Dynamics of Indirectly Transmitted Infectious Diseases with Immunological Threshold
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- Joh, R.I., Wang, H., Weiss, H. et al. Bull. Math. Biol. (2009) 71: 845. doi:10.1007/s11538-008-9384-4
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There are numerous examples of human pathogens which persist in environmental reservoirs while infectious outbreaks remain rare. In this manuscript, we consider the dynamics of infectious diseases for which the primary mode of transmission is indirect and mediated by contact with a contaminated reservoir. We evaluate the realistic scenario in which the number of ingested pathogens must be above a critical threshold to cause infection in susceptible individuals. This minimal infectious dose is a consequence of the clearance effect of the innate immune system. Infected individuals shed pathogens back into the aquatic reservoir, indirectly increasing the transmittability of the pathogen to the susceptible. Building upon prior works in the study of cholera dynamics, we introduce and analyze a family of reservoir mediated SIR models with a threshold pathogen density for infection. Analyzing this family of models, we show that an outbreak can result from noninfinitesimal introductions of either infected individuals or additional pathogens in the reservoir. We devise two new measures of how likely it is that an environmentally persistent pathogen will cause an outbreak: (i) the minimum fraction of infected individuals; and (ii) the minimum fluctuation size of in-reservoir pathogens. We find an additional control parameter involving the shedding rate of infected individuals, which we term the pathogen enhancement ratio, which determines whether outbreaks lead to epidemics or endemic disease states. Thus, the ultimate outcome of disease is controlled by the strength of fluctuations and the global stability of a nonlinear dynamical system, as opposed to conventional analysis in which disease reflects the linear destabilization of a disease free equilibrium. Our model predicts that in the case of waterborne diseases, suppressing the pathogen density in aquatic reservoirs may be more effective than minimizing the number of infected individuals.