## Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease

This work is not based on one comprehensive model that captures all essential features of our community, all plausible characteristics of the infection and all available interventions. Development of such a model and its use to address the numerous aims would be an enormous undertaking, which is quite infeasible in the timeline available for this project. Instead, we have developed a number of models each of which is formulated to contain the essential features needed to address specific issues of interest. There is some overlap in the issues these models can address and this provides a useful check on the results. We now outline some of the features and assumptions that are common to the modeling approaches.

Individuals are assumed to be susceptible at first. Upon infection they are classified as *exposed,* to indicate that they have entered the latent stage of their infection. There follows a period when they are infectious. After this they become removed, by death or recovery with acquired immunity.

Our models of this kind can be classified into two types, referred to as SEIR and SEIRH. In models of the type SEIRH the community has a household structure, while the simpler SEIR model ignores the household structure. The SEIR model is easier to work with and therefore enables an analysis in a shorter timeframe. Some features of the various modeling approaches are listed below.

First, ordinary differential equations were used to model the transitions between the progression states (Sections 4.9, 5.6, 6.2 and 6.3). Such models are the most common form of infectious disease model found in the literature. This approach models the actual numbers and/or quantities in different categories (e.g. anti-viral stockpile) continuously in time, using equations for the rates of transitions. These are deterministic models (i.e. each run of the model produces the same result) and hence do not capture the variability in outcomes. Such variability may be substantial, especially early in an epidemic.

Second, the transitions between disease states were modeled by difference equations (Sections 4.8 and 5.5), whereby quantities are updated at fixed time steps (e.g. daily). This approach allows complexity to be accommodated (e.g. household size and the disease status of individuals within households on a day-by-day basis) in a conceptually simple manner, using a time scale that is natural for collecting data (e.g. daily incidence). We used deterministic models of this form.

Third, the movement between disease states was modeled using the mean numbers in successive generations (Sections 4.6, 4.11, 5.2–5.4 and 6.1). Generations are defined as follows: The initial case (or cases) makes up the first generation. The second generation consists of all cases that had an infectious contact with a case from the first generation. Similarly for the third, and later, generations. In these models the deletion of susceptibles is ignored, so they are suitable only for studying the effect of interventions on the reproduction number (see Section 2.2) and on the early dynamics. However, the approach can incorporate considerable underlying complexity in transmission between different types of individual in a community (e.g. general public, general practitioners, health care workers), and allows the effect(s) of interventions (e.g. use of AVs, closing schools) either singly or in combination to be easily explored in terms of the effect on the reproduction number (*R*) and the mean dynamics of the early stages of an epidemic.

Fourth, branching processes were used to incorporate the chance element of transmission between individuals, which is especially relevant in the early stages of an epidemic where chance plays a large role in whether epidemics take off or fade-out (Sections 3.2–3.8). This type of model is most appropriately applied during the early stages of an epidemic when there is little competition for susceptibles. It is particularly useful for assessing whether an infected individual will successfully initiate an epidemic. We assume that during the early stages of an epidemic, the offspring distribution that describes the number of secondary cases that each infected case generates is Poisson with mean equal to the effective reproduction number operating at the time.

Fifth, a stochastic household model was developed in terms of generations of infected individuals (Sections 4.2-4.5, 4.10, 5.7) that allows for depletion of susceptibles, and so can be used to investigate the size and timing of the outbreak. The model includes adults and schoolchildren as separate types, and was used to consider interventions (such as closing schools) that differentially affect adults and children. The stochastic nature of the model provides insight into the variability in the size and timing of epidemic outbreaks that arises from chance. Transmission within the household follows the Reed-Frost model (Bailey, 1975), while transmission outside the household is according to a mixing matrix that has been calibrated to relative attack rates in adults and children seen in past influenza pandemics.

Finally, stochastic population-based models were used to investigate the spread of infection between cities (Section 4.7). A stochastic approach is valuable in capturing the effects of random variation on the timing of city to city spread. Transmission between cities follows the standard diffusive model, in which the degree of transmission is proportional to the volume of travel between the cities.

For all approaches, the larger structure of the Australian community is not explicitly taken into account (e.g. demographic differences between cities), hence most of the results presented apply at a local level. We have been consistent in the parameters values used in the models (see Appendix B for default parameter values and their justification). Throughout, we allow influenza infected individuals to become symptomatic part of the way through their infectious period. The possibility that individuals may still be susceptible after recovery, due to acquiring only partial immunity is not allowed for, unless explicitly stated.

Assumptions specific to each modeling approach are presented in the sections where they arise.

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