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Math 442 Assignment 2 - Spring 2015 Due Friday, February 6th Directions: Use LATEX to generate the following output. You will print your solutions and hand these in on Friday, February 6th in class. 1. George Box said “Essentially, all models are wrong, but some are useful”. 2. L’Hôpital’s Rule states that if f and g are differentiable functions such that lim |f (x)| = x→a lim |g(x)| where both limits equal 0 or ∞, then x→a f (x) f 0 (x) = lim 0 . x→a g(x) x→a g (x) lim 3. MATLAB can be used to verify that Z ∞ 2 e−x dx = √ (1) π. (2) (x − a)n , (3) −∞ 4. The Taylor series for f (x) centered at x = a is f (x) = ∞ X f (n) (a) n=0 n! where f (n) (a) is the nth derivative of f at x = a. 5. The model consists of the following system of difference equations dS dt dI dt dR dt dP dt = Λ − dS − β1 SI − β2 SP + νR (4a) = β1 SI + β2 SP − (d + µ + γ)I (4b) = γI − (d + ν)R P = αI + rP 1 − − (ξ + δ)P K (4c) (4d) with non-negative initial conditions S(0), I(0), R(0), P (0) ≥ 0. 6. The feasible region is Γ= (S, I, R, P ) ∈ R4+ |S Λ +I +R≤ , P ≤C d (5) where C is the unique positive root of the quadratic αΛ/d − (ξ + δ − r)P − (r/K)P 2 . 1 7. Linearization about the disease-free equilibrium (DFE) gives dI~ ~ = J I~ = (F − V )I, dt where J is the Jacobian matrix evaluated at the DFE. The matrices F and V are β1 S̄ β2 S̄ d+µ+γ 0 F = V = . α r 0 ξ+δ (Hint: The spacing between the matrices F and V is made with the command \qquad.) 8. Each of the measurements x1 < x2 < · · · < xr occurs p1 , p2 , . . . , pr times. The mean value and standard deviation are then v u r r X u1 X 1 t x= p i xi , s= pi (xi − x)2 n i=1 n i=1 where n = p1 + p2 + · · · + pr . 9. The following figure was generated by MATLAB. 1 0.8 0.6 y 0.4 0.2 0 -0.2 -0.4 -10 -5 0 5 10 x Figure 1: Graph of f (x) = sin x on [−10, 10]. x (Hint: Use MATLAB to generate this figure and save it as an Encapsulated Post Script file. These types of figures look best in LATEX prepared documents.) 2 10. The following table lists the parameter values used for numerical simulations. Table 1: Parameter values for our model. Description Recruitment rate Natural death rate Direct transmission Indirect transmission Disease-related death rate Recovery rate Immune period Shedding rate Pathogen replication rate Pathogen death rate Decontamination rate Carrying capacity 3 Parameter Value Λ 0.1 d 0.001 β1 0.0006 β2 1.3 × 10−10 µ 0.001 γ 0.1 1/ν 100 α 5 × 107 r 0.2 ξ 0.051 δ 1.2 K 1.95 × 1014