In this paper, the main biological aspects of infectious diseases and their mathematical translation for modeling their transmission dynamics are revised. In particular, some heterogeneity factors which could influence the fitting of the model to reality are pointed out. Mathematical tools and methods needed to qualitatively analyze deterministic continuous-time models, formulated by ordinary differential equations, are also introduced, while its discrete-time counterparts are properly referenced. In addition, some simulation techniques to validate a mathematical model and to estimate the model parameters are shown. Finally, we present some control strategies usually considered to prevent epidemic outbreaks and their implementation in the model.

#### Keywords

- Epidemiology
- Dynamical systems in biology
- Fixed points and periodic points of dynamical systems
- Stability of solutions
- Bifurcations
- Lyapunov global stability and functions
- Parameters estimation
- Simulation
- Disease control

#### MSC 2010

- 92D30
- 37N25
- 37C25
- 37C75
- 39A30
- 93D05

Infectious or communicable diseases are caused by a biological pathogen such as virus, bacteria, protozoa, toxins, etc. Different transmission mechanisms affect the spread of the disease as, for instance, direct physical contact, aerosol droplets of an infected individual, passive vectors (water, food, etc.), active vectors (mosquitoes, ticks, etc.). In addition to these ways of horizontal transmission, there could be vertical transmission, i.e. from a mother to her child before or at the moment of the birth.

A mathematical model of an infectious disease is a (mathematical) formula which describes the transmission process of the disease. To formulate a mathematical model, the understanding of the main biological features of the disease (states and parameters involved) and the relation among them is essential. This is usually informally reflected by a graphical representation called the

The mathematical analysis of the model provides a qualitative long-term view of the transmission process, while simulations help to understand quantitatively the short-term behavior of the disease. The model parameters can be calibrated by comparing with empirical data. Subsequent validation of a model also allows us to test its underlying hypotheses. This task may be difficult or even impossible to be done, depending on the quality of the available data. Understanding the transmission process characteristics of a specific disease can help to decide what control measures can be taken in order to prevent its outbreaks.

The purpose of this review is to illustrate the key biological features that should be considered to formulate and analyze infectious processes. While reviewing the existing literature, we provide the main biological aspects and their mathematical translation, although this may be someway incomplete. We will mainly focus on deterministic models in continuous-time, while discrete-time counterparts will be properly referenced.

The review is organized as follows: Section 2 presents the most relevant biological features, which lead to different compartmental approaches and their corresponding transfer diagrams. In Section 3, we revise the most relevant classes of mathematical models, according to the main biological aspects affecting the disease dynamics. In particular, we deal with the inclusion of heterogeneity in infectious disease models. That is, special factors that could be relevant for the modeling of the phenomena, although they make the model more complex than when homogeneously considered. Section 4 briefly describes the mathematical tools and techniques that can be used for a qualitative study of an infectious disease model formulated by a system of ordinary differential equations, while referencing their counterparts for a system of difference equations. Section 5 is devoted to shown some simulation techniques to validate the mathematical model and to estimate the model parameters. In Section 6 we present some control strategies usually considered to prevent epidemic outbreaks and their implementation within the model through new variables or parameters. Finally, we present the main conclusions from this review.

In this section, we review the most important (biological) features that should be taken into account to reflect the disease behavior. More specifically, we describe the main (possible) states of a disease and the forms of transition among them in relation to fundamental parameters. This allows us to establish the class of the disease, according to the states and transitions initially observed. Likewise, other relevant features as demography or other heterogeneity factors are introduced. Some transfer diagrams are shown, as examples of schematic representation. A transfer diagram should be always obtained as a conclusion of the observed biological features.

In compartmental approaches, the population (

Concerning the class

The most common approaches frequently found in the literature are as follows: SI (Susceptible–Infected), SIS (Susceptible–Infected–Susceptible), SIR (Susceptible–Infected–Recovered), SIRS (Susceptible–Infected–Recovered–Susceptible), SEIR (Susceptible–Exposed–Infected–Recovered), SEIRS (Susceptible–Exposed–Infected–Recovered–Susceptible), together with other variants, such as SEIRV, SEIQR [91], etc. For life lasting infections, which describe viral diseases like HIV/SIDA, the usual approach is of type SI. The disease for which a recovered individual acquires lifelong immunity, like measles, rubella, mumps or smallpox are treated via an SIR approach. Diseases caused by bacterial agents (meningococcal meningitis, pest, sexually transmitted diseases, etc.) and protozoa (malaria, zika, dengue, etc.) or even a simple cold, for which no immunity after the infection exists, are dealt with an SIS approach [4, 65, 91].

