infected Equation. Explain why (4) The summation of these rates of changes will always be zero. The population remains constant, thus there is no rate of change of the whole.   We can use calculus to take the derivative with respect to time of both sides. We are left with the summation of the rates of change, and 0 on the other side. What assumption about the model does this reflect? Now explain carefully how each component of the equation (5) follows from what you have done thus far. In particular, Why are there two terms? the rate of change of the fraction of infected is represented by the susceptible plus the recovered.    We can use algebra to subtract ds/dt and dr/dt from the left side of the infected equation, and we are left with  di/dt = - (ds/dt + dr/dt) Why is it reasonable that the rate of flow from the infected population to the recovered population should depend only on  i(t) ? the infected fraction of the population, as well as the contact they make with the susceptible fraction of the population, is directly proportional to the rate of flow from the infected population to the recovered population. Essentially, the more that is infected, the more that can recover.    as stated above, the interaction between the infected and recovered population is negligible. Thus, the flow of individuals from infected to recovered solely depend on a k constant that represents the recovery time of the infection. Where did the minus sign come from? Using algebraic manipulation, a negative can be pulled out to give the di/dt equation given above.   di/dt = - (ds/dt + dr/dt)   di/dt = -ds/dt - dr/dt di/dt = bs(t)i(t) - ki(t)  ,   where there is a double negative in front of the susceptible equation What do you think about the relatively low level of infection at the peak of the epidemic? One could see it as wider spread of the disease. A high, sharp peak will have a sharp decline. That may also mean a very quick recovery rate. Can you see how a low peak level of infection can nevertheless lead to more than half the population getting sick? Explain. The summation will nevertheless be 1. Thus, if the peak of the infection function is at 0.2, r(t) could be 0.5 and s(t) at 0.3.   Half the population could have gotten sick by the time of a small peak.