  The spread of infectious diseases, rumors, fashions, innovations are complex
contagion processes, embedded both in networked and spatial contexts. Here we
investigate the pattern dynamics of a complex contagion, where two agents, say
$A$ and $B$, interact with each other and diffuse simultaneously in the
geographic space. The contagion process for each follows the classical
susceptible-infected-susceptible kinetics, and their interaction introduces a
potential change in the secondary infection propensity compared to the baseline
reproduction ratio $R_0$. We show that nontrivial spatial infection patterns
arise, when the susceptible move faster than the infected and the interaction
between the two agents is neither too competitive nor too cooperative.
Interestingly, the system exhibits pattern hysteresis phenomena that quite
different parameter regions allowing for patterns exist in the direction of
increasing $R_0$ and in the direction of eradication by its reduction. The
latter shows a remarkable enhancement in the contagion prevalence, meaning that
the infection eradication now becomes extremely difficult compared to the
single-agent scenario and to the coinfection without space. Linearization
analysis supports our observations, and we identified the required elements and
dynamical mechanism behind the emergence of a pattern. These findings call for
further investigation for their close relevance, both in biological and social
contagions.
