In this activity, we are trying to simulate an outbreak of a new infectious disease that our population of people has not been exposed to before. This means that we are starting with a single case, everyone else is susceptible to the disease, and no one is yet immune or recovered. This can for example reflect a situation where an infected person introduces a new disease into a geographically isolated population, like on an island, or even when an infections "spill over" from other animals into a human population. In terms of the initial conditions for our model, we can define:  and .

Remember, the differential equations for the simple SIR model look like this:

Initial number of people in each compartment
and

Parameters:

Remember that in a simple homogenous SIR model, is directly related to the proportion of the population that is susceptible:

The effective reproduction number is highest when everyone is susceptible: at the beginning,  = . At this point in our example, every infected cases causes an average of 4 secondary infections. Over the course of the epidemic,  declines in proportion to susceptibility.

The peak of the epidemic happens when  goes down to 1 (in the example here, after 50 days). As decreases further below 1, the epidemic prevalence goes into decline. This is exactly what you would expect, given your understanding of the meaning of  once the epidemic reaches the point where every infected case cannot cause at least one more infected case (that is, when  < 1), the epidemic cannot sustain itself and comes to an end.

Prevalence is simply the value of at a given point in time. Now we can see that the incidence is the number of new cases arriving in the  compartment in a given interval of time. The way we represent this mathematically is by taking the integral of new cases over a given duration.

For example, if we wanted to calculate the incidence from day 7 to 14,

Assign all the subsequent rows

Tabulate through the DocumentTools

Solve the following differential equation

with initial condition

Solve the following differential equation

with initial condition

Solve the following differential equation

with initial condition

Solve the following differential equation

with initial condition

Solve the following differential equation

with initial condition and

Solve the following differential equation

with initial condition

LINEAR AND NON-LINEAR OSCILLATORS

Athanasios Paraskevopoulos

Imagine a block of mass m attached to a spring. The spring is fixed at one end, and the block is free to move horizontally on a frictionless surface.
 When the block is displaced from its equilibrium position and released, it oscillates back and forth

According to Hooke's Law, the restoring force exerted by the spring is proportional to the displacement
 from the equilibrium position and acts in the opposite direction:

Using Newton's second law,
, where   is the acceleration, we get:

Since acceleration  is the second derivative of displacement with respect to time, the equation becomes:

Rearranging terms, we get:


Letting where ω is the angular frequency, the equation simplifies to

In a linear simple harmonic oscillator (SHO), the Hamiltonian energy represents the total energy of the system, which is a combination of its kinetic and potential energy.

Kinetic energy depends on the mass of the object and its velocity. In physics, it's expressed as:

where:

Potential energy in a simple harmonic oscillator arises from the restoring force of the spring. The potential energy is given by:

This means that the further the mass is displaced from equilibrium, the greater the potential energy stored in the system.

Suppose you have a spring with a spring constant and a mass . If the mass is displaced by  from its equilibrium position and released, it will undergo SHM

Damped Oscillators:

Consider a mass-spring-damper system, where a block of mass  is attached to a spring (with spring constant ) and a damper (with damping coefficient ).
The damper provides resistance to the motion, simulating friction or air resistance.

Damped Oscillator Equation:

This is in the form of a homogeneous second order differential equation and has a solution of the form

Substituting this form gives an auxiliary equation for

The roots of the quadratic auxiliary equation are

λ=

The three resulting cases for the damped oscillator are

The period  of a damped oscillator is primarily determined by the damped angular frequency

Suppose a mass  is attached to a spring with  and a damper with a damping coefficient

Mass-Spring System with a Non-Linear Spring

The period of oscillation will not be constant and depends on the applitude of oscillations and sign of

First case

In this non-linear mass-spring-damper system (without damping), the Hamiltonian  is essentially the total energy of the system, as there is no energy loss.

Second Case

Introducing Chaos:

If damping  is added to system, we call the EOM  Duffing Equation,  without a driving force

If damping (and external force) is added to system, we call the EOM (Forced) Duffing Equation

This exploration demonstrates the evolution from simple linear oscillations to chaotic Duffing dynamics. By adjusting parameters, oscillators transition through phases: linear, damped, nonlinear, forced, and ultimately chaotic. The Duffing oscillator serves as a powerful model to illustrate the sensitivity and complexity of nonlinear systems, showing how slight changes can push oscillators into new regimes of motion.