2.3: What is a model? - Biology

2.3: What is a model? - Biology

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Science strives for simplicity, and models are part of the process. What is a model? It is just a simplified view of something more complex.

The word “model” is used here essentially as it’s used in everyday English. For example, in ordinary English, “modeling clay” can be used to make simplified miniatures of three-dimensional images of animals, automobiles, buildings, or even full-scale three-dimensional images of objects like the human heart. A “model airplane” can be rendered to show at a glance the physical appearance of a large aircraft, and can even be constructed to fly so as to test aerodynamics under proper rescaling. A “model organism” is a simpler organism that may respond to medical tests or treatments in ways similar to those of a more complex organism.

Even the fashion model on the runway meets this definition of a simplified view of something more complex. The infinite complexity of the human spirit is not relevant on the runway; all that is relevant in this context is the person as a realistic way to display fashions.

This book focuses on computational and mathematical models of ecological systems. What is left out of these models is as important as what is put in. Simplification is key.

If you have a complex natural system you don’t understand, and you construct a computer model incorporating everything you can about that natural system, you now have two systems you don’t understand. — after Chris Payola, UMN

A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away. — Antoine de Saint-Exupery

Two different simplifications of time are commonly used in ecological models:

  • Discrete time — Events happen at periodic time steps, as if time is non-existent in between.
  • Continuous time — Events happen smoothly and at all times.

In addition, there are two different classes of models:

  • Macroscale — Individual organisms are not tracked, but are measured in aggregate and represented by composite variables such as N.
  • Microscale — Individual organisms are tracked separately. These are also known as agent-based or individual-based models.

Macroscale models can be handled either by computers or mathematics, but microscale models are usually restricted to computers. Keep in mind that all four categories are only approximations of reality. Later in this book we will also explore mechanistic versus phenomenological models.

ABC Model of Flower Development | Plants

In this article we will discuss about the ABC model of flower development.

The ABC model of flower development in angiosperm demonstrates the presence of three classes of genes that regulate the development of floral organs. The genes are referred to as class A genes, class B genes and class C gene. These genes and the interaction between them induce the development of floral organs.

Many literatures on molecular genetics and Internet Websites provide articles on ABC model. In the following essay the basic concept of ABC model will be discussed in brief. The analysis of ABC model is based on the use of molecular genetics and formulated on the observation that mutants induce right floral organs to develop in wrong whorls.

In the flower of angiosperms there are usually four concentric whorls of organs, i.e. sepal, petal, stamen and carpel that are formed in whorl 1, whorl 2, whorl 3 and whorl 4 respectively, the whorl 1 being on the peripheral side.

In the whorl 1 class A genes when expressed induce the development of sepals. The interaction between class A and class B genes induce the development of petals in the whorl 2. Stamens are formed in the whorl 3 as a result of interaction between class B and class C genes.

In the whorl 4 class C gene induces the formation of carpel. So the summary of ABC model is: class A genes together and class C gene alone are responsible for the development of sepals and carpel respectively. The class B genes and class A genes function cooperatively to determine the development of petals. The class B genes and class C gene act together to induce the development of stamens (Fig. 30.12).

Coen et al. (1991) formulated the ABC model. While analyzing the mutations affecting flower structure Coen et al. identified the class ABC genes that direct flower development. They also formulated the molecular models of how floral meristem and organ identity may be specified. They have shown that the distantly related angiosperm plants use homologous mechanisms in pattern formation of floral organs. Ex. Arabidopsis thaliana and Antirrhinum majus.

The following two have led to formulate ABC model:

(1) The discovery of homeotic mutants (homeotic genes identify specific floral organs and help the organ to develop in respective whorl. The homeotic mutant has inappropriate expression—that is, it induces right organ to develop in wrong whorl. As for example — petals emerge in the whorl where normally stamens develop).

(2) The observation that each of the genes that induce the formation of an organ in a flower has an effect on two groups of floral organs, i.e. sepal and petals or petals and stamens.

Class A, B and C genes are homeotic genes. They determine the identity of different floral organs and induce the organs to develop in their respective whorls.

The homeotic mutants have defects in floral organ development and induce the right organs to develop in wrong whorls/place, i.e. one floral organ develops in the whorl, which is the normal position of another floral organs. Petals, for example, develop in the whorl where stamens are normally to be formed.

In each whorl of a flower there is one or more homeotic genes and their cooperative functions determine the organ to be formed in that whorl. For example, the activity of class A genes is restricted to whorls 1 and 2. The class B genes have function in whorls 2 and 3. The class C gene functions in whorls 3 and 4.

Another way of describing the function of class A, B and C genes is that—in whorl 1, the class A gene-function alone determines the formation of sepals in whorl 2, class A and B gene-functions both determine the formation of petals in whorl 3, class B and C gene-functions both determine the emergence of stamens and in whorl 4, class C gene-function alone determines the carpel formation.

In Arabidopsis there are two genes in class A, two genes in class B and one gene in class C (Table 30.1). The most characteristic feature of these homeotic genes is in the identification of floral organs and in the determinacy of position / whorl of their emergence in a floral meristem. The two genes of class A and the two genes of class B act cooperatively.

