DEVELOPING THE MODEL STRUCTURE

There is no clear guidance on the best structure of an economic model but rather, evidence suggests how the components of a modelling problem influences the most appropriate modelling approach which further directs the development of public health interventions (505-508). Models can be constructed allowing simulations to occur at cohort or aggregate level, or to allow the behaviour of individuals to be followed independently. Cohort-level models (e.g. Cohort Markov models) allocate individuals to compartments that dictate that individuals within a compartment are homogenous. Such compartment models are simpler and less resource-intensive to construct than individual-level models, but understandably have several drawbacks. For instance, the homogeneity assumption is not satisfied if future model states are determined by an individuals’ history and cohort models tend to be rather complex once several comorbidities are captured (502). Most modelling exercises tend to adopt a simple approach, with the notable exception of certain infectious disease modelling studies (508, 509). The modelling approach used depends on various factors, including the decision-maker’s requirements and the disease process being considered (507, 510, 511).

5.1.1 Decision tree modelling

Decision trees are a simple, commonly used decision modelling technique that is effective for uncomplicated scenarios being evaluated (503, 512). The interventions are displayed graphically with a series of pathways or branches (Figure 10). Terminal nodes indicate the end points of each pathway using triangular symbols; to which values or pay-offs, such as costs, life years or QALYs, are assigned (512). Once the transition probabilities and pay-offs have been incorporated into the tree, the tree is averaged out to determine the calculation of the expected values of each option (513). While decision trees remain a simplistic and transparent technique of evaluation, they are limited by their lack of explicit time variable and their inability to handle recursion or looping within the tree such that chronic diseases marked by recurring events dramatically increase the complexity of the analysis (105, 502).

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Figure 10. Illustrative example of a decision tree model

The model was adapted from Salinas-Escudero et al. (514), and does represent the actual findings. A decision tree comprises modes, branches and outcomes. Decision node (□) - describes the problem, Chance node (○) – represents the point at which several possible events can occur, terminal node (∆) – represents the end of a tree with a pay-off attached. Branches from a chance node represent possible events patients may experience at that point in the tree. Branch probabilities represent the likelihood of each event. The sequence of chance nodes from left to right usually follows the sequence of events. The events stemming from a chance node must be mutually exclusive and probabilities should sum to 1.

RDS - Respiratory Distress Syndrome

5.1.2 Markov models

Markov models are more adaptable than decision tree modelling and have been widely used to determine the costs and health outcomes of health related interventions, particularly recursive, complex or chronic disease (471, 515-517). In a Markov model, the disease being studied is categorized into distinct states, described by the transition of disease in nature, within a stochastic framework, over a specified period of time (referred to as the Markov cycle) (105, 470). Each potential health condition in the model is referred to as a Markov state. The Markov states are

80 intended to represent important clinical and economic events that occur to patients over time, achieved by the allocation of costs and utilities allocated to each health state (105, 503, 518). The model then simulates the transition of a hypothetical cohort of individuals through the Markov model over time, allowing the analyst to estimate the costs and outcomes (517). In each cycle, this is achieved by summing costs and outcomes across health states that are weighted by the proportion of the cohort expected to be in each state, and then finally summing across cycles (470). A time horizon of one year or more requires the application of discounting to generate the present values of expected costs and outcomes (502). The probability of remaining in a specified state or moving to another one in each cycle is governed by defined transition probabilities (502).

Defining the health states; and determining the number of health states needed and duration of the cycles are dictated by the nature of the health problem (e.g. gastro-oesophageal reflux disease may require monthly cycles, whereas cervical cancer may need annual cycles) (519, 520). To end a Markov process, termination conditions need to be set. This could be specified as a particular number of cycles, the proportion passing through or accumulating in a particular state, or the defined population reaching a state that cannot be left (e.g. death) that is referred to as an absorbing state (502).

Cohort simulation represents the most simplistic application of the Markov process. The simulation commences with a proposed cohort of participants (e.g. 1000 individuals) that are initially assigned to different states and then transition from between states during each cycle, thereby establishing a redistribution of the initial cohort during each cycle (503). During the cycle, the data on how many patients have remained in different states are captured and the processes are repeated for several cycles to obtain summary results on patients’ spending in different states (503).

