This paper introduces a novel extension of the edge-based compartmental model to epidemics where the transmission and recovery processes are driven by general independent probability distributions. Edge-based compartmental modelling is just one of many different approaches used to model the spread of an infectious disease on a network; the major result of this paper is the rigorous proof that the edge-based compartmental model and the message passing models are equivalent for general independent transmission and recovery processes. This implies that the new model is exact on the ensemble of configuration model networks of infinite size. For the case of Markovian transmission the message passing model is re-parametrised into a pairwise-like model which is then used to derive many well-known pairwise models for regular networks, or when the infectious period is exponentially distributed or is of a fixed length.

Toxicokinetic-toxicodynamic (TKTD) modelling is essential to make sense of the time dependence of toxic effects, and to interpret and predict consequences of time-varying exposure. These advantages have been recognised in the regulatory arena, especially for environmental risk assessment (ERA) of pesticides, where time-varying exposure is the norm. We critically evaluate the link between the modelled variables in TKTD models and the observations from laboratory ecotoxicity tests. For the endpoint reproduction, this link is far from trivial. The relevant TKTD models for sub-lethal effects are based on Dynamic-Energy Budget (DEB) theory, which specifies a continuous investment flux into reproduction. In contrast, experimental tests score egg or offspring release by the mother. The link between model and data is particularly troublesome when a species reproduces in discrete clutches, and even more so when eggs are incubated in the mother's brood pouch (and release of neonates is scored in the test). This situation is quite common among aquatic invertebrates (e.g., cladocerans, amphipods, mysids), including many popular test species. We discuss these and other issues with reproduction data, reflect on their potential impact on DEB-TKTD analysis, and provide preliminary recommendations to correct them. Both modellers and users of model results need to be aware of these complications, as ignoring them could easily lead to unnecessary failure of DEB-TKTD models during calibration, or when validating them against independent data for other exposure scenarios.

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This paper investigates the dynamics of infectious diseases with a non-exponentially distributed infectious period. This is achieved by considering a multi-stage infection model on networks. Using pairwise approximation with a standard closure, a number of important characteristics of disease dynamics are derived analytically, including the final size of an epidemic and a threshold for epidemic outbreaks, and it is shown how these quantities depend on disease characteristics, as well as the number of disease stages. Stochastic simulations of dynamics on networks are performed and compared to output of pairwise models for several realistic examples of infectious diseases to illustrate the role played by the number of stages in the disease dynamics. These results show that a higher number of disease stages results in faster epidemic outbreaks with a higher peak prevalence and a larger final size of the epidemic. The agreement between the pairwise and simulation models is excellent in the cases we consider.

This paper presents a compact pairwise model that describes the spread of multi-stage epidemics on networks. The multi-stage model corresponds to a gamma-distributed infectious period which interpolates between the classical Markovian models with exponentially distributed infectious period and epidemics with a constant infectious period. We show how the compact approach leads to a system of equations whose size is independent of the range of node degrees, thus significantly reducing the complexity of the model. Network clustering is incorporated into the model to provide a more accurate representation of realistic contact networks, and the accuracy of proposed closures is analysed for different levels of clustering and number of infection stages. Our results support recent findings that standard closure techniques are likely to perform better when the infectious period is constant.