The function of class A genes is confined to whorls 1 and 2. Similarly the function of class C gene is restricted in whorls 3 and 4. This can be interpreted in another way. In the whorls 1 and 2 the function of class A genes prevents class C gene from functioning in the same whorls. Similarly the function of class C gene prevents class A genes from functioning in the whorls 3 and 4.

Any mutation in class A genes with defects in floral organ development will invite class C gene to express in whorls 1 and 2. The class C gene, in class A mutants, will express in whorls 1 and 2 in addition to the normal whorls 3 and 4.

Similarly any mutation in class C gene with defects in floral organ development will lead to the encroachment of the function of class A genes. The class A genes will express in the whorls 3 and 4 in addition to the normal whorls 1 and 2.

The following three examples of homeotic mutant genes will illustrate the above discussion (Fig. 30.13):

(1) The flower of Arabidopsis with class A mutants, such as apetala 1(ap 1) shows the following pattern of floral organs (Fig. 3.13.II): whorl 1 shows bract-like structure with carpelloid characteristics whorl 2 shows stamens whorl 3 shows stamens and whorl 4 shows carpel.

The pattern of floral organ formation in whorls 1 and 2 is changed. In ap 1 mutants the activity of two genes of class A is lost. So the class C gene expressed in whorls 1 and 2 in addition to whorls 3 and 4. As a result carpelloid organ developed in whorl 1 and stamens formed in whorl 2. In the whorls 3 and 4 stamens and carpel respectively are formed similar to wild type (Fig. 30.13.I).

(2) Example: Flower of Arabidopsis with class B mutant, such as apetala 3 (ap 3): The flower shows sepals only both in whorls 1 and 2, while the whorls 3 and 4 show carpel only (Fig. 30.13III). Class B mutant contains loss-of-function genes and as a result class A genes express in whorls 1 and 2 and class C gene alone expresses in whorls 3 and 4. In ap 3 mutants in whorl 2, sepals are formed instead of petals and in whorl 3, carpel is formed instead of stamens.

(3) In Arabidopsis the class C gene contains the sole gene agamous (ag). Arabidopsis flower with agamous (ag) mutant consists of many sepals and petals. The reproductive organs – stamens and carpel are not formed in the whorls 3 and 4. Class C gene with ag mutant contains loss-of-function gene. As a result class A genes express in whorls 3 and 4 in addition to 1 and 2. In ag mutant sepals and petals are formed in whorls 3 and 4 instead of stamens and carpel. The literature of Howell provides the scan electron micrograph of flower phenotypes of the floral homeotic mutants of class A, B and C genes.

In Arabidopsis it was observed that in all the mutants one homeotic gene remains functional in each whorl. The flower with class ABC triple mutant shows sepals in each whorl. In ABC triple mutant, the genes required for floral organ formation become nonfunctional. As a result sepals or leaves are formed in each whorl, as homeotic mutants specify no floral organs. This observation led Botanists to regard ‘flowers as modified leaves’ on the basis of molecular genetics.

The important feature of ABC model is that it can predict the type of floral organ to be induced to develop in any whorl. Krizek et al. (1996) was successful to induce any one of the four different floral parts in whorl 1 of Arabidopsis flower. This became possible by genetic manipulations of right combination of homeotic selector genes.

The ABC model appears to be simple, but a completely different picture is obtained when it is analyzed on the basis of molecular genetics and in molecular terms.

The analysis includes the structure of different classes of homeotic genes, the homeotic mutants, the co-operative function between homeotic genes, mutual exclusion in the expression of class A and C genes in the same whorl, the identification of floral homeotic genes and their isolation by cloning, the production of MADS box protein by homeotic mutants, the study of genes that mediate the interaction between floral meristem and floral organ development, presence or absence of different classes of transcription factors etc., the details of which can be obtained in the literatures on molecular genetics.

Arabidopsis thaliana belongs to the family Brassicaceae and has become the model organism for understanding the genetics and molecular biology of flowering plants like mice and Drosophila in animal researches due to following reasons:

(i) It has five chromosomes (n = 5) and so this small-size-genome is advantageous in gene mapping and sequencing.

(ii) The size of plant is small and so can be cultivated in a small space and requires modest indoor facilities.

(iii) It has rapid life cycle and takes about six weeks from germination to mature seeds.

(iv) An individual plant produces several thousand seeds.

(v) ‘The Arabidopsis genome is among the smallest in higher plants, with a haploid size of about 100 megabases (mb) of DNA. With a small genome size it was expected that there would be fewer problems with gene duplication’— Howell.

(vi) It is easily transformable with T-DNA mediated transformation.

In 2004 ABCE model has been formulated. The characterization of sepallata 1, 2, 3 triple mutants in Arabidopsis has led to the above formulation. It is regarded that the class E genes have important role in the development of floral organs.

Celiac Disease—An Autoimmune Entity?

