# Epidemiological Model of the Spread of Pertussis

## Introduction

Pertussis (whooping cough) resurges every few years onto the epidemic stage, infecting millions and killing hundreds of thousands globally. Pertussis is currently endemic in the human population, with epidemic phases every four years. One of the shortcomings with current diagnosis of pertussis is the lack of a cheap, fast and reliable method. Our method of detection, where phages are used to indicate the presence of pathogens, fix these shortcomings.

To project the impact of our invention, we created a mathematical model that progresses through a pertussis epidemic and predicts beneficial effects on the health of a population. The model uses a population of 1000 individuals, a small number (n=10) of whom are initially infected with the pertussis, and forecasts the spread of the disease through the rest of the population.

Individuals are categorized into five compartments: Susceptible (S), Exposed (E), Infected (I), Treated (T) and Recovered (R). Each individual can occupy one compartment at a time. An individual with no immunity or exposure would start in the susceptible compartment and upon exposure to infective bacteria would progress to the exposed compartment. After a brief latent period of infection, the individual progresses to the Infected compartment where the individual could either follow the nature course of infection and progress directly to the Recovered compartment, or the individual could be treated (after diagnosis) with anti-bacterial factors that would shorten the infective period as well is decrease the spread of pertussis to other hosts. Finally, the individual’s acquired immunity to pertussis would wane over time and the treated or recovered individuals would return back to the susceptible compartment.

## Compartmental Flow Diagram

The flow diagram shows the movement of individuals through each compartment at a rate indicated on each arrow. The rates are derived through logical intuition consisting of parametric coefficients. The flow rate between S and E is derived from the standard incidence function involving the force of infection, λ = b*(I/N), where (I/N) is the probability of infective exposure and b is the contact rate. The rates between compartments E and I, I and R, and R and S consist of a parametric coefficients a, v and k respectively, which represent the rates of outflow from the corresponding compartments. All compartments experience a general death rate of a human population proportional to µ, while the S compartment experiences the inflow of newborns to the human population also proportional to µ.

Each compartment experiences a net flux of people over time that can be represented mathematically with the following ordinary differential equations (ODEs):

These system of equations are derived from the compartmental flow diagram, where each compartment experiences inflow and outflow of population at the rates shown on the arrows. An inflow would lead a positive term in the ODE for the compartment, while an outflow would be represented by a negative term.

The deterministic compartmental model uses the system of ordinary differential equations in an initial value problem to progress through an epidemic scenario. To predict the outcome of an epidemic, we integrate the system using a stiff ODE solver.

The following graph shows the changing population of each compartment where ɑ = 0.2701 and all other parameters are initialized as indicated in the next section.

## Assumptions

Our model assumes a non-heterogeneous mixture of people; however, extensive population gradients would be expected in a real scenario. To be more representative of this element, spatial abstraction would be required.

Another assumption of the model is that age distribution is generalized as rectangular, which more closely represents the distribution of developed countries than the pyramidal distribution of developing countries. Age is relevant because all individuals in developed countries are vaccinated in their early years, and acquired immunity wanes over an individual’s lifetime. This would be less true in developing countries where vaccination rates are lower.

## Results

Our invention will provide cheap, reliable and quick diagnosis of pertussis which would allow medical facilities to treat infected individuals with antibiotics that shorten the infection period and, in fact, stop the individual’s contagiousness upon delivery of antibiotics. This produces an effect in the model where the infection period is generally decreased than the natural course of infection. Although, the efficacy of an antibiotic treatment must be considered as it can vary depending on how soon the antibiotics are delivered and there is also a general uncertainty with the impact of antibiotics in its reduction of infection period. Thus, it would be appropriate of find the progression of infection over time in relation to a range of infection periods.

The following 3d graph shows the progression of epidemic over time over a range of infection periods. The surface graph shows the infected individuals (color gradient where more red means more infected) over time over different infection periods. Two conclusions can be drawn from the graph:
1. Shorter infection period due to administration of antibiotics decrease overall level of infected individuals.
2. Shorter infection period causes the maximum of the infected individuals to delay to a later time in in an epidemic (the maximum is shifted to a later time).

The graph displays a typical population of the infected compartment in an epidemic, over varying infection period. The current course of epidemic without any intervention, with infection period of 15 days, is shown to have more infected population. Reduction in infected population and delay in peak of epidemic is seen when infection period is reduced with intervention.

## Spatial analysis

Spatial abstraction is achieved by partitioning of space where separate instances of the system of equations are solved for each partition of space. Subsequently, over each discrete time step, the populations are mixed in respect to possible travel between the spaces. This was made using an opensource project, STEM.

A simulation involving an epidemic in Chad is run where a small number of people are infected in Mangalme district (0.005% of the population), which sets off an epidemic throughout the whole country. The desease infects a large amount of people in the starting area and then spills through to neighboring regions until the whole area is covered. After saturation, the infection decreases since most of the population have acquired immunity by exposure to the pathogen.

If the model is run with faster recovery of the infected individuals, a possible result of our invention (since infected individuals would be treated), the impact of disease (amount of total people infected) is reduced and delayed (maximum occurs later). The following video shows the Spread of Pertussis in Chad without any intervention on the left and with intervention possible through our invention on the right.