Chapter 8 Modelling Epidemics [First Draft CBPrice November 20th 2011] 8.1 Introduction Disease results in more deaths than road-traffic accidents, wars and famine combined. The 14th Century Black Death resulted in the death of about one third of the population in Europe at that time (around 85 million). Our current understanding on the causes of disease focus on various micro-organisms, such as viruses, bacteria, parasites and fungi. In this chapter we describe one classical model for the population dynamics of these “disease agents”. Such a model may serve to help us understand how the disease spreads and also to engineer strategies for disease control, such as vaccination. What is an “Epidemic”? The traditional definition of an epidemic is that the occurrence of a particular disease is “in excess of normal expectancy”, for infectious diseases an epidemic is characterised by a sharp increase in the number of cases over time. A good example is influenza where person to person contact results in a rapid increase in the incidence of this disease. In this chapter we shall take a dynamical systems approach, where we consider a population which is exposed to infection and we shall study under what conditions the infected individuals changes with time. Our definition of an “epidemic” is a situation where the number of infected individuals increases above the initial number of infected individuals. We shall discuss two models. First a model which does not include the contact between individuals in space. This model focusses on the state of the individuals, whether they are in a “susceptible”, “infected” or “recovered” (removed) state. The second model explicitly looks at the actual interactions between individuals, and considers how the disease is spread through inter-individual contact in space. 8.2 The SIR Epidemic Model (Deterministic, Non-Spatial) This model does not focus on actual interactions between people in space, but looks at a whole population of people which may be in one of three states: Within this total population there are a number of distinct classes: (i) The susceptibles S who can catch the disease, (ii) the infected I who have caught the disease, and (iii) the removed R who may have had the disease and have become immune, or who have the disease and have been isolated (until recovered). The movement of each individual through this model is indicated in Figure 1. This model involves setting up a set of differential equations to determine how the sub-populations S,I,R vary with time, hence the “SIR Epidemic Model”. Let’s think about how to construct the differential equations. The rate at which the disease spreads depends on the contacts between the infected and the susceptible individuals. We can write this mathematically as , where r is a parameter. To understand this product, consider the situations shown in Figure 2. In Fig2(a) we contrast two cases where we have just one infected (red) but on the left a few susceptibles and on the right many susceptibles. Clearly the rate of infection will be greater on the right since we have more susceptibles, so that’s why S appears in the expression . Now, looking at Fig.2(b) we contrast two cases where we have the same number of susceptibles. On the left we have a few infected and on the right we have many. Clearly the rate of infection will be greater on the right since there is more chance of the infection spreading. That’s why I appears in the expression . Therefore the change in time of the infected has a term and its expression starts off as The removal of the infected (due to isolation or treatment) depends on the number of infected. This is given by the term –aI where a is a parameter which is the “removal rate” of the infected. Adding this into the above expression for the rate of change of infected gives us Now we have to formulate two additional differential equations. First for the rate of change of the susceptibles. Since susceptibles become infected at a rate , then the susceptibles must decrease with this rate, in other words Finally the increase of the removed is the same as the rate they are removed from the infected which is –aI. So we have In conclusion, the set of three coupled ordinary differential equations which describes the SIR model is We can apply the Euler algorithm to these equations to give which we can re-write as which in turn could yield the computer code suscep += -r*suscept*infect*deltaT; infect += (r*suscept*infect – a*infect)*deltaT; recovd += a*infect*deltaT; 8.3 Analytical solutions to the SIR Epidemic Model It is easy to obtain some useful mathematical analytical solutions to the above system of three ODEs. Let’s first consider the conditions needed for an epidemic to be produced. Remember our definition of an epidemic is that the number of infected should increase (at least initially) with time. The rate of change of the infected is of course , so if this increases then we need . So from the differential equation for I we conclude that or in other words In this case, I initially increases and we have an epidemic. This inequality makes sense. If a (the rate of recovery) is large then we need more susceptibles for the infection to spread. If r (the rate of infection) is large, then we need fewer suscpetibles for the infection to spread. This inequality is of high significance in understanding how infections spread and how to prevent this. A further analytical solution to the system of three ODEs can be found by dividing the second ODE by the first. This eliminates time and allows us to obtain a relationship between infected I and susceptibles S at any moment in time. We do not explain the maths here (phew!) but merely state the result of our analysis: Here I and S refer to the infected and susceptibles at any time, and and refer to the initial numbers of infected and susceptibles at the start of the simulation. 