In this post we'll dip our toes into the waters of epidemological dynamics models, using R and simecol, as we have done in the previous two posts of this series. These models of epidemics are interesting in that they introduce us to a more general class of models called compartment models, commonly used in the study of biological systems. This "compartment point of view" will prove to be an useful tool in modeling, as we shall see in future posts. As usual, after a brief theoretic and mathematical rundown of the various types of epidemic models, we shall fit one of those models to some data using R's simecol package.
Infectious or Epidemic Processes
When studying the spread of infectious diseases, we must take into account the possible states of a host with respect to the disease:
 Susceptible (S) These are the individuals susceptible to contracting the disease if they're exposed to it.
 Exposed (E) These individuals have contracted the disease but are not yet infectious, thus still not able to pass the disease unto the susceptible (S) group.
 Infectious (I) After having been exposed, these individuals are now abel to pass the disease unto other individuals in the S group. This group should not be confused with the infected group of individuals, which are those that are either exposed (E) or infectious (I).
 Recovered (R) Having gone through the previous phases of the disease, individuals recover because they have developed an immunity to the disease (permanent or temporary). This group is neither susceptible to the disease nor infectious^{1}.
Not all epidemological models will include all four of the above classes or groups, and some models with greater degree of sophistication may include more (Berec, 2010). For example, there may be diseases for which it is useful to consider a group of chronic carriers. Moreover, infected mothers may have been previously infected with a disease and, having developed antibodies, pass these to their newborn infants through the placenta. These newborn infants are temporarily immune to the disease, but later move into the susceptible (S) group after the maternal antibodies disappear. These newborn infants would then constitute a group of passively immune (M) individuals. On the other hand, it is posible that all these groups may not be as homogenous as one may think. Different social, cultural, demographic or geographical factors may affect or be related to the mode of transmission, rate of contact, infectious control, and overall genetic susceptibility and resistance.
Based on the selection or ommision of these classes or compartments, which in itself implies some assumptions about the characteristics of the specific disease we're trying to model, there are several accronyms used to name these models. For example, in an SI model, also known as a simple epidemic model, the host never recovers. An SIS model is one in which the susceptible population becomes infected and then after recovering from the disease and its symptoms, becomes susceptible again. The SIS model implies that the host population never becomes immune to the disease. An SEIR model is one in which there is an incubation period: susceptible individuals first become exposed (but not yet infectious), later enter the infectious group when the disease is incubated, and finally, they enter the R group when they cease to become infectious and develop immunity. An SIR model is basically the same as the SEIR model, but without an incubation period, etc.
Compartment models
We can see from the last paragraph on different epidemic models that these attempt to describe how the individuals in a population leave one group and enter another. This is characteristic of compartment, or "box" models, in which individuals in a population are classified at any given time into different compartments or boxes according to the state in which they find themselves. Compartment models are commonly used to study biological or chemical systems in which the main interest is the transport or change of state of materials or substances over time^{2}.
In the process of dynamic model building, it is often useful to think of the state variables as boxes or compartments that "hold" a certain amount of substance at any given time, even though "substance" or "material" may refer to discrete entities such as individuals. In fact, because system dynamic models are governed by differential equations, modeling discrete entities through quantities that vary continously is inevitable. Notwithstanding, thinking in terms of compartments is useful because it forces us to reflect on the mechanisms that affect how individuals enter or leave a compartment. For example, in the Lanchester model of the Battle of Iwo Jima there were two compartments: American troops and Japanese troops. In that case, individuals would not leave one compartment to enter another, because we were not contemplating soldiers defecting from their original army to fight for the other side. In the epidemic models, however, we do contemplate individuals leaving one compartment and entering another as they progresss through the various stages of the disease. The utility of the "compartment point of view" in modeling lies in the principle that if there are group of entities whose dynamic behavior is different from other groups, even though they are apparently the same type of entity, they should each have their own compartment. We shall explore this idea in more detail when we cover population growth models in a future post in this blog.
SIS models
SIS models of epidemics comprise two compartments: S (susceptible) and I (infected). In what follows we will assume that:
 There is a finite population \(N\) that remains constant at all times.
 The epidemic cycle is sufficiently short to assume that births, deaths, immigration/emmigration or any other event that modifies the population do not occur.
 Individuals transition from being susceptible to the disease, then to being infected, and back again to being susceptible as they recover. The disease is such that no permanent or temporary immunity is developped, and there are no mortalities.
 The number of infections of susceptible individuals that occur is proportional to the number of contacts between infected and susceptible individuals.
 At each time unit, a certain percentage \(\kappa\) of infected individuals recover from the disease and become susceptible again. This amounts to stating that the mean duration of the infected/infectious period is \(1/\kappa\) time units.