Nitza Lahat , . Aaron Lerner , in The Decade of Autoimmunity , 1999

2.3 Non-HLA Genes

Genetic models based on results obtained from research of CD families implicated non-HLA genes in genetic predisposition and etiology of CD ( Houlston and Ford, 1996 ). As CD is described as an abnormal T-cell mediated response to exogenous antigens, genes outside the HLA system which could potentially contribute to disease susceptibility, are those which influence this immune response. Although no association has been found between CD and T-cell receptor polymorphism, or TAP alleles within the HLA system, other genes involved in determining the T-cell immune response, such as the genes coding for cytokines, cell adhesion molecules, etc., could be implicated in CD. Indeed, association of CD with microsatellite polymorphism within the Tumor necrosis factor (TNF) genetic locus, in the HLA class III locus was recently observed ( McManus et al., 1996 ). TNFβ3 and TNFα2 were expressed in significantly higher percentage of CD patients compared to controls. As TNFα2 has been correlated with high TNF production, its elevated expression could have functional significance in CD. However, in the absence of concrete evidence for a particular HLA-unlinked gene, and no reliable model for the inheritance of CD, the best approach for identifying causative non-HLA linked genes seems to be through a genome-wide linkage search. So far, a genetic model of inheritance cannot be inferred, as the mode of CD inheritance is unknown. Further genetic analysis should be performed using nonparametric methods ( Terwilliger and Ott, 1994 ).

2.3: What is a model? - Biology

If you never thought that sex appeal could be calculated mathematically, think again.

Male fiddler crabs (Uca pugnax) possess an enlarged major claw for fighting or threatening other males. In addition, males with larger claws attract more female mates.

The sex appeal (claw size) of a particular species of fiddler crab is determined by the following allometric equation:

where Mc represents the mass of the major claw and Mb represents the body mass of the crab (assume body mass equals the total mass of the crab minus the mass of the major claw) [1] . Before we discuss this equation in detail, we will define and discuss allometry and allometric equations.

  • a 10 kg organism may need a 0.75 kg skeleton,
  • a 60 kg organism may need a 5.3 kg skeleton, and yet
  • a 110 kg organism may need a 10.2 kg skeleton.

As you can see by inspecting these numbers, heavier bodies need relatively beefier skeletons to support them. There is not a constant increase in skeletal mass for each 50 kg increase in body mass skeletal mass increases out of proportion to body mass [2].

Allometric scaling laws are derived from empirical data. Scientists interested in uncovering these laws measure a common attribute, such as body mass and brain size of adult mammals, across many taxa . The data are then mined for relationships from which equations are written.

f (s) = c s d ,

  • If d > 1, the attribute given by f (s) increases out of proportion to the attribute given by s. For example, if s represents body size, then f (s) is relatively larger for larger bodies than for smaller bodies.
  • If 0 < d < 1, the attribute f (s) increases with attribute s, but does so at a slower rate than that of proportionality.
  • If d = 1, then attribute f (s) changes as a constant proportion of attribute s. This special case is called isometry, rather than allometry.

Using Allometric Equations

Notice that (1) is a power function not an exponential equation (the constant d is in the exponent position instead of the variable s). Unlike other applications where we need logarithms to help us solve the equation, here we use logarithms to simplify the allometric equation into a linear equation.

We rewrite (1) as a logarithmic equation of the form,

When we change variables by letting,

Therefore, transforming an allometric equation into its logarithmic equivalent gives rise to a linear equation.

By rewriting the allometric equation into a logarithmic equation, we can easily calculate the values of the constants c and d from a set of experimental data. If we plot log s on the x-axis and log f on the y-axis, we should see a line with slope equal to d and y-intercept equal to log c. Remember, the variables x and y are really on a logarithmic scale (since x = log s and y = log f). We call such a plot a log-log plot.

Because allometric equations are derived from empirical data, one should be cautious about data scattered around a line of best fit in the xy-plane of a log-log plot. Small deviations from a line of best fit are actually larger than they may appear. Remember, since the x and y variables are on the logarithmic scale, linear changes in the output variables (x and y) correspond to exponential changes in the input variables (f (s) and s). Since we are ultimately interested in a relationship between f and s, we need to be concerned with even small deviations from a line of best fit.

Practical Importance of Osmosis

Now that you understand the basic processes of osmosis, and what different conditions will cause osmosis to occur, you will be able to see the value of this process in so many areas for every form of life.

For plants, osmosis is responsible for the movement of water into the root system, which allows the plant to grow and survive. The root hairs of plants are the key point where minerals and water are taken into the organism. The concentration of water molecules are less in the root hairs than in the soil (hypertonic solution), so water moves into the cells of the root hairs osmosis continues through numerous layers of cells (cell-to-cell movement) until that water reaches the xylem tubes &ndash equivalent to human veins.

On a related note, when water is taken into the cells of plants, the pressure caused by that osmotic movement is called turgidity. When equilibrium is achieved, those plant cells should be full of water, as well as firm and turgid. This prevents leaves from wilting, allowing them to increase their surface area for sunlight capture. Osmosis also helps protect plants against drought and frost damage, as well as in regulating the opening and closing of stomata.

For animals (humans), some of the key osmotic functions relate to the balance of water content in the blood versus the surrounding tissues. Similarly, in the kidneys, osmosis controls the amount of waste buildup by increasing fluid flow into that organ. When the solute concentration is higher in the kidney cells (hypertonic solution), water is pulled from the body&rsquos bloodstream into the kidneys (nephrons), which will eventually stimulate the need to urinate in a person/animal, thus eliminating those unwanted waste products.