A disease

The

Of course, other rates or coefficients different from the incidence rate and the recovery rate could be considered as model parameters, if we observe transference between other compartments or states of the diseases under study.

Usually, a compartmental approach leads to a schematic representation by means of a transfer diagram. Each compartment in a transfer diagram is represented by a box labeled with the initial capital of the corresponding population class. Then, arrows indicate the movement of individuals from one compartment to another, depending on the above indicated transference rate, as shown in figure 1 [91]. The transfer diagram in figure 1 represents one of the first compartmental models of direct transmission. It was proposed by Kermack-MacKendrick [81,82,83]. It is a simple epidemic model of SIR type, predicting the evolution of the number of cases of an infectious disease, as it spreads through a population. It represents the first compartmental model properly studied [26,91]. For a historical review of the development of models for epidemics, there exists a wide variety of papers and books describing the different views, such as [16] or [56].

The transmission of the infection is usually firstly described by a

In particular, variations of the total population are referred as

In view of the homogeneous mixing and homogeneous infectiveness oversimplifications, which do not capture other more subtle features of reality, in the last years, other approaches have been developed based on

For instance, one can find in the recent literature models with spatial heterogeneity [14, 22, 49, 88, 97, 103, 105], age-structured models, [4, 7, 15, 31, 33, 67, 74, 91, 117, 119], or seasonal models [17, 76, 79].

Likewise, some infectious agents, such as bacteria, can replicate outside their hosts in environmental reservoirs (e.g. water, food, etc.). Humans acquire the infection indirectly from the pathogen-contaminated environmental reservoir, as is the case of cholera (see [30,57] and the references therein). In fact, even common airborne infections can spread indirectly by a virus transferred through an intermediate object, for instance, contaminated hands or fomites [25]. To reflect this feature, a box is added in the transfer diagram, for the environmental pathogen concentration, which is denoted by _{h}S_{h}I_{M}/N_{M}_{M}S_{M}I_{h}/N_{h}_{h}I_{h}

In addition, other models where, in the transmission of the disease, the virus is eliminated from the infected individual and is acquired by the susceptible [20, 25, 28, 98], models with a saturation effect in the transmission rate, for which a higher density of infected individuals decreases their per capita infectiousness, and situations where multiple exposures to an infected individual are required for an effective transmission to occur [84], have been considered.

This section presents the different formulations to (mathematically) model the main biological aspects mentioned in the previous section. Likewise, this leads to a mathematical model classification, so differentiating deterministic versus stochastic, continuous versus discrete, etc. In addition, we show how the different heterogeneous aspects, which could influence their evolution (e.g. seasonality, age structure, spatial mobility, etc.) can be implemented using various mathematical tools.

Deterministic models are those whose parameters have fixed (proportional) values and whose state variables evolve due to fixed rules, which are continuous functions of time. They are usually established by continuous or discrete-time formulations, i.e. differential or difference equations (see, for instance, [7, 17, 27, 29, 32, 44, 65, 79,91]). They are appropriate if the epidemics occurs in large populations, but they describe the global behavior of complex systems consisting of multiple elements, without taking into account the local interactions between individuals.

Stochastic models instead are more effective for small populations [79]. To formulate such models, one needs to specify a probability law for the time and type transition. In contrast with determinist ones, they allow us to capture the stochastic nature of one-to-one transmission. They make use primarily of Markov chains and probability distributions, with continuous or discrete states. They better account for individual interactions: most of such models are based on cellular automata [118], individual agents [101] and networks [98, 100]. A detailed introduction to stochastic epidemic models and their statistical treatment can be found in [3] and [8], respectively.

On the other hand, in [5], one can find a good comparison between deterministic and stochastic SIS and SIR models in discrete-time.

In the rest of the paper, we focus on deterministic models and mainly in those given by differential equations, although we reference properly the counterparts for those given by difference equations.

