Decision tree

Decision tree is the simplest way of representing decision modelling.⁴ Graphic resources are used to draw possible pathways that patients would follow in case they were under the strategies or interventions investigated. In these pathways, the events and their respective probabilities of occurrence are included. In the end, costs and health outcomes are assigned to each itinerary taken by the individuals.

Given the limited circumstances of this type of modelling, regarding its use, only acute short-term diseases are considered here. For situations where it is necessary to represent recurring events and long time horizons - chronic diseases, for example -, there are other options of decision models, as follows.

Markov model

The Markov model has three essential characteristics. The first one is about the restricted number of clinically and economically relevant events that a cohort of individuals may experience during the evolution of the disease. These events are called health states. The second element refers to the period of time in which individuals remain in each health state, called Markov cycle. The third element relies on the transition probabilities from one health state to others. For instance, when considering breast cancer, possible health state would be: (i) local cancer; (ii) metastasis; (iii) cure; and (iv) death. Each cycle would last six months, i.e., every six months there would be a change in the health state (from local cancer to metastasis) or remaining in the same stage (local cancer to local cancer). If the time horizon were ten years, 20 cycles would be repeated, and, at the end of each cycle costs and health outcomes would be determined - these topics were discussed in the two last articles of this series.^5,6

These characteristics of Markov model are appealing for chronic diseases analyses,⁷ since the recurring events and the explanation of time through cycles can be easily included. Usually, Markov models are calculated through hypothetical closed cohorts of individuals,⁸ cases in which its use presents some limitations. The first one is about the strictness of cycles' definition by Markov, which are always constant. In real life, the length of each cycle tends to vary with time, due to the natural history of the disease. The second limitation is about the memoryless between two cycles, that is, the past is not taken into consideration. For instance, there would be the same possibility of transition from metastasis to death, regardless of the individual has been with metastasis for one or four cycles, which is not supported by clinical data. An individual with metastasis for twenty-four months (a total of four cycles) has higher probability of evolving to death than an individual that has been with metastasis for only six months (only one cycle). The third limitation of Markov model is related to its interaction between individuals and groups, a characteristic that appears in the next models presented.

Discrete event simulation

The model of discrete event simulation is adequate for cases in which interactions between individuals or groups should be considered, when time-to-event is best described stochastically, when individual pathways through the model are influenced by multiple characteristics of the individuals, and when there are resource constraints.⁹ Different characteristics mean the possibility of including patients with different traits, such as several age groups, various family histories concerning diseases, comorbidities. The resource constraint represents a situation in which there is no intervention enough for all the individuals who need it, in a certain moment. This occurs, for example, when the health system is not able to offer a satisfactory amount of elective surgeries to the population, resulting in waiting lists. Time, in turn, may influence both the characteristics of the individuals and the events. Keeping the example of elective surgery, the longer the individual waits, the worse his/her health situation will become, that is, the individual's characteristics will get worse. A long period waiting for care also increases the probability of occurrence of events, such as reaching a more advanced stage of the disease.

Dynamic model

The dynamic model is especially appropriate to the analysis of strategies or interventions that aim to control infectious diseases.¹⁰ In this models, the direct and indirect effects of transmissions between groups are introduced, also known as externalities of the disease. The transition probabilities depend on the health situation of other individuals. For instance, if a vaccination campaign reduces the number of cases in the population, then the risk of transmission to non-ill people will be lower. In the case of chronic diseases, such as those represented by Markov models, this characteristic does not exist, because the reduction in the prevalence of cardiac diseases, for example, does not affect the individual risk of having heart problems.

For dynamic models, an important measure is the basic reproduction number, an indicator of the disease spread among the population. Values higher than one indicate an exponential growth in the number of infected cases among the susceptible population, that is, in a higher proportion. For instance, when the basic number reproduction is 3, it means that, on average, a new case generates new three cases, and each of them will generate three new cases, resulting in nine new cases, and those, in 27 new cases, and so on.