# The population at any time t. When these

The Ebola virus is one of the latest viruses to address global fears of a pandemic.  Over the course of time, we have seen many pandemics, including the Black Death in Europe during the Middle Ages, and the Spanish Flu in the early 20th century. Recently, HIV/AIDS has been responsible for several million deaths.  How contagious is Ebola, and how can mathematics be used to predict and model its spread?When modeling infectious diseases, the basic model that is used is the Susceptible, Infectious and Recovered (SIR) model. In this model, the population is divided into three groups: the population susceptible to infection, the infected population, and the population that has recovered from the disease. Due to the lack of a cure for Ebola, and therefore its high mortality rates, the recovered population in our model will instead represent the population that has died from the disease. Disease spread models are affected by various factors, including the per-capita death rate of an individual that is infected, as well as the per-capita contact rate of the disease.The contagiousness or infectiousness of a disease is shown by the basic reproductive number, denoted by R0. The basic reproductive number of the Ebola virus is between 1.5 and 2.5. The table to the left compares the R0 values of a few other common diseases.Individuals are not permanently fixed in the Susceptible and Infected categories; they can move from Susceptible to Infected, and from Infected to Dead, as everyone in the susceptible population has a chance of becoming infected. As previously stated, the model is affected by the per-capita death rate of an infected individual, ?, and the per-capita contact rate, ?, which can vary from and exponential decay function to a constant function depending on the size of the infected population at any time t. When these parameters have been found, the system modeling the data is solved using Mathematica, or any program that can use the fourth-order Runge-Kutta algorithm.We will use S(t) to represent the susceptible population, I(t) to represent the infected population, and R(t) to represent the dead population at time t. This SIR model assumes that during the outbreak, there is a constant population, meaning that there will not be deaths due to factors other than Ebola, and that births will be ignored. All together, the susceptible, the infected, and the dead make up the total population (denoted by N), so N = S(t) + I(t) + R(t) at every time t. To model the spread of Ebola, we will use the following equations.dSdt=-?SI/NdIdt=?SI/N-?IdRdt=?IThe number of infected individuals due to direct contact is t= ?SIN, where ? is the contact rate of the disease. The death rate is ?I, where ? is the per-capita death rate. The incubation period, which is the time from infection to the showing of symptoms, is 2-21 days, and the infectious period, the time it takes for an infected person to die, is 4-10 days. It will take a person between 6 and 31 days after initial infection for that person to die. This means that ?, our per-capita death rate, is a range between 1/6 and 1/31. Now we will bring our basic reproductive number, R0, into play. The reproductive number can be denoted as R0 =??. We will use 2 as our R0 value, since it is in the middle of our range for R0 values of the Ebola virus. After substituting 2 as our R0 value, our resulting equation is 2=??. Now we can substitute our range of values for ? into the equation and obtain 231