Figure 2: C. elegans anatomy (artwork by Altun and Hall, © Wormatlas)

C. elegans has a simple anatomy with a small number of tissues and internal organs (see Figures 2 and 3). The head contains the brain and the prominent feeding organ - the pharynx. The main body is filled with the intestine and - in the case of an adult hermaphrodite - the gonad consisting of the uterus and spermatheka. Embryos start to develop inside the mother and are laid through the vulva around gastrulation stage.

Figure 3: C. elegans anatomy - cross section through the main body (artwork by Altun and Hall, © Wormatlas)

The body is cylindrical, surrounded by a sheet of epithelial cells (hypodermis) and protected by a secreted cuticle consisting of a variety of proteins (mainly collagens). Body wall muscle cells are arranged in four rows, two each on the ventral and dorsal sides (green in Figure 3). Major nerve cords run along the entire length of the body on the dorsal and ventral midline. The interior is a fluid-filled space (pseudocoelomic space) surrounding the intestine and gonad. C. elegans completely lacks skeletal elements and has no circulatory system. Adult animals are only 1mm in length about 0.2mm in diameter, small enough to allow oxygen from the air to diffuse through the body. Nutrients from the gut are simply released into the pseudocoelmic space and taken up by other cells. The animals are under internal hydrostatic pressure, which acts as 'hydrostatic skeleton'. Muscle cells are tightly connected to the external cuticle through the hypodermal cells. Contraction of muscle cells on one side leads to bending of the rigid body. Coordinated contractions allow movement in elegant sinusoidal waves (hence the name C. elegans). When worms dry out they loose their internal pressure and the ability to move (think of a ballon no longer able to maintain a rigid shape when loosing air). C. elegans critically depends on a moist environment and has little protection against desiccation.

Conceptual Models are qualitative models that help highlight important connections in real world systems and processes. They are used as a first step in the development of more complex models.

Interactive Lecture Demonstrations Interactive demonstrations are physical models of systems that can be easily observed and manipulated and which have characteristics similar to key features of more complex systems in the real world. These models can help bridge the gap between conceptual models and models of more complex real world systems.

Several additional quotes relevant to using models and developing theories include:

Designing Biological Circuits: Synthetic Biology Within the Operon Model and Beyond

In 1961, Jacob and Monod proposed the operon model of gene regulation. At the model's core was the modular assembly of regulators, operators, and structural genes. To illustrate the composability of these elements, Jacob and Monod linked phenotypic diversity to the architectures of regulatory circuits. In this review, we examine how the circuit blueprints imagined by Jacob and Monod laid the foundation for the first synthetic gene networks that launched the field of synthetic biology in 2000. We discuss the influences of the operon model and its broader theoretical framework on the first generation of synthetic biological circuits, which were predominantly transcriptional and posttranscriptional circuits. We also describe how recent advances in molecular biology beyond the operon model—namely, programmable DNA- and RNA-binding molecules as well as models of epigenetic and posttranslational regulation—are expanding the synthetic biology toolkit and enabling the design of more complex biological circuits.

NET Life Science Model Question Paper 2015: Biology MCQ-8: Biochemistry: Amino Acids: Part 4

1). Which group of a fully protonated glycine (NH3+ – CH2 – COOH) first release a ‘proton’ when it is titrated against – OH- ions?
a. Carboxyl group
b. Amino group
c. Both at the same time
d. It cannot be predicted

2). pKa is the measure of a group to __________ proton.
a. Take up
b. Release
c. Combine
d. Consume

3). Which of the following amino acid bears a guanidine group in the side chain?
a. Lysine
b. Arginine
c. Histidine
d. Proline

4). The precursor of glycine synthesis in microbes and plants is_______.
a. Serine
b. Leucine
c. Valine
d. None of these

5). Single letter code of selenocysteine is _____.

6). Which of the following amino acid have an imino group in the side chain?
a. Proline
b. Asparagine
c. Glutamate
d. Histidine

7). 4-hydroxy proline (a derivative of proline) is abundantly present in _______.
a. Keratin
b. Myoglobin
c. Hemoglobin
d. Collagen

Biology MCQ-8: Biology/Life Science Multiple Choice Questions (MCQ) / Model Questions with answers and explanations in Biochemistry: Amino Acids Part 4 for preparing CSIR JRF NET Life Science Examination and also for other competitive examinations in Life Science / Biological Science such as ICMR JRF Entrance, DBT JRF, GATE Life Science, GATE Biotechnology, ICAR, University PG Entrance Exam, JAM, GRE, Medical Entrance Examination etc. This set of practice questions for JRF/NET Life Science will help to build your confidence to face the real examination. A large quantum of questions in our practice MCQ is taken from previous year NET life science question papers. Please take advantage of our NET Lecture Notes , PPTs , Previous Year Questions and Mock Tests for you preparation. You can download these NET study material for free from our Slideshare account (link given below).

