These are systems that, thanks to a knowledge base and an inference engine, can draw conclusions or make suggestions based on facts or clinical data. They replicate the reasoning ability of a specialist in a specific field.

MYCIN, developed at Stanford University in 1977 [2] , is historically considered one of the most representative medical expert systems. It is a clinical consultation system equipped with a specialized inference engine dedicated to selecting antimicrobial therapies for patients with severe infections.

In a more specific context, Konan et al. [3] developed Metrad+, an expert system focused on traditional African medicine. Coming from an expert system generator designed for the use of an iconic language, it offers translation practitioners the possibility of structuring and storing their knowledge in digital tools adapted to their context.

Although deductive expert systems remain an essential element of decision support in medicine thanks to their modular structure and reasoning logic, the research reviewed shows that they excel in rule-based interpretation and explanatory capabilities.

However, as the articles reviewed point out, these systems often fall short in their ability to learn autonomously from new data or predict future changes in health conditions.

These are systems that use machine learning algorithms to make predictions.

Basma Boukenze and her colleagues [4] use the C4.5 algorithm to predict chronic kidney disease, with remarkable accuracy (396/400 cases correctly classified) and a minimal error rate (0.37%).

In Senegal, Boubacar Sow and his colleagues [5] examined four machine learning algorithms (KNN, Random Forests, SVM, Naive Bayes) for predicting anemia and malaria in children. SVM showed the highest accuracy for both classifications.

These systems generally provide accurate predictions. However, two notable flaws remain: on the one hand, their inability to explain the underlying reasons for their predictions reduces their transparency and the confidence that healthcare professionals might have in them; on the other hand, they do not take sufficient account of the in-depth deductive analysis of medical results, which is essential for comprehensive and reliable clinical decision-making. These issues highlight the need for a complementary strategy.

These are systems that combine the two methods mentioned above.

The work of Abdelhak Mansoul et al. [6] suggests a theoretical perspective on medical decision support systems, based solely on case-based reasoning (CBR) combined with a multi-criteria approach. The aim is to overcome the limitations of traditional expert systems by giving the system the ability to learn from concrete clinical cases and manage several indicators simultaneously. The hybrid system, called MCDS (Multi-Criteria Decision System), aims to improve decision-making by combining previous experience with multi-criteria judgments, representing a form of learning fueled by an organized approach.

The hybrid model developed by Syed Imran Ali and his colleagues [7] assists practitioners in managing the treatment of bone and mineral disorders associated with chronic kidney disease (CKD-MBD), combining medical knowledge and clinical case studies, with an accuracy rate of 78% based on the evaluation of 250 cases. This illustrates the importance of merging organized knowledge with case expertise for explanation and more effective adaptation. However, its ability to generalize is limited due to the small number of cases available.

The hybrid system consists of a predictive module and a deductive module. We will therefore translate each module mathematically.

The predictive module is represented by the following function:

Ŷ^ML = f^ML (H, θ)

where:

This module produces a predictive output based on the data and learned models, without direct intervention from expert rules. It measures the expected response or probable diagnosis based on the inputs X and the history H.

The deductive module is represented by the following function:

Ŷ^EX = f^EX (X, R)

Because it produces an output (decision or hypothesis) based on both the input data X and the set of clinical rules R, i.e., formalized expert knowledge.

To design a high-performance hybrid system, it is crucial to select an appropriate coupling technique, as the quality of the system depends directly on it. The main couplings studied are:

Thus, the mathematical modeling of our hybrid system is based on the introduction of a parameter α, representing the chosen coupling strategy. This parameter formalizes this strategy and allows the system’s operation to be dynamically adjusted according to the context. We have defined α as follows:

α = C^M/(C^ML + C^EX)

where:

Meaning of α values:

In concrete terms, confidence levels C^ML and C^EX can be estimated using performance indicators. For the machine learning module, C^ML can be derived from metrics such as AUC-ROC, precision, or recall. For the expert module, C^EX can be based on rule validation or consensus among doctors.

In view of the above, our final mathematical model translates into the following objective function:

Yfinal = α × Ŷ^ML + (1 − α) × Ŷ^EX

The hybrid Yfinal function is based on two separate decision opinions:

Each module is associated with a confidence level that reflects its relevance or reliability in a given context.

The final function acts as an automatic arbitration mechanism, combining the two decisions based on the confidence measured for each module.

The combination of machine learning and a traditional expert system not only enhances the reliability of decisions, but also helps clinicians anticipate factors that they might not perceive due to the complexity of the data or other factors. The following figure ( Figure 1 ) illustrates this synergy:

In light of the above, our final mathematical model translates as follows: