Mathematical Modeling — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Parts of questions to be attempted together. compartments experience a natural death rate. Further, suppose that susceptible individuals become exposed through contact with infectious individuals, and exposed individuals progress to the infectious class before eventually recovering. (i) Write down the system of first-order differential equations describing the SEIR model under these assumptions. (i) Derive the differential equation governing the total population N(t) and obtain its solution. (iii) Construct a discretized version of the SEIR model using a Nonstandard Finite Difference (NSFD) scheme. 2. (a) Construct a dieting model for body weight dynamics by assuming that the body mass of an individual at time t is denoted by M(t). Let F(t) represent the food or nutritional intake, E(t) denote the physical activity level, and let the metabolic rate be described by a function W(M) that depends only on the body Imass. Using an appropriate balance between intake, expenditure, and metabolism, obtain a differential equation governing the evolution of body mass. Obtain an analytic approximation solution for M(t) by assuming two different ansatz function. (b) Construct the susceptible-infected-recovered (SIR) epidemic model using the SIR methodology. Obtain a relation between the infected population /(t) and the recovered population R(t). Hence determine the threshold value S* and discuss the following cases: (i)S. > S* that this condition leads epidemic. (ii) So < S* : Show that under this condition no epidemic occurs. (c) Examine an epidemic model in which the susceptible, infected, and recovered population are denoted by (t), /(t), and R (t), respectively. The transmission process is assumed to depend on the square-root interaction between susceptible and infected individuals, while recovery occurs at a rate proportional to the square root of the infected population. The governing equations of the ds dR at = -BVSVI đi = BSVI - YVT, = yVI, Here, B and y are positive constants. 3.(a) Consider that the populations (t) and y (t) satisfies the competition system ax dt= 14x-2x2-xy dy = 16y-2y2-xy (i) Show that the critical points are (0,0), (0,8), (7,0) and (4,6). (ii) Find the linearization of the above system at each of the critical points. (il) Show that the behavior of the point (4,6) is a nodal sink. (b) Find the critical points of the system (c) Show that the following system of differential equations is a case of survival of single Species by investigating the behavior of the critical points (0,0), (0,32), (28,0) and (12,8) (b) Consider the following system of differential equation dx at =-3x + x2 - xy dy at = -sy + xy Show that the linearization of the system at (5,2) is u' = 5u - 5v, v' = 2u. Find the eigen values corresponding to the coefficient matrix of the linear system and hence confirm the nature of the critical points.