8). Desmosine is a complex derivative of five ____________ residues.
a. Lysine
b. Arginine
c. Histidine
d. Methionine

9). Isoelectric pH is designated as______.
a. pKa
b. pI
c. Pi
d. None of these

10). Which of the following amino acid is biosynthesized from Ribose 5-phposphate?
a. Histidine
b. Serine
c. Glycine
d. All of these

11). Amino acid biosynthesized from Pyruvate of glycolysis is _____.
a. Alanine
b. Valine
c. Leucine
d. All of these

12). Aromatic amino acids (Phenylalaine, Tyrosine, and Tryptophan) are derived from Phosphoenol pyruvate and __________.
a. Ribose 5-phosphate
b. Erythrose 4-phosphate
c. Oxaloacetate
d. α-ketoglutarate

13). Isoleucine is derived from ___________.

a. Methionine
b. Threonine
c. Lysine
d. Leucine

14). Blood clotting protein thrombin usually contain which of the following modified amino acid?

a. 4-hydroxy proline
b. 5-hydroxy lysine
c. 6-N-methyl lysine
d. γ-carboxy glutamate

15). Cysteine is not an essential amino acid in human, since we have the machinery to synthesize cysteine from other two amino acids namely _______ and serine.

a. Methionine
b. Selenocysteine
c. Citrulline
d. Hydroxyproline

16). Which of the following contain a disulfide bridge?

a. Cysteine
b. Cystine
c. Methionine
d. None of these
e. All of these

17). Which of the following protein contain a modified amino acid – desmosine?

a. Keratin
b. Gelatin
c. Elastin
d. Collagen

180. During biosynthesis, Methionine and Threonine are derived from a common intermediate:

a. Chorismate
b. Citrulline
c. Homoserine
d. Cystathione

19). 6-N-methyl lysine is a derivative of lysine, present in__________.

a. Keratin
b. Collagen
c. Myosin
d. Myoglobin

20). A common intermediate branch point in the synthesis of all aromatic amino acids such as Tryptophan, Phenylalanine and Tyrosine is _____.

a. Homoserine
b. Chorismate
c. Cystathione
d. None of these

21). Sarcosine, a ubiquitous non protenacius amino acid in animals and plants is ____.

a. N-methylglycine
b. N-methylvaline
c. N-methylserine
d. N-methylmethionine

Answers and Explanations:

1. Ans. (a). Carboxyl group

First COOH group will release H+ ions and it will combine with OH- ions to form water. Only after complete ionization of all the COOH groups, the NH3 + release H + ions.

2. Ans. (b). Release

pKa is the negative logarithm of Ka. Ka is the dissociation constant of an ionization reaction such as the ionization of acetic acid. Ka is similar to equilibrium constant of any chemical reaction and it is calculated by dividing the concentration of products divided by concentration of its reactants. Ka denotes the strength of an acid. Strong acids will have a higher value of Ka where as a weaker acid will have a lesser values of Ka. The stronger the tendency to dissociate a proton, the stronger is the acid and the lower its pKa (Since pKa is negative logarithm of Ka i.e., reciprocal of ka).

3. Ans. (b). Arginine

4. Ans. (a). Serine

6. Ans. (a). Proline

7. Ans. (d). Collagen

8. Ans. (a). Lysine

10. Ans. (a). Histidine

11. Ans. (d). All of these

12. Ans. (b). Erythrose 4-phosphate

13. Ans. (b). Threonine

14. Ans. (d) γ-carboxy glutamate

15. Ans. (a). Methionine

Methionine provide sulfur

Serine provide the backbone of cysteine

16. Ans. (b). Cystine

17. Ans. (c). Elastin

18. Ans. (c). Homoserine

19. Ans. (c). Myosin

20. Ans. (b). Chorismate

21. Ans. (a). N-methylglycine

The answer key is prepared with best of our knowledge.
Please feel free to inform the Admin if you find any mistakes in the answer key..

Age separation dramatically reduces COVID-19 mortality rate in a computational model of a large population

COVID-19 pandemic has caused a global lockdown in many countries throughout the world. Faced with a new reality, and until a vaccine or efficient treatment is found, humanity must figure out ways to keep the economy going, on one hand, while keeping the population safe, on the other hand, especially those that are susceptible to this virus. Here, we use a Watts–Strogatz network simulation, with parameters that were drawn from what is already known about the virus, to explore five different scenarios of partial lockdown release in two geographical locations with different age distributions. We find that separating age groups by reducing interactions between them protects the general population and reduces mortality rates. Furthermore, the addition of new connections within the same age group to compensate for the lost connections outside the age group still has a strong beneficial influence and reduces the total death toll by about 62%. While complete isolation from society may be the most protective scenario for the elderly population, it would have an emotional and possibly cognitive impact that might outweigh its benefit. Therefore, we propose creating age-related social recommendations or even restrictions, thereby allowing social connections while still offering strong protection for the older population.

1. Introduction

The COVID-19 pandemic started in late December 2019 with mysterious pneumonia in Wuhan, China, and was declared a global pandemic by the World Health Organization (WHO) on 11 March 2020. The disease, caused by the SARS-CoV-2 pathogen, has quickly spread throughout 6 continents and over 210 countries. COVID-19 causes respiratory disease and is considered to be much more contagious than influenza [1]. Common symptoms include fever, dry cough, fatigue, shortness of breath and loss of smell or taste [2–5]. COVID-19-related complications include pneumonia and acute respiratory distress syndrome that may develop into a severe respiratory failure, septic shock and death [6,7]. In addition to being more contagious than influenza, COVID-19 has longer incubation periods compared with influenza. During the incubation period, the patients may be contagious [8–11]. Reports present incubation periods with a mean of 5–6 days, during which the patients are contagious [8,12]. Additionally, the mortality rate from COVID-19 disease is higher than the mortality rate from influenza complications [13]. These conditions instigated a rapid spread of the disease, causing over 100 countries to declare lockdowns and curfews, and causing an estimated global economic loss of one trillion US dollars in 2020 [14]. Till October 2020, over 36 million people were reported to be infected with COVID-19 and more than one million people died from virus-related complications.