In continuous-time models, their variables can take an infinite number of values within a given range.

The vast majority of compartmental deterministic models for studying the spread of diseases are based on the use of non-linear differential equations. For instance, the

However, due to the constraint on the total population size, the evolution of the number of individuals in the infected class

The behavior of this model depends fundamentally on the threshold parameter _{0}, which determines the stability of the disease-free and the endemic equilibria. Another important asymptotic property in epidemic models is the size of the epidemic, defined as the total number of individuals infected at the end of an epidemic, namely _{0} < 1 the infection will be reduced, the number of infected individuals will decrease until disease eradication, and the system will attain an equilibrium state with no infected. If, on the contrary, _{0} > 1, then the infection will spread, i.e. the number of infected individuals will grow, to reach a stable endemic equilibrium, where the disease persists. The number _{0} depends on the parameters involved in the system: for the model (1), it is _{0} = _{0}, during its average infectious period, 1/_{0} ≤ 1, it follows _{0} > 1, from _{m}^{−1}. This maximum is given by
_{0} = (_{0} = _{0}. As claimed earlier, at the initial time the basic reproductive number and the number of replacements are equal.

Based on the model of Kermack and McKendrick, many other types of models have been developed, incorporating more features and details to describe the transmission dynamics of the infectious diseases [44, 65]. A more complex model than (1) may include demographic aspects in the population, i.e. birth and death rates assumed to be equal to each other, _{0} = ^{−1}. Thus, the value of _{0} will depend on the epidemiological characteristics of the disease and the demographic features of the population,

The previous two models consider only direct horizontal transmission, i.e. contacts among susceptible and infected individuals. But for certain diseases such as AIDS, toxoplasmosis or the Zika virus the infected mother transfers the infection to her offsprings, i.e. vertical transmission occurs [32]. Model (2) can be modified as follows to account for this feature. A fraction _{0}, namely

The first model incorporating a nonlinear transmission rate appears in [34]. Formulated to understand the rapid curbing of the cholera epidemics of 1973 in Bari, Italy, it contains a contact rate that decreases as the infected population grows, such as a gamma function

In this way the individuals response to the spread of epidemics is modeled, for which suitable measures are taken to avoid possible contacts with infected individuals.

A good reason for working with epidemic models in discrete-time is that data are collected and/or reported in units of discrete-time and the models can be adjusted to the collected data [2]. In addition, numerical scanning of discrete-time epidemic models is quite simple and can, therefore, be easily implemented also by nonmathematical experts [27]. The advantage of discrete-time over continuous-time modeling is that the former one allows an easier comparison of their results with empirical data, which are given discretely. Further, these types of models are best adapted for populations that have several generations a year. For instance, insects reproduce several times in a summer, so that _{k}

A discrete model to describe the evolution of West Nile encephalitis with weekly data from New York City in 1999 is proposed in [120]. It allows us to determine the specific amounts of periodic fumigation which are needed for virus eradication. A comparison of the similarities between single-outbreaks in classical continuous-time epidemic models and their discrete-time counterparts, as well as the final size of an epidemic are studied in [29]. A discrete model for the transmission of Babesiosis between cattle and ticks shows that similar conclusions can be obtained for both discrete and continuous models. However, the former one needs some parametric restrictions that are not present in the continuous version. But the parameter constraints are natural and reflect the epidemiological robustness of the model [12]. The model is described by a system (4) of non-linear difference equations

The transmission and control of the SARS infection in China was studied in [133] via a discrete mathematical model whose numerical simulations fitted the real data well. An age-structured model for the spread of tuberculosis in China, where the discrete-time intervals correspond to the reproductive periods or the average duration of a particular stage, can be found in [33].

While global models study the behaviour of complex systems composed of multiple elements, disregarding the local interactions between individuals, individual-based models, on the contrary, take these individuals interactions into account. Network modeling approaches can be useful to incorporate a heterogeneous contact structure. They are most useful when each individual is in contact with a small fraction of the population [20,78].

