Sic Transit Gloria Mundi: A Mathematical Theory of Popularity Waves Based on a SIIRR Model of Epidemic Spread

We discuss the spread of epidemics caused by two viruses which cannot infect the same individual at the same time. The mathematical modeling of this epidemic leads to a kind of SIIRR model with two groups of infected individuals and two groups of recovered individuals. An additional assumption is that after recovering from one of the viruses, the individual cannot be infected by the other virus. The mathematical model consists of five equations which can be reduced to a system of three differential equations for the susceptible and for the recovered individuals. This system has analytical solutions for the case when one of the viruses infects many more individuals than the other virus. Cases which are more complicated than this one can be studied numerically. The theory is applied to the study of waves of popularity of an individual/groups of individuals or of an idea/group of ideas in the case of the presence of two opposite opinions about the popularity of the corresponding individual/group of individuals or idea/group of ideas. We consider two cases for the initial values of the infected individuals: (a) the initial value of the individuals infected with one of the viruses is much larger than the initial values of the individuals infected by the second virus, and (b) the two initial values of the infected individuals are the same. The following effects connected to the evolution of the numbers of infected individuals are observed: 1. arising of bell-shaped profiles of the numbers of infected individuals; 2. suppression of popularity; 3. faster increase-faster decrease effect for the peaks of the bell-shaped profiles; 4. peak shift in the time; 5. effect of forgetting; 6. window of dominance; 7. short-term win-long-term loss effect; 8. effect of the single peak. The proposed SIIRR model is used to build a mathematical theory of popularity waves where a person or idea can have positive and negative popularity at the same time and these popularities evolve with time.