## 2 Overview of Mathematical Models to Predict Infectious Diseases  
Epidemiology has been considered the gold standard for investigating diseases' dynamics in varied populations, focusing mainly on the distribution and determinants of disease, to better prevent and control these diseases. Moreover, epidemiology could be considered the first mathematical model applied to the prevention and control of diseases [17]. Epidemic mathematical models have usually been applied to the prediction of outbreaks, epidemics, and transmission, as well as the resurgence of infectious disease [18]. Guidelines and vaccination programmes have been established to prevent and control transmission. Although these first epidemiological models were \"static\" and representative of neither different social or geographical spaces, nor their evolution over time [17; 19], they have served as a model to develop dynamic models that evolve over time (Figure 2). Bayesian models are based on dynamic probabilistic models, in which a \"theorem\" describes the probability of any event supported by prior knowledge related to that particular event. The variables are random, and uncertainty can be measured [20]. The applicability of Bayesian models [21], networks [22; 23; 24], or successful combinations [25], e.g., with gaussian variables [26], is too broad. Certainly, Bayesian methods are more difficult to implement than traditional methods, especially in epidemiology and infection diseases [27]. However, they have been applied in malaria studies [28], to evaluate interventions applied to prevent human immunodeficiency virus (HIV) infection and its collateral risk [29], as well as to evaluate the population dynamics of HIV [30]. Epidemiology (or non-epidemic) models are based on different statistical mathematical models, such as (i) regression, mainly used to detect outbreaks [31], as well as (ii) the autoregressive integrated moving average model, ARIMA, and seasonal SARIMA, both used to predict outbreaks and risk factors [32; 33]. Currently, more complex models are gaining popularity, including [34] (iii) models based on a time series, which are more useful in antimicrobial studies to unravel trends [35], (iv) methods  
Figure 2: Schematic classification of mathematical models applied to epidemiology and medicine. The figure shows the evolution from a simpler statistical fixed model, such as a regression model, to a more sophisticated and dynamic learning machine or artificial intelligence models. SIR, Susceptible-Infection-Recover. including cumulative sum and an exponentially weighted moving average (for example, in the ARIBACA project, were this model was implemented to detect and forecast selected Caribbean diseases, such as dengue [36]), and (v) spatial models, which are more sophisticated, including place-specific models requiring multivariate techniques to model the spatial heterogeneity of all the infection's covariates [37] (Table A1). Mathematical or mechanist models are more complex state-space models based on compartments (S = susceptible, E = exposed, and I = infectious and recovered populations). The compartmental or deterministic approach treats each different status of an epidemic as a different sub-population or compartment of a population [38]. The stochastic approach takes random variables into consideration. Susceptible-Infection-Recover (SIR) compartments and derivatives models are mainly used for predictions about the spread of infectious diseases [39], vaccination impact [40], or both [41; 42]. Importantly, population dynamics are inexorably subjected to environmental background and natural phenomena, and rather than following random oscillations, they strictly follow deterministic laws. The stochastic or random SIR model has the advantage of introducing \"the random or the chance\" as a variable [43]. Then, stochastic models can more realistically predict epidemics [44], vaccination impact on herd immunity [45], or the dynamics of protein-protein interactions (PPIs) during infections [46].