The first network-based epidemic approaches were developed to study the spread of sexually transmitted diseases, the so-called

The mobility of the population affects to a large degree the speed of transmission of the disease. To study the relation between the mobility of the population and the propagation of the disease, space must be included in the mathematical description [14,97]. In a compartmental model, this can be reflected by partitioning the population into spatially different sites (e.g. cities, countries, towns), called

Metapopulation models were first introduced in ecology in 1997 [64]. The patches are connected through transference of individuals. This approach has been effectively extended also to modeling infectious diseases [14,79]. For epidemic metapopulation models, an adequate modified version of the SIR epidemic model is given in (5)

Space can be incorporated as an independent variable on a continuous or discrete basis. Continuous spatial models are based mainly on partial differential equations, concretely, on reaction-diffusion equations [22, 97, 105]. However, more recently, in reaction-diffusion models, the nearest neighbor contact, as the main infection transmission mechanism, has been considered. In this approach, the infection is considered in a population of immobile individuals, and the propagation occurs by a local contact mechanism between individuals and the part of the population nearest to them [103].

To describe the spatial structure in the SIR model (1) discussed above, we can consider that the individuals move randomly in a limited one-dimensional habitat 0 ≤

Spatial heterogeneity leads to consider the possibility of heterogeneous contacts Recently, different approaches based on epidemiological networks have been developed to explicitly model the heterogeneity of disease contacts independently of host contact rates. Each individual has a finite number of potential interactions by which the infection can be transmitted. This contact structure profoundly affects the dynamics of the infection [20, 78, 100].

Diseases affecting interacting populations have been reviewed in [129].

Several characteristics could contribute to population heterogeneity. A disease can present a differential degree of infectivity depending on the social, ethnic, sexual or age group. In particular, age usually gives a source of heterogeneity and, due to that, age structure models are commonly studied. One of the most important characteristics in the modeling of infectious diseases is the age, because individuals respond differently to diseases according to their age. Age, of course, usually influences the single individual reproductive and survival capacities. But this occurs also for diseases. In general, older individuals are indeed weaker than young ones to fight against an infection. The same could be said for newborns. As diseases may have different infection rates and mortality rates for different age groups, the risk of acquiring and transmitting a contagious disease through multiple pathways changes with the biological age [4, 79, 91]. We must not confuse biological age of individuals, with the “age” of the disease, i.e. the time elapsed from the moment in which the individual has become infected. A model that considers both biological age as well as the age of the disease, i.e. the stage of the disease is presented in [128].

Models incorporating population heterogeneity may exhibit fundamentally different behaviors compared to homogeneous infectivity models. These models can be used to compare, for instance, different alternative age-specific immunization programs (see Figure 5). A broad spectrum of age-structured compartmental models can be found in [4, 79].

Age structure can be described as a continuous variable. In fact, this description already appears in the first epidemic model [93]. This approach leads to hyperbolic partial differential equations models [7, 44, 90]. Thus, these equations are analytically more difficult to be handled than ordinary differential equations. However, a numerical solution can be obtained by finite difference schemes [91].

Let us introduce the population density in age and time _{1},_{2}] at time ^{0}(^{0}(^{0}(

Assuming intracohort mixing [79], the infection rate is given by _{1}(_{2}(

Compartmental models can be adapted to reflect age groups [15, 79, 117]. Figure 5 shows a compartmental model with just two age stages; juvenile and adult. This age-structured deterministic SIR model is represented by the following system (8) of differential equations:

A broad spectrum of models using this approach can be found for instance in [4, 79, 108,109,110].

Although in traditional literature it does not appear as an heterogeneity factor, it is obvious that vertical transmission and indirect transmission by means of vectors or fomites, leads to a non-homogeneous infectivity. This is the case of the majority of vector-host diseases where, although homogeneously distributed, infected hosts cannot infects other hosts, but only vectors (see for instance [11, 92, 106]. Nevertheless, the modeling of these kinds of heterogeneity can be usually set out by means of new state variables and incidence rates when transmission is possible. For instance, the following system (9) of ordinary differential equations was the first established one for modeling the Babesiosis on bovine and ticks populations [11].