8.4 Phase Plane Description of the SIR Epidemic Model Typically, a study of the SIR system of equations will include graphs of how S, I, and R change with time. But we can make other plots, for example for the SIR system a useful graph is obtained by plotting I against S. This is shown in Figure 3 for a two experiments. Here the results of two simulation runs are shown. Simulation (1) (red) shows an epidemic. The initial values of S and I are shown on the right. As the system evolves, it passes along the curve in the direction indicated by the arrows. Clearly the number of infected initially rises until it reaches a maximum then decays to a smaller value where I is zero and the epidemic has ended. Simulation (2)(green) shows the evolution of S and I which corresponds to a non-epidemic. As you follow the curve (arrows) the number of infected always decreases. Remember that the initial value of S must be large enough for an epidemic to ensue. [Next revision: Show the N = I + S line and a family of orbits] 8.5 The SIR Epidemic Model (Spatial, Stochastic) The model presented in 8.2 above, and developed in 8.3 and 8.4 did not take into account the spatial distribution of individuals involved in an epidemic. In other words there was no explicit modelling of the transmission of the disease between individuals. In this section we address this issue and present a simple, though effective model of the transmission of the disease between individuals. A typical model of disease transmission involves the direct contact between individuals, and this occurs when the individuals are present in the same place. The model presented here divides continuous space (a room, a town, a country) into a grid of cells. Each cell may contain a number of individuals, though to maintain simplicity, we assume here that each cell contains one individual. The difference between continuous space and discrete space is shown in Figure 4. The model assumes that at the beginning of the simulation, each individual in each cell is found in one of three states, susceptible, infected or removed. A typical starting configuration is shown in Figure 5, where green represents susceptible and red represents infected. The simulation advances in time by computing the chance that an individual in a particular state may transit to a new state. For example, an individual who is in the state susceptible may transit to the state infected if surrounded by two or more individuals who are in the state infected. On the other hand an individual who is in the state susceptible and who is not surrounded by any infected will surely remain in the state susceptible. To develop this model consider a single susceptible individual. Let us denote the chance of infection by having contact with one infected neighbour as . So the chance of NOT getting infected with this single contact is (1 - . So if there are N infected neighbours the chance of NOT getting infected with all of these is (1 - . (N times) ie So the chance of getting infected is 1 – the chance of not getting infected which is simply So if the individual is in a susceptible state then we move to the infected state with this chance. This is done in the following lines of code. infectionProb = 1 - (1 - rho)**nrInfected; randNum = RandRange(0.0,1.0); if(infectionProb > randNum) nextState = 'infected'; else nextState = 'susceptible'; The first line calculates the infection probability (chance). The second line returns a random number between 0.0 and 1.0. Every random number has an equal chance of being returned. The if statement decides whether or not to infect. Let’s see how this works. Say that randNum is 0.5 and infectionProb is 0.6. Then we choose to infect. But if randNum is 0.7 and infectionProb is 0.6 then we choose not to infect. Since randNum when called say 100 times will return a value between 0.0 and 1.0, then we expect 60 of these values to lie below 0.6 and 40 of these values above. So the number of times we will infect is 60 out of 100, which of course is the probability 0.6. The removal of infected occurs with a fixed probability . This is done with the following lines of code removedProb = alpha; randNum = RandRange(0.0,1.0); if(rmovedProb > randNum) nextState = 'removed'; The same argument presented above applies to this code. 8.6 Correspondence between the Deterministic and Stochastic Models.
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