Model Overview
This model tracks susceptible hosts, hosts infected with either pathogen 1, pathogen 2 or both, and individuals recovered from infection with pathogen 1, pathogen 2 or both. As usual, infected hosts are assumed to be infectious.
The following compartments are included:
- S - uninfected and susceptible individuals.
- I1 - individuals who are infected with pathogen 1.
- I2 - individuals who are infected with pathogen 2.
- I12 - individuals who are infected with both pathogen 1 and 2.
- R1 - individuals recovered from infection with pathogen 1, still susceptible to pathogen 2.
- R2 - individuals recovered from infection with pathogen 2, still susceptible to pathogen 1.
- I1X - individuals previously infected with pathogen 2 and now infected with pathogen 1.
- I2X - individuals previously infected with pathogen 1 and now infected with pathogen 2.
- R12 - individuals recovered from infection with both pathogen 1 and 2.
The included processes/mechanisms are the following:
- Susceptible individuals can become infected with either pathogen 1 or 2.
- Hosts infected with pathogen 1 or 2 can become infected with the other pathogen.
- Hosts infected with pathogen 1, 2 or both infect susceptibles or hosts infected with the other pathogen at rates b1/b2/b12. For simplicity, we assume that the rates are independent of the status of the host that becomes infected (e.g. an S host and a I1 can become infected at the same rates with pathogen 2). It also does not matter if a host has previously recovered from an infection with the other pathogen.
- Hosts infected with both pathogens infect susceptibles with either pathogen 1 or 2 at fractions a/(1-a). We do not consider the possibility that both pathogens are transmitted at the same time.
- Infected hosts recover after some time at rates g1/g2/g12. The rate of recovery does not depend on previous infections.
- Recovered individuals are immune to reinfection by the pathogen they have already been infected with (either at the same time or consecutively) but can still be infected with the pathogen they have not yet been infected with.
Model Implementation
The flow diagram and the set of differential equations for the mathematical model implemented in this app are as follows:
\[\dot S = - (b_{1} (I_1+I_{1X}) + b_{2} (I_2+I_{2X}) + b_{12}I_{12}) S \] \[\dot I_1 = (b_{1} (I_1+I_{1X}) + ab_{12} I_{12})S - (g_1 + b_{2} (I_2+I_{2X}) + b_{12} I_{12}) I_1\] \[\dot I_2 = (b_{2} (I_2+I_{2X}) + (1-a) b_{12} I_{12})S - (g_2 + b_{1}(I_1 + I_{1X}) + b_{12} I_{12}) I_2\] \[\dot I_{12} = (b_{2} (I_2+I_{2X}) + b_{12} I_{12}) I_1 + (b_{1}(I_1 + I_{1X}) + b_{12} I_{12}) I_2 - g_{12} I_{12}\] \[\dot R_1 = g_1 I_1 - (b_2 (I_2 + I_{2X}) + b_{12} I_{12}) R_1\] \[\dot R_2 = g_2 I_2 - (b_1 (I_1 + I_{1X}) + b_{12} I_{12}) R_2\] \[\dot I_{1X} = (b_1 (I_1 + I_{1X}) + b_{12} I_{12}) R_2 - g_{1} I_{1X}\] \[\dot I_{2X} = (b_2 (I_2 + I_{2X}) + b_{12} I_{12}) R_1 - g_{2} I_{2X}\] \[\dot R_{12} = g_{1} I_{1X} + g_{2} I_{2X} + g_{12} I_{12} \]