Basic Mathematical Modeling of Disease - Essay Example

Basic Mathematical Modeling of Disease: In many occasions, similar principle acts as the basis for the formation of most mathematical model. To start with, population should be divided into different classes, dependent on their relation to disease. To make it simple there are three division groups that classify as recovered, infectious and susceptible. This is known to modern mathematics as ‘SIR’ model. Some individuals are considered to be susceptible. Although they don’t have the disease, there is still chance that they might get infected, for example, if they were in contact with infected person. So if the transition happens they move into infectious group. Consequently infectious group is the one that spreads it back to susceptible for certain period of time, which is known as ‘infectious period’ after that period they are a considered to be immune for life if recovered. The following picture depicts a basic SIR model used analyzing an infectious disease Picture 1: Basic SIR model. Susceptible people once infected turn into infectious group and then if recovered from the disease into recovered group. On the picture β represents the contact rate (transmission rate) and . The Differential Equation used in SIR Model. Using notation from our SIR model there are some equations that can be formed in order to find answer to my IA question. When modelling SIR models it is very important to identify the independent and dependant variables. As in majority of the mathematical models time ‘t’ is going to be independent variable and it is going to be measured in days. Equation with the basis notation: N = S+I+R S= S (t) number of susceptible individuals I = I (t) number of infected individuals R = R (t) number of recovered individuals N - represents the total population of a chosen country. T is time measured in days. Important to note: The differentiation stands for the rate of change of susceptible with relation to time (next day increase in . The differentiation stands for the rate of change of the infected people with relation to time. (next day increase in ). The differentiation stands for the rate of change of the recovered people with relation to time. (next day increase in ). In mathematics differentiation is defined as a rate of change, in our investigation it is rate of change in three variables with relation to time β -represents the contact rate. The rate at which susceptible meet infected measured as individuals per unit time (day).2 - represents ; is calculated by Three basic differential equations: 1.= ) – 3.= Following equations are made from application of the simple S, I, R model explained before. Now I am going to explain how I derived those equations. Deriving the equations: Equation 1: = More people are getting infected when there is a contact between infected people and susceptible. In our equations represents number of contacts infective person has each day. If we decided that I represents number of all infected people than represents number for all infected contacts per day. But infected people come in contact only with susceptible ones therefore we need to multiply (susceptible fraction of the population) we get: This expression looks like first differential equation, but in our equation of change in susceptible class is negative. It is negative because people from that class are getting removed into the infected class. Next day increase in = In order to represent those equations as a derivation they should be expressed with relation to our dependant and independent variables. In order to represent the rate of change as a derivation every dependant variable such as S, I, R should be represented with relation to time. When talking about infectious disease such function of time as ‘next day’ can be represented as: S(t+t) – S(t). Applying those changes to our equations we get: S [t+t] = S [t] –( ) t Following mathematics rules those equations can be written as: = S [t] –() Dividing by N on both sides to get rid of the denominator: = –() Using mathematics rules we can cancel N’s in denominators: == Setting our limit to 0, because we are dealing with time: == Finally we get what we started with: = Equation 2: ) – Following the same principle as we used explaining ‘Equation 1’ next day increase in I can be represented by finding all the cases that can happen tomorrow i.e. and subtracting tomorrow increase in individuals recovered bI from it. Next day increase in - I Following same procedures as we did explaining Equation 1 we should express following equation with relation to our independent variable t(time). Once again next day can be represented as: I [t+t] - I [t]. Applying the change we get: I [t+t] = I [t] + ( – I[t]) t Following mathematics rules those equations can be written as: = I [t] + (– I[t]) Dividing by N on both sides to get rid of the denominator: = () – Using mathematics rules we can cancel N’s in denominators: == ) – Setting our limit to 0, because we are dealing with time: == Finally we get what we started with: ) – Equation 3: = In the following equation there is no need to show any working out. The equation represents change in the Recovered class and it is simply calculated by g (recovery rate constant) by number of the Infected people. It represents the people that were recovered from the disease and now healthy (immune for life). Working Example using Ebola spread in Sierra Leone. Lets use data from Sierra Leone to plot those equations and discuss. Applying data from “WHO situation Report 12 November” we get this data. Assuming that everyone in Sierra Leone is a suspect 620000. Susceptible Infected Recovered T=0 6,192,123 4,523 3,354 N – population of Sierra Leone is 6,200,000 as estimated of 2014. - Of Ebola is approximately = β - Of Ebola is 0.128 SIR equations for Ebola. Using eqn 1 for “susceptible”:= = Using eqn 2 for “infected”: ) – – Using eqn 3 for “recovered”: = = Calculating basic reproductive ratio,  R0 is the sum of secondary infections resulting from a single infection entering a population totally made of susceptible population. From this definition, we can derive the equation, in which case is contact rate (β) divided by recovery rate constant (g).   Hence R0 = 0.128/0.005 = 25.6 Using a computer application we can plot those equations. (http://www.public.asu.edu/~hnesse/classes/sir.html Graph of Population vs Time for the three group levels Discussion The differentiated values obtained from the above three equations, were entered in a computer simulation program that then gave the above graph. This were used as the variables for the graph, essentially representing the number of people in the three groups. As shown in the calculations, R0 is greater than 1 meaning that the disease can enter a fully susceptible population just as proved by the graph. As shown in the graph, the disease has entered a population and therefore mostly occupied by “susceptible” individuals as shown by the yellow curve at 6200000. Consequently, because of the susceptibility, the disease spreads very fast through the population leading to a larger percentage of the population infected within a very short time. This situation is well illustrated by the graph, which shows the curve for “infected” reaching its optimum within the first 100 days. The infected cases abruptly rise due to the infection. The graph also shows that as the result of increasing “infected” cases the “susceptible” population decreases to the point that the former becomes more than the latter. After the 100 days, the graph starts to decline to the zero level. In which case, the graph depicts a clear relationship between the number of susceptible individuals and the infected cases. However, as time elapses a good part of the population enters the “recovered” as shown by the red curve. At the same time the “susceptible” cases becomes fewer leading to reduction in the spread of the disease. In which case, the graph above signifies the reduction of spread by the decline in the number of infected cases as the recovering cases increases. Eventually, the number of cases reduces hence validating the fact that not everyone will contract the infection before the disease ends. Read More