The mortality rate from COVID-19 is strongly age biased, affecting the older population to a much greater extent [15–17]. In fact, it is thought that the younger population is usually asymptomatic or experience mild symptoms, even when infected with the virus [11,18]. For those symptomatic patients, the incubation period is the same regardless of their age. Recovery is reported to be 28 ± 14 days [19].

When the global lockdown is released, a slow release of the population back to their daily routine will occur. There is thought to be a psychological toll [20–22] due to the isolation of the population. Since COVID-19 is lethal mostly to the elderly population, suggestions of reopening the curfews, but keeping the elderly population isolated, have been proposed [23–25]. The psychological, emotional and even cognitive impacts may be stronger on this population, as social interactions are known to be essential for preventing a cognitive and physical decline [20,24,25]. Therefore, if there could be a solution where social interactions would not be prevented for the elderly population while keeping the population at low risk from the virus, this may be a preferred solution to this population.

The lower chances of the older population to be asymptomatic or to present mild symptoms compared with the younger population means that older COVID-19 patients are contagious for shorter periods of time compared with the younger patients. Asymptomatic (or weakly symptomatic) patients may be contagious all the way until full recovery (28 ± 14), whereas a symptomatic patient will be contagious for 6.4 ± 2.3 days (after which he will be isolated from society). This makes a young individual a stronger candidate for infecting others compared with an older individual. Here, we used these assumptions to derive a model of a large population to see what happens if we allow social interactions in the elderly population, but only among their own age group. This type of restriction will allow for the social interactions that are so important for the elderly age group. The model that we developed confirms that there would be a drastic reduction in mortality rate compared with allowing social interactions with other age groups (between 62% and 93%, depending on the scenario in a younger population distribution such as in Israel, and between 54% and 99%, depending on the scenario in an older population distribution such as in Italy), even when we keep the total number of connections the same by adding new connections within the elderly population for every lost connection with the younger population.

2. Methods

2.1. Population and network connectivity

We simulated a population of 50 000 individuals using network theory and a Watts–Strogatz model network with a degree distribution of 15 [26]. The Watts–Strogatz model was chosen because this ‘small-world’ model captures the features of high clustering and has a small average number of degrees of separation between any two individuals, which have been widely observed in real-world networks. Additionally, we simulated an Erdős–Rényi model [27] since it is a simple random graph, easy to generate, with a fixed number of nodes. Each node has a similar number of connections (degree). However, random graphs lack some of the crucial properties of real social networks such as ‘clustering’, in which the probability of two people knowing one another is greatly increased if they have a common acquaintance. In a random graph, by contrast, the probability of there being a connection between any two people is uniform. The results with this model are presented in the electronic supplementary material for reference (electronic supplementary material, figures S1–S6). We separated the population into four age groups: 0–14, 15–34, 35–54 and 55+ years. We ran our models in two types of populations. One is a population that has a younger distribution. For this, the Israeli population distribution was chosen as presented by the Central Bureau of Statistics in Israel [28]. The second is an older distribution for which the Italian distribution was chosen as presented by the CIA World Factbook [29]. We chose the distributions as were published in the references mentioned and chose a normal distribution with an expectation of the middle point of the age group and a standard deviation of half of the range of that age group. The final distributions are presented in figure 1 and when integrating over all the age points in every age group, the percentages in each of the age group is similar to the one that we based it on [28] with up to 0.5% error. The derivation of the final distribution from the published distributions is presented in electronic supplementary material, figure S7A for the Israeli population and S7B for the Italian distribution.

Figure 1. The distribution of the population by their age. (a) The population distribution in Israel. (b) The population distribution in Italy.

We chose the size of the families (or households) to be normally distributed with a mean and a standard deviation of 4 ± 1. Families were constructed in the following manner: After randomly grouping the entire population into families using a normal distribution with a mean of 4 and a standard deviation of 1, the family members were filled in according to the size of the family. If the family size was larger than 2, a pair of parents aged 20–60 were randomly taken from the population pool and the remaining members of the family were randomly selected with an age of 0–20. If the family had only two members, two adults with an age of 55–100 were randomly selected, and if the family size was 1, one adult with an age of 20–100 was randomly selected from the general population. These constraints resulted in the following family sizes according to the age groups: 5.2 ± 0.99 for the first age group, 5.1 ± 1.05 for the second group, 4.9 ± 1.03 for the third age group and 3.2 ± 1.74 for the fourth age group in the Israeli population. In the Italian population, this resulted in family sizes of 4.4 ± 0.91 for the first age group, 4.4 ± 0.90 for the second group, 4.1 ± 0.92 for the third age group and 3.3 ± 1.41 in the fourth age group. The populations' distributions are plotted for the different age groups in figure 2 (Israeli families on the left and Italian families on the right). The results for Erdős–Rényi model are similarly presented in electronic supplementary material, figure S1.