In this model, the variables with subindex _{B} represent the bovine population, while the variables with subindex _{B}_{T}_{B}_{T}_{B}_{B}

The transmission of childhood infectious diseases has been proven to vary seasonally, peaking at the beginning of the school year and decreasing significantly in the summer months [17]. Human influenza exhibits seasonality in temperate zones and tropical climates, showing seasonal patterns of infection. Influenza and measles epidemics tend to occur during the winter, and in tropical regions. Hence, these diseases have a seasonal variation that seems to have a strong relationship with the local factor. Differences in the efficiency of influenza transmission at different levels of absolute humidity have been postulated as a possible reason for seasonal variation in influenza transmission. Possible drivers of seasonality are often location-specific. Diseases transmitted by vectors (malaria or dengue) are often highly seasonal [50]. In [79], a detailed literature review of this compartmental modeling approach is provided. It is therefore generally assumed that climatic conditions or seasonal variations lead to seasonal variation in incidence.

The seasonality of propagation is modeled by incorporating periodicity, between susceptible and infected individuals, by a variable contact rate

Let

The reformulation of the SIR model (1) via a system of fractional order differential equations becomes as shown in (10).

This section reviews some mathematical tools and methods used to carry out a qualitative study of a (deterministic) epidemic model, mainly of those given by a system of differential equations.

For the analysis of epidemic models described by differential equations, where _{0} representing the basic reproduction number does not exceed the unity, _{0} < 1, the only equilibrium is the disease-free equilibrium and this equilibrium is asymptotically stable. On the contrary, if _{0} > 1, the usual situation is that the disease-free equilibrium is unstable and there is a unique endemic equilibrium that is asymptotically stable. This is commonly known as the

We present a brief review of the main mathematical tools and methods needed in epidemiology, focusing just on the models described by a equation like in (11). For further information, see for instance [68, 102] and the references therein. We will deal with the following topics: positively invariant regions, equilibria and their local stability, global stability and finally bifurcation and existence of periodic solutions.

A solution for the system (11) is a function ^{n} such that _{0}),_{0}),_{0}),_{0}) ≥ 0 for the system (11). Let

The positive invariance of the non-negative orthant
_{i}_{1},_{2},...,0,...,_{n}

More generally, let Ω be a region in

When analyzing a system of the form (11), the first step is usually to determine its equilibria. An equilibrium is a (constant) point ^{*} satisfying the equation. Since it does not vary in time, it can be found by solving the nonlinear (algebraic) system ^{*}) = 0, obtained by annihilating the derivatives on the left hand side of (11).

Secondly, one must assess the stability of the equilibria. An equilibrium ^{*} of (11) is stable if any solution starting at _{0} sufficiently near this equilibrium point remains in its neighborhood (stability) or tends back to the equilibrium (asymptotic stability) as

Note that exogenous disturbances have important effects on equilibria, but they are absent in the case of autonomous systems, as it is the case here. In general, time-dependent forcing functions alter the system behavior and determine whether or not the steady state will persist under the inevitable disturbances that occur in any real situation being modeled.

^{*} be an equilibrium of the system (11) and let ^{*}) = _{x*} denote the Jacobian matrix of ^{*}. The ^{*} is

For the assessment of the nature of the roots of (13) the Routh-Hurwitz criterion can be used [72,91,107,125]. They give a set of necessary and sufficient conditions for stability. In particular, for the case ^{*}) of 2 × 2 have real negative parts if and only if
_{2} denotes the sum of all the principal minors of order 2 of ^{*}). In the discrete-time case, the Jury criterion offers similar conditions to determine the modulus of the eigenvalues at an equilibrium or fixed point [27, 77, 91].

In compartmental approaches of diseases, three important disease _{0}, the _{0} represents the most important threshold for many epidemic models, being defined as the average number of secondary infections produced by an infected individual in a totally susceptible population. This parameter often represents the threshold determining whether a disease can invade and persist in a population [127]. More specifically, if _{0} > 1 an epidemic outbreak will occur, while if _{0} < 1 the disease will become extinguished. If _{0} = 1, each individual will replace himself and there will be no epidemic outbreak [48]. After the spread of the disease, the fraction of susceptible individuals is less than 1. Thus, not all suitable contacts turn out to be a new case. When populations include infected individuals of different types, a sophisticated approach defines _{0} as the maximum eigenvalue of the _{0} is calculated at the beginning of the spreading of the disease, while _{0} ≥

Given a continuous system of the form (11) in the plane, i.e.

