Have portions of part A and B complete, however I'm not sure on how to graph the phase potraits along with the inital conditions. Ultimately needing help with parts a through c thanks! ### Understanding Epidemics: Susceptible-Infectious ModelIn this section, we explore the dynamics of infectious diseases through a mathematical model. Specifically, we'll investigate a common epidemic model, which considers two vital populations: susceptible and infected individuals.#### Key Assumptions:1. **Immediate Infectivity:** Individuals become immediately infectious after contracting the disease.2. **Average Death Rate:** On average, an infected individual has a lifespan of 10 days post-infection.3. **Single Initial Infection:** The epidemic starts with just one infected person.4. **Birth Rate of Susceptibles:** Susceptible individuals reproduce at a rate of 0.0003 per person per year, and their offspring are also susceptible.#### Differential Equations ModelLet \( S(t) \) represent the number of susceptible individuals and \( I(t) \) represent the number of infected individuals over time \( t \). The following system of differential equations describes the interaction between these two populations:\[\frac{dS}{dt} = -\alpha SI + 0.0003S \quad \text{(2.2.7)}\]\[\frac{dI}{dt} = \alpha SI - 0.1I \quad \text{(2.2.8)}\]Here, \( \alpha \) is a constant representing the relative infectivity of the disease.#### Case Studies for Different Infectivity Rates**Case (a): High Infectivity (\(\alpha = 0.05\))**- **Task:** Draw the phase portrait for \(\alpha = 0.05\).- **Instructions:** Identify and label all nullclines and equilibrium solutions. Assume initial conditions \( S(0) = 1000 \) and \( I(0) = 1 \).- **Question:** What happens to the solutions as \( t \) approaches infinity?**Case (b): Low Infectivity (\(\alpha = 0.000001\))**- **Task:** Draw the phase portrait for \(\alpha = 0.000001\).- **Instructions:** Identify and label all nullclines and equilibrium solutions. Assume initial conditions \( S(0) = 30000 \) and \( I(0) = 1 \).- **Question:** What happens to the solutions as \( t \) approaches infinity?#### Discussion**Case (c): Comparative Analysis ### Understanding Epidemics in Isolated PopulationsConsider an epidemic that moves through an isolated population. We will make the following assumptions about the epidemic:- **Infection Rate:** Individuals are infected at a rate proportional to the product of the number of infected and susceptible individuals. We assume that the constant of proportionality is \( \alpha \).- **Incubation Period:** The length of the incubation period is negligible, and infectious in-This excerpt provides a foundational understanding of how an epidemic might spread in an isolated population by considering specific assumptions related to the infection rate and incubation period. These assumptions are critical for constructing mathematical models to predict the behavior of such epidemics over time. **Note:** The text seems to be incomplete at the end, and it abruptly terminates after mentioning the incubation period.

A condition brought on by the human immunodeficiency virus (HIV) known as acquired immunodeficiency…

Math

Statistics

(c) What conclusions can you draw about the behavior of the two dif- ferent epidemics?

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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