Figure 2. The distribution of family size by age group. (a) The family size distribution in age group 1 (0–14) in Israel. (b) The family size distribution in age group 2 (15–34) in Israel. (c) The family size distribution in age group 3 (35–54) in Israel. (d) The family size distribution in age group 3 (55+) in Israel. (e) The family size distribution in age group 1 (0–14) in Italy. (f) The family size distribution in age group 2 (15–34) in Italy. (g) The family size distribution in age group 3 (35–54) in Italy. (h) The family size distribution in age group 3 (55+) in Italy.

Due to the family members’ selection, the connectivity level in the different age groups also varied slightly and consisted of 14 ± 2.25 connections for the first age group, 14 ± 2.28 connections for the second age group, 14 ± 2.30 for the third age group and 13 ± 2.27 connections for the fourth age group in the Israeli population. Figure 3a shows the distribution of the number of connections according to the different age groups (these conditions will later be defined as state 1) in the Israeli population. Figure 3d similarly presents the number of connections in the Italian population showing a similar number of connections to the Israeli population. Figure 4 presents example connections within a general 50 000 individuals' population of three individuals (figure 4a) and of 30 individuals (figure 4b). Electronic supplementary material, figures S2 and S3 present the results for Erdős–Rényi model, respectively.

Figure 3. The distribution of the number of connections per individual in the different age groups in states 1–3 shows similar connectivity between these states. Simulations with an age distribution similar to the Israeli population are presented in (a–c), and similar to the Italian population is presented in (d–f). (a) The distribution of the connections in state 1 in the different age groups (Israel). (b) The distribution of the connections in state 2 in the different age groups (Israel). (c) The distribution of the connections in state 3 in the different age groups (Israel). (d) The distribution of the connections in state 1 in the different age groups (Italy). (e) The distribution of the connections in state 2 in the different age groups (Italy). (f) The distribution of the connections in state 3 in the different age groups (Italy). The number of connections did not change much between the states, indicating that disease evolution in these states changes mainly due to the age separation and not due to changes in the connectivity of the network.

Figure 4. Example connections within a sample of the population. (a) Example of connections between 3 subjects. (b) Example connections between 30 subjects.

2.2. Infection

The model assumes different infection rates at different scenarios, but what is common is that once an individual becomes symptomatic, he is assumed to be under full quarantine and is therefore removed from the network. The highest infection rate occurs within the family and is 10% daily each day each infected individual will infect another individual within his family with a probability of 10%. The infection rate in public is reduced to 1%, assuming people keep social distancing, thus reducing dramatically the infection rate.

Once infected, symptoms may appear in all age groups under a different probability. There is a higher probability to be asymptomatic (or presenting mild symptoms) for younger people. We used 80% chance to be asymptomatic for the 0–14 age group, 60% chance to be asymptomatic for the 15–34 age group, 40% chance for the 35–54 age group and 20% chance for the 55+ age group. These numbers were based on several reports [18,30,31]. For those patients that will become symptomatic, the number of days until presenting symptoms has a Weibull distribution with a mean of 6.4 days and standard deviation of 2.3 days [12], and this does not vary between the age groups.

Mortality rates vary dramatically among age groups and are 0% for 0–14, 0.15% for 15–34, 1% for 35–54 and 24% for ages 55+, similar to what was reported in [15]. Once showing symptoms the recovery rate (for those who recover) is 28 ± 14 days (normal distribution) in all age groups [19].

2.3. Creating multiple sample paths

We have run our model multiple times (n = 10) to achieve different sample paths in the model. This was done to get a better understanding of the average and the distribution of the results of a stochastic model such as the Watts–Strogatz. The algorithm for one infection simulation is described in electronic supplementary material, figure S4. Each run starts with a random and different five patients that are the first carriers. This allows for different realizations of the model and the results. To graphically show this, we added in figure 6 a plot that describes these 10 different instances by plotting the mean (solid line) and a 95% confidence interval for each of the states for a 250 days simulation run.

3. Results

We have defined five different states. The first state complies with all the above assumptions and with no other restrictions. Electronic supplementary material, videos 1 and 2 show the spread over time of COVID-19 demonstrating graphically the connections and infections in individuals in a population of 10 000 people for state 1 in Israel and Italy, respectively. Blue dots represent susceptible individuals, orange dots represent carrier individuals, red dots represent infected individuals and black dots are individuals who died from COVID-19. The shape of the dots marks the age groups (circle, 0–14 triangle, 15–34 square, 35–54 cross, 55+). We next wanted to test how age separation changes infection and mortality rates in the population. For this, we defined four more states with different scenarios of age separation. In state 2, we did not allow interactions between the different age groups, such that the interactions among family members remained without a change, but all other connections were eliminated. This reduced the connectivity of the network. Since we were interested to see the effect of the age separation and not the effect of reducing the connectivity, we added in these state new random connections within the same age group to replace any connection that was eliminated. The new number of connections for each group resembles the original conditions, and is shown in figure 3b for the Israeli population and in figure 3e for the Italian population (electronic supplementary material, figure S2 for the Israeli population using Erdős–Rényi model). The new number of connections for both populations is similar to the original number of connections (the family size distribution remains since we had not changed the family connections).

