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.
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