Limitations to the Markov model

Similar to any model, Markov models have limitations that must be overcome as models become more complex. An important limitation of the Markov model is what is referred to as the Markov assumption or the ‘memoryless’ feature of the Markov model where the transition probabilities depend on only the current health state, independent of historical experience (502, 521).

81 Additionally, the Markov model may apply two useful but infrequently used states viz. the tunnel and temporary. Temporary states are used when events have short, significant effects where participants remain in that state for at least a single cycle. Temporary states allows for the assigning of state specific transition probabilities and further allows for adjusted utilities and costs (503, 521, 522). A tunnel state, where patients transition in a pre-determined sequence, is likened to the passage through a tunnel, and is generally applied when the temporary state persists for more than a single cycle (503). In the situation where a life-threatening disease is being modelled, future events would depend on past events, which is often lost in the ‘memoryless’ nature of the Markov model (470). The ‘tunnel state’ serves to circumvent this issue by enabling the integration of health experiences from the previous cycles, thus implementing a degree of time-dependency into the model (523).

The half-cycle correction

The use of decision analytic software to implement discrete Markov models requires that transitions occur between simulated health states either at the beginning or at the end of each cycle (524). However, the usual assumption is that, on average, people will often transit between health states halfway through the cycle implying a systematic overestimation at the beginning of the cycle or a systematic underestimation when measured at the end (524-527). The half-cycle correction (HCC), a method used to deal with the inaccuracy caused by inadequate cycle length in Markov models, appears to be the gold standard correction to address this situation (524, 525, 528, 529). The benefits (and futility) of HCCs have been widely published in the international literature (525, 528). Although widely accepted, data shows that very few models actually incorporate the HCC (528). It was often the case that the ICER has changed by less than one percent when the HCC was employed and has had minimal effect on the net health benefit under certain circumstances, compared with the base-case scenario (524, 528). Additionally, standardizing the approach to the HCC remains problematic. Discounting presents a difficult prospect as the population distributions across the states becomes difficult to determine with larger populations in the first cycle and lower membership in all the others (524, 525). The result is that the discounted stream of populations within a state will always be too high (525)). Naimark et al. considers the best alternative to the HCC as no correction at all. Failing which, the adoption of a life-table approach or a correction based on Simpson’s rule (an arithmetical rule and method for numerical integration based on

82 estimating the area under a curve) (524). Barendregt considers the HCC correction to be “inelegant and baffling” in most circumstances and postulates the use of the life-table method as superior and easier to explain (525).

5.1.3 Alternative approaches to the discussed cohort models

Although Markov models, alone or in combination with decision trees, are widely applied in economic evaluations, several other approaches exist.

Patient level simulation (or microsimulation)

As the name suggests, patient level simulations track the progression of individuals rather than hypothetical cohorts within a model (502). In these models, the progression of potentially heterogeneous individuals and the accumulated history of each individual is used to determine the transitions, costs, and health outcomes (470, 507). Patient levels simulations are able to simulate the time to next event, rather than prescribing to equal cycle lengths, and additionally, are able to simulate multiple events occurring in parallel (507).

Discrete event simulations

Discrete event simulations (DES) describes the analyses of the disease progression of individuals through a resource constrained health care system with the aim of improving the organization of delivered services (530). The characteristics and outcomes are described over unrestricted time periods (507). The DES are not limited by the Markovian assumption (507). Unlike patient level simulation models, DES allows for individuals to interact with each other (e.g. in a transplant situation where organs are considered scarce, the transplant decisions and outcomes for any individual affects every individual in the queue) (510). The DES is limited by its computational complexity (often resulting in attempts to gain insights translating into futile, ambiguous models), the integration of intense randomness into the simulation making it difficult to distinguish whether an observation is attributed to system interrelationships or merely randomness and lastly, the models can be time consuming and expensive (531).

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Dynamic transmission models

Dynamic transmission models (abbreviated to dynamic models) are able to reproduce the evolving direct and indirect effects (e.g. herd immunity) associated with communicable disease control programmes (532). The model allows for the internal feedback loops and time delays that impact the behaviour of the entire health system or population associated with the communicable disease process (533). They differ from largely static models that assume a constant risk of infection, thus changing the likelihood of infection over time and more effectively representing the progression of disease in reality (534).