Infectious diseases are of the most fatal threats in human history. Many studies have been made to investigate the way these diseases become epidemic and how it is possible to prevent them from spreading to the whole society. Of the most interesting properties of the phenomenon is the outbreak of the disease which resembles a phase transition. There are some examples of coinections that two or more pathogens cooperate in infecting individuals; i.e. if one is infected by one disease, the chance to become infected by the second one will be much more. Recently such phenomenon has been modeled using two interacting SIR model and discontinuous transitions, in contrast to single SIR dynamics, were observed . However the meanfield approach equations were only solved numerically and the insight for how the transition is occurred was not complete. Here we solve the equations analytically and show that the transition occurs when something like a saddle node bifurcation happens in the system (see figure 1). Through this analytical solution we can derive the exact diagram of the order parameter as a function of infection rates. Though the solution could be expressed in terms of the roots of a non-algebraic equation, in certain limits we can provide simple, yet relatively accurate expression for the order parameter as function of infection rates.
Figure 1: The transition occurs when two roots of X(T)=0 vanish.
We have also generalized the above model to the case that we have n interacting SIR dynamics.
Where is the fraction of agents that have experienced exactly diseases and is the fraction of agents that has one particular disease. Again the meanfield equations can be solved analytically and with usual initial condition the same phenomenon arises.