This is a very simple model and some of the above assumptions may be unrealistic or inapplicable to all SIS epidemic processes but nevertheless, it is an useful model for illustrating the emmergent behavior of some systems. These assumptions are usually expressed in the literature in differential equation form as follows:
\[\frac{dS}{dt}=\kappa I  \alpha\gamma S\cdot I\]
\[\frac{dI}{dt}=\kappa I + \alpha\gamma S\cdot I\]
As already mentioned, the parameter \(\kappa\) represents the percentage of infected individuals that recover at each time unit and become susceptible again. Parameter \(\alpha\) represents the percentage of contacts that result in an infection and \(\gamma\) is the rate of contact between infected and susceptible people. Having these two last parameters in a model may render the model impossible to fit using simecol, since there are infinetly many combinations of \(\alpha\) and \(\gamma\) that result in the same value for their product \(\alpha\gamma\)^{3}. Besides, in principle one should strive to have as few parameters as possible in a model  all the more so in this case since the two parameters can be considered as one: \(\beta=\alpha\gamma\), which could be taken to represent the percentage of cases from the overall susceptible and infected populations that effectively result in an infection. So the resulting model is:
\[\frac{dS}{dt}=\kappa I  \beta S\cdot I\]
\[\frac{dI}{dt}=\kappa I + \beta S\cdot I\]
At a glance, one thing becomes apparent in this model: the rate of flow into compartment S is the same as the rate of flow out of compartment I, so that at any two given time instants \(S_{t+a}+I_{t+a}=S_t+I_t\). The total number of individuals that are either in compartment S or in compartment I is invariable, in keeping with one of the assumptions laid down for this model.
Let's try to understand the implications of this assumption intuitively. At one instant, susceptible individuals are becoming infected, but at some later instant, these infected individuals are becoming susceptible again. There is a negative feedback mechanism in place here because each compartment relies on a greater quantity of individuals on the other compartment to grow in numbers. In other words, the lack of susceptible individuals ensures that the population of infected individuals will start to decrease, which can only mean that there will be more susceptible individuals in the future, as the overall population remains constant. Indeed, the SIS model represents a system that works its way towards equilibrium.
And precisely what is this equilibrium? Can we expect the disease to eventually extinguish itself or on the contrary, will everyone become infected? Or will the system perhaps work its way towards a certain fixed amount of infected and susceptible individuals? Answering questions of this type is one of the aims or dynamic system modeling. We usually want to determine the emmergent behavior of the system that results from the mathematical properties hidden deep within the differential equations.
In our case of this simple SIS model of epidemics, we don't have to dig too deep. A simple mathematical procedure will sufice to determine this equilibrium state analitically. In a state of equilibrium, the state variables do not change, so any of the above derivatives can be set to 0:
\[\kappa I  \beta S\cdot I=0\qquad\rightarrow\]
\[\kappa (NS)  \beta S(NS)=0\qquad\rightarrow\]
\[\beta S^2  (\kappa+\beta N)S+\kappa N=0\]
The last equation is a simple cuadratic equation whose roots are \(S=\kappa/\beta\) and \(S=N\) these are the equilibrium states for the number of susceptible individuals. If the equilibrium state for the susceptible population S is greater than or equal to N, everyone will eventually become susceptible and no one will be infected \(I=NN=0\): the disease is extinguished. If on the contrary no one recovers from the infection (\(\kappa=0\)), then everyone will eventually be infected and there will be no susceptibles. Otherwise, \(0\leq\kappa/\beta\leq N\) and the number of steadystate susceptibles will be closer to N for large values of \(\kappa\) (high recovery rates) and low values of \(\beta\) (low rate of effective contagious contacts).
The following R/simecol simulation model, run for three different combinations of the \(\kappa\) and \(\beta\) parameters serve to validate our simple mathematical analysis. Each simulation run starts with \(I=5\) infected individuals and \(S=95\) susceptibles. But regardless of the initial values, we can see that the steadystate susceptible population reaches \(\kappa/\beta\) or \(N\), if \(\kappa/\beta\gt N\) in all simulation runs.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 

We run the SIS model script using different values for the parameters \(\beta\) (b in the code) and \(\kappa\) (k in the code), to illustrate the steady state behavior of the system. Notice that the plot method in line 16, when called with an odeModel object, produces an individual plot for each state variable.
SIS model simulation run with k=1, b=0.003
SIS model simulation run with k=1, b=0.0125
SIS model simulation run with k=1, b=0.05
SIR models
The SIR model of disease was first proposed in 1927 by Kermack and McKendrick, hence the alternative denomination of KermackMcKendrick epidemic model. With this model, researchers sought to answer questions as to why infectious diseases suddenly errupt and expire without leaving everyone infected. In this regard, the KermackMcKendric model succeeded and was considered one of the early triumphs of mathematical epidemology^{4}.
