EPID 634_Lab 3_Answer Key

EPID 634 Instructor: Joe Eisenberg Winter Semester, 2023 Lab #3 Population dynamics of directly transmitted diseases (Solutions) I. The SIS Model (6 points total for section I) dS dt =− SI + I +− S dI dt = SI − I − I Tasks: 1. What is d S /dt + d I /dt equal to? Why? ( 1 point ) dS/dt + dI/dt = 0. The population is assumed to be constant in this model. 2. Simulate the model for each of the following parameter sets Set the initial conditions for S and I to 0.9 and 0.1 respectively. Remember to set the simulation time sufficiently long so that the solution reaches steady-state. 2.1 Keep track of the associated steady-state values for the state variables. ( 1 point for answers in columns 5 and 6 ) The following table contains the steady-state solutions ( S ∞ and I ∞ ) for the different parameter sets. b g M S ∞ I ∞ S 90% T 90% 0.9 0.1 0.1 0.22 0.78 0.29 6 0.9 0.2 0.1 0.33 0.67 0.39 7 0.9 0.2 0.2 0.44 0.56 0.49 8 0.1 0.1 0.1 1 0 0.99 22 9.0 1.0 0.1 0.12 0.88 0.2 .6 2.0 1.0 0.1 0.55 0.45 0.59 4 1.0 1.0 0.1 1 0 0.99 17 1 Lab3sol_634_2020.docx 2/20/2023

EPID 634 Instructor: Joe Eisenberg Winter Semester, 2023 The last two columns are answers to Task 3 below. T(90%) is the time it takes for S to reach 90% of its final value and S(90%) is the value of S at T(90%). 2.2 Using the above equations for the SIS model solve for the steady state (or equilibrium points) ( 1 point ) There are two possible steady states: the endemic case in which S ∞ = γ + μ β and I ∞ = β − γ + μ β ; and the case in which the disease dies out (S = 1 and I = 0). 2.3 Verify your S ∞ and I ∞ values are correct. ( 1 point ) The fourth and seventh simulations produced solutions at the later steady-state, whereas the rest produced endemic solutions. 2.4 What is the parametric constraint for obtaining an endemic condition (i.e. I ∞ > 0)? ( 1 point ) There are two possible steady state values depending on the value of β γ + μ 2.5 Show that this is consistent with the above table. How does this relate to the reproductive number? ( 1 point ) This parametric relationship is called the reproduction ratio. The criteria for an endemic condition is β γ + μ > 1 3. How do the parameters, b, g, and m, affect the time it takes to reach equilibrium (i.e., which parameters increase/decrease the rate of change in prevalence)? (1 point) To answer this question, add an additional column to the above table to record the time to 90% of steady-state value, based on your simulations. See above table (columns 6 and 7). The parameter β increases the rate of obtaining the steady-state value, whereas both parameters γ and μ decrease the rate. 2 Lab3sol_634_2020.docx 2/20/2023

EPID 634 Instructor: Joe Eisenberg Winter Semester, 2023 2.2 Why do you see this relationship in an SIS model? See the description of the incidence and contact pattern data described in Hethcote (1984). ( 1 point ) The transmission rate peaks at t = 8.75 whereas the number of cases peaks at t = 9. This corresponds to a delay of 0.25 years or 3 months. Therefore, if the transmission rate peaks in July we would expect a peak in the number of cases in October. III. The SIR Model, without vital dynamics (the epidemic model) (4 points) dS dt =− SI dI dt = SI − δI dR dt = δI Tasks: 1. Explore the relationship between the parameter values (b and g), the fraction of initial susceptibles (S o ), the existence of an epidemic, the time to peak fraction of cases (T Imax ), the maximum fraction of cases (I max ), and the fraction of susceptibles that remain after the end of the epidemic (S ∞ ). ( 2 points for filling out table ) B g S o Epidemic (Y/N) T I max I max S ∞ 1 2 0.9 N 1 1 0.9 N 1 0.5 0.9 Y 4 0.21 0.175 4 Lab3sol_634_2020.docx 2/20/2023

EPID 634 Instructor: Joe Eisenberg Winter Semester, 2023 1 0.1 0.9 Y 5 0.675 0 0.2 0.1 0.9 Y 18 0.21 0.175 1 0.5 0.5 N 2 0.5 0.5 Y 0.62 0.55 .01 3 0.5 0.5 Y 0.56 0.65 0 5 0.5 0.5 Y 1 0.73 0 2. Explain how the two parameters and one initial condition affect these four output variables. Use the above table as a guide, though you don’t necessarily need to be limited to these selections. ( 2 points for including 3 of 5 answers ) 1. An epidemic occurs only if S o > g/b. 2. Increasing g decreases the time of infectiousness and therefore decreases the impact of the epidemic. For this reason the epidemic occurs sooner and generates fewer cases 3. Increasing b increases the impact of the epidemic. Therefore it decreases the time to an epidemic and increases the maximum number of cases. 4. The final number of susceptible individuals after an endemic ends increases for higher values of g and lower values of b. 5. Higher number of initial of susceptibles result in a lower number of final susceptibles. The following table summarizes the relationship between the parameters (g and b) and three characteristics of the epidemic (I max , T Imax , and S ∞ ). I max T Imax S ∞ g b So IV. The SIR Model with vital dynamics (the endemic model) (4 points) dS dt = μ − SI − μS dI dt = SI − δI − μI 5 Lab3sol_634_2020.docx 2/20/2023

EPID 634 Instructor: Joe Eisenberg Winter Semester, 2023 year, as illustrated in the last graph. The second time scale is the rate of incoming susceptible individuals, which occurs over the course of 10 - 20 years, as illustrated by the first graph. Once the number of susceptible individuals reaches a threshold level, an epidemic occurs, as indicated by the sharp decrease in the plot of fraction susceptible and the epidemic curves in the plot of fraction infectious. The final endemic level is close to zero, the steady-state values of S and R are both approximately 0.5. 2. Solve the following equations, which come from setting dS/dt = 0 and dI/dt = 0: The solution for S and I are steady-state solution for the number of susceptibles and infected as time increases to infinity. 2.1 Solve for I* and S* (the steady-state solutions). ( 1 point ) The steady-state solution for the endemic case is S ∞ = δ + μ β and I ∞ = μ β ¿ ) . 2.2 Plug in the parameter values used in the above simulation to verify that the simulated and calculated steady-state values are the same. ( 1 point ) You should get a value that is close to 0.5 for the S*. It is not exactly 0.5 because the simulation hasn’t reach steady state.. 7 Lab3sol_634_2020.docx 2/20/2023