Problem 16 - Advanced Engineering Mathematics

Question

(a) If we know \(s(t)\) and \(i(t)\) then we can determine \(r(t)\) from \(s+i+r=n\) (b) In this case the system is \\[\begin{array}{l}\frac{d s}{d t}=-0.2 s i \\\\\frac{d i}{d t}=-0.7 i+0.2 s i\end{array}\\] We also note that when \(i(0)=i_{0}, s(0)=10-i_{0}\) since \(r(0)=0\) and \(i(t)+s(t)+r(t)=0\) for all values of \(t .\) Now \(k_{2} / k_{1}=0.7 / 0.2=3.5,\) so we consider initial conditions \(s(0)=2, i(0)=8 ; s(0)=3.4, i(0)=6.6\) \(s(0)=7, i(0)=3 ;\) and \(s(0)=9, i(0)=1\). We see that an initial susceptible population greater than \(k_{2} / k_{1}\) results in an epidemic in the sense that the number of infected persons increases to a maximum before decreasing to \(0 .\) On the other hand, when \(s(0)< k_{2} / k_{1},\) the number of infected persons decreases from the start and there is no epidemic.

Step-by-Step Solution

Answer

"An epidemic occurs if initial susceptibles, \(s(0)\), exceed 3.5."

Step 1: Understand the Equation Relationship

Given the relationship from part (a), use the equation \(s(t) + i(t) + r(t) = n\), to find \(r(t)\) when you know \(s(t)\) and \(i(t)\). Rearranging gives \(r(t) = n - s(t) - i(t)\), where \(n\) is the total population which remains constant.

Step 2: Interpret Differential Equations

For part (b), recognize the system of differential equations. The equations are \(\frac{ds}{dt} = -0.2si\) and \(\frac{di}{dt} = -0.7i + 0.2si\). These describe the change in susceptible \(s\) and infected \(i\) over time \(t\).

Step 3: Apply Initial Conditions

Use the information \(i(0) = i_0\), \(s(0) = 10 - i_0\) and \(r(0) = 0\) to interpret initial conditions: \(s(0) + i(0) + r(0) = 10\). This means that the initial conditions sum to the total population \(n = 10\).

Step 4: Calculate Critical Threshold

Calculate \(k_2/k_1 = 0.7 / 0.2 = 3.5\), which is the threshold for predicting an epidemic. If the initial number of susceptibles \(s(0)\) is greater than this threshold, an epidemic is likely. Otherwise, it's not likely.

Step 5: Analyze Different Initial Conditions

Given the initial conditions:- \(s(0) = 2, i(0) = 8\)- \(s(0) = 3.4, i(0) = 6.6\) - \(s(0) = 7, i(0) = 3\) - \(s(0) = 9, i(0) = 1\)Compare each \(s(0)\) with \(3.5\). If \(s(0) > 3.5\), an epidemic occurs; otherwise, it doesn't.

Key Concepts

Differential Equations

Differential equations are mathematical tools for describing how things change over time. In epidemic modeling, they help predict the spread of a disease by defining relationships between different groups in a population. For example, in a simple model, we might use a differential equation to specify how quickly the number of infected individuals changes based on the current number of infections and interactions with susceptible people.
These equations can express how the number of susceptible (\(s(t)\)), infected (\(i(t)\)), and recovered (\(r(t)\)) individuals change over time (\(t\)). They consider various rates, like the rate of infection and recovery, which are crucial for understanding how an epidemic progresses. Differential equations provide a structure that helps predict future trends by showing how small changes at any point lead to overall outcomes.
By solving these equations, we can model real-world scenarios, anticipate peaks in disease spread, and make informed decisions about public health interventions.

Susceptible-Infected-Recovered (SIR) Model

The SIR model is a foundational concept in epidemic modeling. It divides a population into three groups based on disease status: Susceptible (S), Infected (I), and Recovered (R). This model is a simple yet powerful way to understand the dynamics of infectious diseases.

• Susceptible: Individuals who are healthy but can catch the disease.
• Infected: Individuals who have the disease and can spread it to susceptible persons.
• Recovered: Individuals who had the disease and have gained immunity or are no longer infectious.

The SIR model uses differential equations to describe how these populations change over time. It assumes that recovered individuals gain permanent immunity, meaning they don't return to the susceptible group. The equations involve parameters representing how easily the disease spreads and how quickly people recover, allowing us to simulate different scenarios and outcomes.
It's a critical tool used to predict whether a disease will die out quickly or lead to a substantial outbreak, helping public health officials to plan and respond effectively.

Initial Conditions

Initial conditions in epidemic modeling set the stage for how an analysis or simulation begins. These conditions refer to the initial numbers of susceptible, infected, and recovered individuals at the start of the observation period (\(t = 0\)). They are essential in solving differential equations because they provide the starting point from which changes can be mapped over time.
In the original problem, the initial conditions are given by the values of \(s(0)\), \(i(0)\), and \(r(0)\). Together, they sum up to the total population size (\(n\)), establishing a consistent baseline. The choices of initial conditions directly influence the behavior of an epidemic model because they dictate how quickly an infection can spread or diminish based on how many individuals are initially infected or at risk.
Understanding and accurately setting initial conditions allows for better predictions about the potential outcomes of an epidemic, making it possible to assess various public health intervention strategies before they are implemented.

Critical Threshold

The critical threshold is a key concept in determining the potential growth of an epidemic. It is a calculated benchmark that helps us understand when an outbreak might escalate into a full-blown epidemic. This threshold depends on the balance between the rate at which a disease spreads and the rate at which infected individuals recover.
In the case study with differential equations, the critical threshold is represented by the ratio \(\frac{k_2}{k_1} = 3.5\). This ratio compares the recovery rate to the infection rate. An epidemic is likely to occur if the initial number of susceptible individuals, \(s(0)\), exceeds this ratio.
Conversely, if \(s(0)\) is less than 3.5, the disease will likely decline without causing an epidemic. The critical threshold thus acts as a predictive tool, helping determine the intensity of intervention needed to prevent a major outbreak and guiding vaccination or other preventative measures.
By assessing the initial conditions against this critical value, we gain insights into whether an epidemic will grow or recede, enabling proactive and informed decisions.