### Cooperative Contagion Phenomena

We live in a world full of infectious pathogens. When pathogens spread and proliferate in a host population they can interact, i.e. the infection with one pathogen may impact a persons susceptibility towards an infection with a second disease. For example, the during the 1918 Spanish Flu pandemic, the majority of people suffered secondary bacterial infections and died of pneumonia. Other examples include of course secondary infections with hepatitis, tuberculosis, syphillis of HIV infected individuals with compromised immune systems. One way to view this “positive” feedback between different pathogens is within the context of cooperative coinfections.

What basic dynamical features can we expect in multiple-epidemic spreading with disease-disease interactions?

To what extent does coinfection change the classic outbreak scenario of single diseases, e.g. a classical threshold behavior?

How does coinfection dynamics shape the generic features of spatial propagation?

In this project, we approach these problems with a structurally parsimonious model of two cooperating epidemics, say A and B, that spread within a structured host population. The model distinguishes two transmission rates: the rate at which the initial infection is acquired and the rate for acquiring a secondary infection. Cooperative coinfection occurs when the second rate is larger than the basic transmission rate. The underlying structure of the population is chosen to be a lattice or a random network. Each node represents an individual and infections can be passed on along the links.

What basic dynamical features can we expect in multiple-epidemic spreading with disease-disease interactions?

To what extent does coinfection change the classic outbreak scenario of single diseases, e.g. a classical threshold behavior?

How does coinfection dynamics shape the generic features of spatial propagation?

In this project, we approach these problems with a structurally parsimonious model of two cooperating epidemics, say A and B, that spread within a structured host population. The model distinguishes two transmission rates: the rate at which the initial infection is acquired and the rate for acquiring a secondary infection. Cooperative coinfection occurs when the second rate is larger than the basic transmission rate. The underlying structure of the population is chosen to be a lattice or a random network. Each node represents an individual and infections can be passed on along the links.

**Numerical experiments.**When simulated on a lattice for example, the outbreak typically exhibits discontinuous phase transition if the pathogens cooperate, contrary to the continuous transition of single disease dynamics. In fact, the system exhibits two different critical thresholds for the basic transmission rate, one outbreak threshold and a smaller one for eradication. This implies that when the basic transmission rate increases beyond a the outbreak threshold leading to an outbreak one cannot eradicate the diseases by descreasing the basic transmission by the same amount. One has to decrease basic transmission below the smaller eradication threshold. In physics terms, the system exhibits hysteresis of outbreak and eradication events. You can click the below image to play around the coinfection within an interactive tool.**Mean field theory.**Dynamical systems and bifurcation theory can be used to understand the basic features of the cooperative coinfection. The population is compartmentalized into four different compartments (S, A, B, AB) defining four different infectious states. The disease dynamics are then analogous to “chemical kinetics”, which can be summarized into a set of ordinary differential equations. Bifurcation analysis of the system provides a global picture of the beahvior of the system identifying different bifurcations as a function of parameters: transcritical, pitchfork, and saddle-node bifurcations, which qualitatively are in good agreement with observations made in numerical simulations.