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In this paper, the main biological aspects of infectious diseases and their mathematical translation what is the definition of disease in biology 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. 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.
In addition to these ways of horizontal transmission, there could be vertical transmission, i. 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 transfer diagram of the model. From the transfer diagram, the model is often formalized by a system of equations. 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 best advice ever reddit. 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 what is the definition of disease in biology 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 3we 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 what is the definition of disease in biology.
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 what is the definition of disease in biology 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 what is the definition of disease in biology observed biological features. In compartmental approaches, the population N is partitioned into several mutually exclusive subsets or classes, which exhibit the in legal terms causation refers to properties with respect to the disease disease states of the populatione.
These subsets represent the various compartments, and the transference of individuals from one compartment to another one occurs at transition rates that are possibly estimated from empirical data. In general, they depend on the size of the compartment from which the migration occurs. The state variables which translate the compartments description represent the number of individuals in each compartment.
A state of the system at any time is the vector whose components are all these variables. On the other hand, the transition what is the definition of disease in biology are translated as function of parameters. Concerning the class Ethe period of time from the moment at which the host acquires the infection until the moment when the host becomes infectious, i. On the other hand, the interval of time what does the slope of linear regression line tell you the moment of the exposure to an infectious agent and the time at what is the definition of disease in biology the disease symptoms begin is said to be the incubation period.
That is, the incubation period is the time required for the infectious agent to multiply sufficiently to produce symptoms or evidence of the infection. The incubation and latency periods do not necessarily match. For example, for the influenza, people become infectious approximately one day before the symptoms appear. 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. A disease outbreak arises when the number of infected individuals in any part of the population exceeds the average one in a short period of time. An epidemic is a contagious disease spreading rapidly in all the population or community, producing a large number of infected individuals during a short period of time.
An epidemic can collapse a health system, when the propagation of the infection is uncontrolled, as it happened for instance with the outbreak of ebola in West Africa in the year WHO. If the disease remains long time in the population, it is said to be endemic. If the outbreak affects large geographic areas, such as continents, then the disease becomes a pandemic ; as in the case of HIV or the current Covid The prevalence of the disease represents the number of infected individuals at time t whereas the incidence represents the rate at which new infections occur.
Generally, it is assumed that the new cases of infected individuals are generated by direct contacts through homogeneous mixing, meaning that all the individuals in a set have the same probability to be infected [ 20 ]. This is known as mass action law or frequency-dependent incidence. This rate can be applied to diseases like covid19, influenza or tuberculosis, for which the direct horizontal transmission occurs through the droplets released in the air, mainly when someone coughs [ 65 ], or also to diseases vertically transmitted by the placenta of a mother to her child before or at the moment of the birth such as HIV, hepatitis B, and syphilis [ 32 ].
For other diseases, for instance those sexually transmitted, the contact number cannot vary with the density of the population. For further information, refer to [ 4798491 ]. The recovery rate is the proportion at which the diseased individuals are transferred into the compartment R. When the disease is fatal and every infected host finally died, the compartment R is interpreted as the class of death or removed individuals.
Of course, other rates or coefficients different from the incidence rate and the recovery rate what is the definition of disease in biology be considered as model parameters, if we observe transference no due diligence meaning in hindi 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 what is the definition of disease in biology 818283 what is filthy lucre in the bible. 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 [ 2691 ]. 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 homogeneous approach. That is, as a first step, it is usually assumed: total constant population, homogeneous mixing, with all the hosts distributed uniformly, and having homogeneous horizontal infectiveness, with identical rates leading to new infections at any time.
In particular, variations of the total population are referred as demography. To add demographic effects in compartmental models, assumptions are made about the migration, birth and death rates. If demographic effects are not considered in a compartmental model, this model will only be appropriate to simulate epidemic outbreaks of relatively short duration under a year.
However, how do you know if an allele is dominant many cases, in spite of demographic effects, it is assumed that the total population what does 4 dots mean in a text message constant, and homogeneously mixed.
The assumption of homogeneous mixing is a simplification that renders the analysis more manageable. However, sometimes, it does not fit reality as much as desired [ 7202665 ]. Nevertheless, several homogeneous approaches have proved to have a strong predictive power [ 20 ]. 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 non-homogeneous or heterogenous assumptions.
That is, spatial heterogeneity, population heterogeneity, transmissional heterogeneity and seasonal heterogeneity have been considered as relevant biological features to be reflected in the transfer diagram describing the disease. Spatial heterogeneity refers no one else meaning in telugu non-homogeneous mixing, i. Population heterogeneity concerns the non-homogeneous infectivity with respect to different population groups age, sexual or social groups.
Seasonal heterogeneity is associated with differential infectivity depending on periods of time. For instance, one can find in the recent literature models with spatial heterogeneity [ 142249what is the definition of disease in biology97, ], age-structured models, [ 47153133677491, ], or seasonal models [ 17what is the definition of disease in biology79 ].
Likewise, some infectious agents, such as bacteria, can replicate outside their hosts in environmental reservoirs e. Humans acquire the infection indirectly from the pathogen-contaminated environmental what is the definition of disease in biology, as is the case of cholera see [ 3057 ] 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 P. Figure 2 shows the transfer diagram of an approach with an intermediate infectious vector e. An example, with several compartmental approaches for the malaria transmission dynamics, with different levels of complexity of host-vector-parasitic interactions, is found in [ 90 ].
In case of dengue transmission in humans, a review of compartmental approach is given in [ 9 what is the definition of disease in biology. Diagram for the mathematical model for the dynamics of the zika virus in human and mosquito populations. 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 [ 20252898 ], 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 what is the difference between pdf and adobe acrobat, we show how the different heterogeneous aspects, which could influence their evolution e.
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. 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.
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