I should mention in passing that the basic SIR or KermackMcKendric model asumes a disease transmission rate given by the \(\beta SI\) term, a form called massaction incidence or density dependent transmission. When the disease transmission rate is \(\beta SI/N\) (\(N\) being the overall population), the disease transmission mechanism implicit in the model is known as standard incidence or frequency dependent transmission.
The use of one or another model of disease transmission between infected hosts and susceptibles and the circumstances under which either one is more appropriate for modeling the onslaught of a disease is a somewhat controversial issue. It has been generally suggested, however, that the massaction incidence form is more appropriate for airborne diseases for example, where doubling the population also doubles the rate at which the disease is transmitted. On the other hand, sexually transmitted diseases are best modeled by the frequency dependent transmission form, because the transmission depends on the mean frequency of sexual contacts per person, which is basically unrelated to the size of the susceptible population.
In view of these considerations, we could appropriately model shortlived and massaction incidence diseases by the SIR model given below:
\[\frac{dS}{dt}=\beta SI\]
\[\frac{dI}{dt}=\beta SI  \kappa I\]
\[\frac{dR}{dt}=\kappa I\]
We will apply this (massaction incidence) SIR model to an epidemic event involving a case of influenza in an English boarding school for boys, as reported by anonymous authors to the British Medical Journal in 1978^{5}. This influenza outbreak apparently began by a single infected student in a population of 763 resident boys (including that student), spanning a period of 15 days. The data furnished refers only to the number of bedridden patients each day these will be taken as the infected population numbers^{6}.
Day  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14 
I  1  3  8  28  75  221  291  255  235  190  125  70  28  12  5 
With this data, we're ready to code the R/simecol script for fitting the SIR model:
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Influenza outbreak in an English boarding school (1978)
The results of the simulation of the influenza outbreak indicate that over 20 students remained susceptible and everyone else was infected. Epidemic diseases do not always affect the entire population whether they do or not depends on the transmission and recovery rates. In fact, looking at the differential equation for the I compartment, we can easily see that the \(dI/dt\) is 0 only when \(I=0\), which is the trivial case when there are no longer infected individuals to infet anyone else and so the epidemic has died out or when \(S=\kappa/\beta\).
This special value \(\kappa/\beta\) is known as the threshold value of the disease because when \(S\) reaches this threshold value, the disease begins to recede (the infected population is decreasing). If at the outset of the epidemic, the susceptible population number is less than the threshold value, the disease will not invade. One can think of this intuitively as the point when the infectious disease "runs out of fuel" because the infected patients are recovering faster than the rate at which the susceptibles are contacting the disease.
If you play around with the value of \(\kappa\), you will see that if you increase this value, there will be less individuals infected and the contrary if you decrease \(\kappa\). This makes sense, as the larger the percentage of the infected population that recovers at each time period, the less the duration of the disease. What this means is that a fatal disease is much more dangerous if it kills slowly and those fatal diseases that kill (remove) their hosts quickly are likely to have a shorter life themselves^{7}.
Notes
 There are indeed mortal diseases from which individuals do not recover. Still, recovered and deceased individuals should be dealt with differently in the model (and in real life), since the purpose of the model is to study the ocurrence of these two very different events.
 See BLOMHØJ et al.
 We hit upon the topic of parameter identifiability, which will be dealt with in a future post.
 See BEREC, p. 12.
 The data can be found in SULSKY.
 See CLARK.
 See "SIR Model of Epidemics" (TASSIER, pp. 56).
Bibliography
 BEREC, L. (2010). Modeling Infectious Diseases in Humans and Animals (Lecture Notes).
 BLOMHØJ, M., KJELDSEN, T.H., and OTTESEN, J. (2014). Compartment Models. Retrieved August 2016, from http://www4.ncsu.edu/~msolufse/Compartmentmodels.pdf.
 CLARK, A. (2005). "SIR Model of Epidemics Part 1  Basic Model and Exapmples". Retrieved June 2011 from www.me.rochester.edu/courses/ME406/webexamp5/sir1.pdf
 PETZOLDT, T. and RINKE , K. (2007). "simecol: An ObjectOriented Framework for Ecological Modeling in R". Journal of Statistical Software, 22(9), pp. 1–31. ISSN 15487660. http://www.jstatsoft.org/v22/i09.
 R DEVELOPMENT CORE TEAM (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3900051070. http://www.Rproject.org.
 SULSKY, D. (2012). "Using Real Data in an SIR Model". Retrieved August 2016 from http://math.unm.edu/~sulsky/mathcamp/ApplyData.pdf
 TASSIER, T. (2005). "Epidemics and Development Policy" Course Lectures: Fordham University. Retrieved June 2011, from http://www.fordham.edu/academics/programs_at_fordham_/economics/faculty/tassier/teaching/epidemics_15958.asp
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