We now present some criteria to prove non-existence of periodic or closed orbits. This situation implies that if there is only one equilibrium point in Ω, it turns out to be globally asymptotically stable [32].

A periodic solution is a function such that ^{n}, this solution is represented by a closed curve because the path starting at _{0}) returns to this point after a time

Consider a bidimensional system. Let _{1}(_{2}(_{n}_{n}_{n}_{n}_{n}_{n}_{n}

If a curve _{0}) can contain a fixed point or closed orbits of the flow.

This theorem can be used to show the global asymptotic stability of an equilibrium point if periodic orbits can be ruled out in a positively invariant set. A technique to achieve this result is the Bendixon-Dulac criterion [68].

^{2}. Then,

As mentioned, the Poincaré-Bendixson theorem holds only in dimension 2. For higher order nonlinear systems, it is not valid. The system can exhibit many more complicated dynamic behaviors. Although in the last years some researchers have obtain generalization for systems with higher dimensions [54, 89, 116], the global analysis of the trajectories is much more difficult to assess.

An efficient method that can sometimes be applied is the

If

If

If

Let ^{n} an open set. A function ^{*} the maximum invariant subset of the set {_{0}) decreases along a system solution of (11).

^{*} = {_{0} ∈

This result is important in the stability analysis of dynamical systems. It provides a tool to prove global asymptotic stability when ^{*} is locally stable and ^{*} is globally asymptotically stable in the region

This method applies when the equilibrium point is located at the origin, but if ^{*} ≠ 0, the change of variables
^{*} shifts to the origin, i.e. ^{*}) = 0. At this point, the method could be applied.

The Lyapunov functions method can be used to establish global stability results for epidemic models [53]. However, it is often difficult to be applied, since there is no general technique to build such Lyapunov functions. One of the general forms of Lyapunov functions used in the literature is
_{i}

Similar techniques have been recently found out to deal with discrete-time systems. One can check these results in detail in [73]. If the disease model possesses a network structure, in particular if it can be considered as a coupled system in which each subsystem has a Lyapunov function, then the

In this section, we address the question of how the qualitative behavior of (11) changes as a function of parameters associated with the description of the vector field _{1},..., _{m}^{m} represents such parameters. If the qualitative behavior remains the same for all nearby vector fields, the system (family) given by ^{m}, is said to be

If a particular vector field

In epidemic models, bifurcation analysis is used to establish the feasible epidemic evolutions. The most common situation is a transcritical bifurcation at the parameter value _{0} = 1, with the appearance of an asymptotically stable endemic equilibrium with an infected individuals amount that continuously depends on _{0} [38, 55, 69, 91]. Here, a change of stability occurs between the disease-free equilibrium, which is asymptotically stable for _{0} < 1 and unstable for _{0} > 1, and the endemic equilibrium, which, in contrast, shows the opposite stability behavior, although it is biologically significant only for _{0} > 1.

Another simple way in which non-linearity can lead to periodic behavior is the

The bifurcation behavior can be described graphically. It contains the size of the dependent variable at equilibrium as a function of one of the bifurcation parameters. For epidemic models, the bifurcation curve is the graph that plots the values of the infectious population _{0}.

In some epidemic models with multiple sets and asymmetry between sets or mechanisms of multiple interactions, it is possible to have different bifurcation behaviors at _{0} = 1 [32]. Also, there can be several positive endemic equilibria for values of _{0} < 1 [80] and a subcritical (also called _{0} = 1 [24].

The qualitative behavior of a system with a subcritical (backward) bifurcation differs from the one of a system with a supercritical (forward) bifurcation [62]. The epidemiological consequence of a subcritical bifurcation lies in the fact that the condition _{0} < 1 is necessary, but not any longer sufficient, to ensure disease eradication. Therefore, the presence of this phenomenon in the disease makes the control of propagation difficult [62]. Figure 6 shows the bifurcation diagrams for supercritical and subcritical bifurcations, respectively [62].

