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During epidemics, individuals may adjust their social behavior in response to the threat. This may affect the course of the epidemic, and, in turn, again modify people's behavior. Game theoretically, the system may end up in a Nash equilibrium, where no member of the population can benefit by unilaterally changing their behavior. Compartmentalized epidemic models can incorporate such endogenous decision making, where individuals try to optimize a utility function via their behavior. Typically, such models can only be solved numerically. Here, we extend a recently discovered analytic solution for time-dependent social distancing and the corresponding epidemic dynamics: now, the probability of an infection taking place can depend on both the susceptible and infectious individual behaviors. We show that the more effectively the susceptible individual can reduce the probability of infection, the more self-organized social distancing is expected to occur. The previously identified heuristic that the strength of rational social distancing is proportional to both the perceived infection cost and prevalence is found to also hold in the generalized model.