Adams et al. 10.1073/pnas.0602768103.

Supporting Information

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Supporting Figure 6
Supporting Text




Supporting Figure 6

Fig. 6. Compartment structure of the model. Black lettering corresponds to the compartments (circles) and fluxes (arrows) described by Eqs. 1-5 and used throughout this study. Gray lettering corresponds to the underlying nine-compartment model. Note that the fluxes due to recovery are too complicated to show accurately, and they have been omitted from this diagram.





Supporting Text

Structure of the Model

An individual host has three possible immune states S (susceptible), I (infected), and R (recovered) with respect to each pathogen strain. Hence, there are nine possible immune states with respect to both pathogen strains: SS, SI, SR, IS, II, IR, RS, RI, and RR, which can be modeled by using nine differential equations. This system can be reduced to six differential equations by introducing the state variables (1) x = SS, y1 = IS + II + IR, y2 = SI + II + RI, z1 = SI + SR, z2 = IS + RS, and z3 = IR + RI + RR. However, it is not necessary to explicitly include the z3 compartment because the IR and RI components already appear in the y compartments and hosts in the RR state cannot be reinfected and so do not influence the dynamics. Thus, only five differential equations are needed:

The new state variables create compartments that sometimes overlap: y1 and y2 overlap because both contain II, y1 and z2 overlap because both contain IS, similarly for y2 and z1.

y

1 and z3 overlap because both contain IR, similarly for y2 and z3. This arrangement is shown in Fig. 6. Note that the nonoverlapping components of the y1 and y2 compartments are, in fact, always empty.

The x compartment is increased by births at a constant rate. All compartments are reduced by natural deaths at a rate proportional to their size. Primary infections connect x to y1 and y2. Secondary infections connect z1 to y2 and z2 to y1. The section of z1 implicitly containing SI connects to the section of y1 containing II. The section of z1 implicitly containing SR connects to section of y1 containing IR. Recovery appears only in the differential equations for the y1 and y2 variables. From y1, recovery connects IS to RS, II to RI, and IR to RR. Recovery from IS to RS reduces y1, but z1 does not change because it already contains both of these components. Recovery from II to RI reduces y1, but y2 does not change because it already contains both of these components. Recovery from IR to RR reduces y1 and increases z3, but this compartment is not explicitly modeled.

1. Kamo M, Sasaki A (2002) Physica D 165:228-241.