An epidemic model has periodic oscillations if and only if the model is cyclical and the disease has temporary immunity by which the return to susceptibility of an individual can be significantly delayed [62]. Several epidemiological mechanisms may lead to periodic solutions. The most direct way in which the periodicity arises is through the contact rate, but periodicity can also emerge autonomously [76]. Infectious disease models with non-linear incidences of certain general forms may have periodic solutions. Some models with variable population size and disease-related deaths have periodic solutions. Most of them are vector-host models, where the vector's life is much shorter than the one of the host [50].

Recently, periodic solutions have numerically been found in age-structured models with cross-immunity between two viral strains [66]. For more details, see also [51, 91].

In general, after a qualitative study of an epidemic model a quantitative study is needed as well, to confront the theoretical results with the real data and to validate the model. To this end, numerical simulations are performed to observe the short-term behavior of the disease spread in the host population. Classical adaptive numerical methods for solving ordinary differential equations can be used, such as the Runge-Kutta-Fehlberg method, or other routines implemented for instance in MATLAB, e.g. ode45 or ode15s for stiff problems [11, 12, 59, 104, 121].

To run simulations, parameter values which have to be estimated directly from epidemiological data are used. In a mathematical model, it is important to investigate the relevance of each parameter for the system outcome. The objective of sensitivity analysis is to assess the most influential parameters on the dynamical evolution of the system [91]. In epidemiological models, this is especially relevant for the disease transmission dynamics. The parameters of an epidemic model, such as the transmission coefficient in the incidence term and the basic reproduction number _{0}, can be estimated from data points collected at successive moments during an outbreak. If reliable annual prevalence data are available for a specific disease, an estimation of these parameters can be obtained by finding the best fit of the model outcome to the available data, using standard optimization techniques. They usually minimize the sum of the square errors (SSE); for this purpose several Matlab or Mathematica routines can be used [91]. An alternative approach to solve the least-squares problem is to use a genetic algorithm [59].

For outbreaks of an emerging disease, the main disadvantage of estimating parameters is the long enough time that must be elapsed for collecting data. Also, stochastic effects are important, and need to be taken into account particularly at the start of an outbreak [45, 91].

Once the dynamics of a model is known, one can search for control strategies that are relevant for each specific epidemic.

One of the most effective ways to prevent and control a disease is to reduce contacts. However, the emergence and re-emergence of infectious diseases, climate change, population growth, unplanned urbanization and the ability of humans to migrate on a large scale have increased. Therefore, it is not easy to reduce interactions among individuals. Vaccines and medical treatments are the two widely used prevention tools that can potentially reduce transmission of a disease and control its spread. Prevention and control measures are used to keep the number of infected individuals as low as possible. Other measures may include isolation or quarantine, education, and biological control. They lead to a reduction in the number of medical treatments, hospitalizations and absences from work due to illness, so contributing to a reduction of total costs, which is the main goal of adopting alternative measures.

Mathematical models can help to quantify the benefits of possible prevention scenarios. Actually, it results as necessary to accurately assess the benefits of these programs. Different mathematical approaches that include vaccination are discussed for different infections in [21]. For the human papilloma virus, vaccination is shown to be effective in reducing the number of infected and the number of fatalities [117].

Another form of intervention is through education-induced behavioral changes. They can alter the dynamics of an outbreak of a disease. For instance, for the Ebola virus in Sudan, a deterministic model for a community partitioned into two types of individuals, those that are educated about Ebola and take precautions to avoid contracting the disease, and other ones not taking precautions, shows the importance of education as a preventive measure [87]. A transfer diagram for the inclusion of two compartments representing the intervention by education in an SIR model is shown in Figure (7). The effects of public health education on the evolution of a disease have also been studied for HIV and the Hepatitis C virus transmission [87].

Isolation and quarantine were proved as possible strategies to stop the spread of SARS in Toronto [37]. Early treatments to significantly reduce the transmission of new HIV infections are discussed in [104]. A mathematical model for the prevention of the avian influenza pandemic with three controls (namely, isolation of infected individuals, antiviral treatment of infected individuals, and elimination of infected birds) is presented in [123].

As we mentioned in the previous sections, _{0} allows us to determine the severity of a disease and to study the qualitative behavior of the epidemic model. It can be quite easy to estimate the average duration of the infection and the initial number of infectious individuals in order to produce an estimation of _{0}.

