The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model

In this paper, we develop the theory of a state-reproduction number for a multistate class age structured epidemic system and apply it to examine the asymptomatic transmission model. We formulate a renewal integral equation system to describe the invasion of infectious diseases into a multistate class age structured host population. We define the state-reproduction number for a class age structured system, which is the net reproduction number of a specific host type and which plays an analogous role to the type-reproduction number [M.G. Roberts, J.A.P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond. B 270 (2003) 1359; J.A.P. Heesterbeek, M.G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci. 206 (2007) 3] in discussing the critical level of public health intervention. The renewal equation formulation permits computations not only of the state-reproduction number, but also of the generation time and the intrinsic growth rate of infectious diseases.Subsequently, the basic theory is applied to capture the dynamics of a directly transmitted disease within two types of infected populations, i.e., asymptomatic and symptomatic individuals, in which the symptomatic class is observable and hence a target host of the majority of interventions. The state-reproduction number of the symptomatic host is derived and expressed as a measurable quantity, leading to discussion on the critical level of case isolation. The serial interval and other epidemiologic indices are computed, clarifying the parameters on which these indices depend. As a practical example, we illustrate the eradication threshold for case isolation of smallpox. The generation time and serial interval are comparatively examined for pandemic influenza.