Different prevention and control strategies can be implemented, e.g. vaccination at the beginning of an outbreak. If a perfect vaccine is available for a specific disease and a fraction

In this context, the disease will not spread if (1 − _{0} < 1. This indicates that if a fraction
_{0} require a large proportion of the population successfully vaccinated in order to be eradicated. Smallpox is the only human disease that currently has been eradicated, due to vaccination campaigns [91].

The different mathematical approaches useful for capturing a diversity of early epidemic growth profiles, obtained from empirical data, ranging from sub-exponential to exponential growth dynamics, are presented in [40]. Early epidemic forecasts consisting of the likely short-term evolution of an ongoing outbreak can help guide the type and intensity of interventions, including diagnostic needs of health infrastructure, isolation of infected individuals, and contact tracing activities.

Recently, optimal control theory has been employed to find the best combination of the different disease-curbing alternatives in such a way that the control, the economic and the social costs are minimized. Generally, the optimal control problems for real life applications found in the literature can only be solved numerically [91]. Using the Pontryagin Maximum Principle, the necessary optimality conditions for the existence are usually derived (see [113]).

The prevention of the avian influenza pandemic has been studied by adjusting three control functions in the human-to-human transmission model to minimize the impact of the disease. They consist of isolation of infected individuals, antiviral treatment of infected individuals, and elimination of infected birds [123]. A new deterministic model to assess the population-level impact by quarantining individuals suspected of being exposed to the disease during the ebola outbreaks in 2014–2015 is presented in [43]. Early treatments to significantly reduce the transmission of new HIV infections are introduced in [104]. The optimal policy for distributing chlorine tablets for water purification during a cholera outbreak to assess the fraction of susceptible individuals who should have access to them in order to minimize the total number of new infections is investigated in [86], with an application to real data of the cholera outbreak in Yemen in 2017–2018.

A preventive control for ebola in the form of an education campaign and two treatment controls applied to an infected and advanced stage of the human population are shown to lead to a reduction of the ebola infection, giving a good framework for planning and designing cost-effective strategies for good interventions in the treatment of this disease [23]. A mathematical model with optimal strategies for dengue reduction and prevention in Cali, Colombia is presented in [114]. A new version of an optimal control problem for a vaccination strategy with two dengue serotypes appears in [41].

Another type of control strategy that can be implemented is the feedback control [39, 42]. The results show that, when appropriately choosing the feedback control variables values, the endemic disease may be eradicated. Alternatively, it may remain endemic, but the level of endemicity can be kept at a low value.

All these measures, including population-level immunity as well, taken jointly or separately, can provide useful information for the proper implementation of effective treatment programs for a pandemic.

In this brief review paper, different compartmental paradigms that are generally used in mathematical epidemiology were discussed. Biological aspects used to make a qualitative analysis of the model were also described to support the relevance of this features for understanding the dynamics of diseases modeled by ordinary differential equations.

The vast majority of the contributions in the literature for the spread of infectious diseases that have been discussed are of a deterministic nature, continuous in time, and modeled by ordinary non-linear differential equations or by difference equations. Furthermore, these models do not simulate the individuals dynamics, for which individual-based or network (stochastic) models would be more appropriate. Nonetheless, the latter are more complex to be analyzed. Stochastic approach is of importance for modeling the probability of contact with the infected as well as other relevant biological features such as the immune system of each host, and so on.

Mathematical models may effectively help in assessing various biological features to implement or suppress heterogeneities, control measures and environment conditions, that are key links in the dynamics of disease transmission models. Consolidating our understanding of the evolution of different modeling approaches helps in the development of new models, by incorporating different biological aspects, and finally contributes to improve predictions and to take more realistic control measures. They will require the development of innovative and manageable mathematical and statistical tools and techniques for their overcoming. Contact networks, spatial heterogeneity and early epidemic growth profiles represent some of these shortcomings at the present state of knowledge.

Collection and use of large empirical data sets for parameter estimation is another relevant current challenge. Efforts to improve the modeling of disease transmission patterns will provide a better understanding and, as a consequence, better prevention strategies, having a positive impact on the calibration of the intervention and contingency plans, in order to face the threats of the emergence of new infectious diseases, and the reemergence of old ones.

_{0} in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28(4), 365–382._{0} in models for infectious diseases in heterogeneous populations

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