We next defined a more plausible constellation of the network. In state 3, we grouped the three age groups 0–14, 15–34 and 35–54, and did not allow connections with the elderly group of 55+ (except if they are in the same household). Such restrictions also reduced the overall number of connections, and we therefore added random connections within the same age group to any connection that was eliminated between the age groups. Figure 3c presents the new distribution of connections in the Israeli population per individual in the different age groups, and the number of connections is similar (or greater) than the original number of connections. Similarly, figure 3f presents the distribution of connections of state 3 in the Italian population, and there too the number of connections in state 3 is similar to the number of connections in state 1.

Since it is also reasonable to assume that restrictions of connections between age groups may reduce network connectivity, we added state 4 in which we keep all the connections, but reduce the connectivity between age groups by reducing by half the infection rates to 0.5% between age groups. We similarly defined state 5, where we completely diminish connections between different age groups (but without adding new connections).

Figure 5a presents the simulation results over a time period of 250 days in a population of 50 000 individuals (with statistics as in the Israeli population) by the categories of susceptible, carrier (has the virus), infected, recovered and deceased for the entire population pooled together in the five different states (electronic supplementary material, figure S5 presents results for the Erdős–Rényi model in the Israeli population). Similarly, figure 5b presents the simulation results over 50 000 individuals with statistics as in the Italian population. We expanded the plots to see in more detail the infection and death rates. The results are presented in figure 6a–c for the Israeli population and in figure 6d–f for the Italian population. For the Israeli population, figure 6a presents the infection over the 250 days for the five different states. Figure 6b similarly shows mortality from the virus over the course of 250 days in the different states. Further comparison of infection and mortality rates (figure 6a,b, respectively) shows dramatic infection rate changes, and most importantly lower mortality in the different states (2–5) compared with state 1. This shows that reducing the interaction between age groups decreases mortality, even when keeping the number of interactions between individuals constant or even increasing this number. Separating just the older group (55+) from interacting with the other age groups (but keeping interactions within the same household, state 3) reduces the overall mortality rate in the population by 62%. This decrease becomes even more pronounced when not adding new connections within the same age group (state 5) with a reduction of 93% in the mortality rate in the overall population. We plotted also the mortality rate in each of the states in the different age groups (figure 6c) for the Israeli population. The plots show that most of the decrease in mortality rate occurs due to a decrease in the mortality of the oldest age group and in state 5 also of the second oldest age group (but to a lower extent). The results for the Italian population are presented in figure 6d,e and a reduction of 54% can be observed in state 3 and of 99% in state 5. Similarly, figure 6f shows that most of the reduction in mortality rate is due to a reduction in the mortality of the oldest age group. Using this strategy in a real-world scenario would probably result in a reduction that is somewhere between these two states (state 3 and state 5) since some interactions will be replaced (such as seats in the theatre), but some will be eliminated (such as meeting distant relatives). The results using Erdős–Rényi model are presented in electronic supplementary material, figure S6.

Figure 5. The total number of susceptible, carriers, infected, recovered and deceased individuals in the entire population of 50 000 people over a period of 250 days for states 1–5 for the Israeli population (a) and states 1–5 for the Italian population (b).

Figure 6. (a) A statistical span of the number of infected individuals in the entire population for the different states in Israel showing multiple realizations over n = 10 runs of the simulation for a better statistical average of the infection rates in the Israeli population (see Methods). (b) Similarly, a statistical span of the number of deceased individuals in the entire population in the different states in Israel (10 simulation runs). While in state 1 in a population distribution similar to the Israeli population 2183 individuals died on average, in state 2 only 1719 individuals died, in state 3 only 829 individuals died. In state 4, 1877 individuals died and in state 5 only 189 individuals died. (c) The distribution of deaths among the age groups in the different states for a population distribution similar to the Israeli distribution shows that the reduction in mortality occurs mainly due to the reduction in mortality rate in age group 4 and in state 5 also in age group 3 (but mostly in age group 4). (d) Similarly, a statistical span of the number of infected individuals in the entire population for the different states with a population with a distribution similar to Italy (10 simulation runs). (e) The total number of deceased in the entire population in the different states in Italy (10 simulation runs representing ten realizations). While in state 1 in a population distribution similar to the Italian population 3772 individuals died on average, in state 2 only 3154 individuals died, in state 3 only 1731 individuals died. In state 4, 3209 individuals died and in state 5 only 34 individuals died. (f) The distribution of deaths among the age groups in the different states for a population distribution similar to the Italian population distribution shows that the reduction in mortality occurs mainly due to the reduction in mortality rate in age group 4 and in state 5 also in age group 3.

Electronic supplementary material, videos S3 and S4 present the spread of the disease in states 1–5 in Israel and Italy, respectively. The colour and shape schemes are the same as in electronic supplementary material, videos S1 and S2. Electronic supplementary material, videos S5 and S6 similarly show the same disease evolution as electronic supplementary material videos S3 and S4, but the blue dots were removed for clearer graphs. A clear reduction in the death rate is exhibited in the new states.

4. Discussion

COVID-19 caught the world unprepared and has infected (by October 2020) over 36 million people (reported) and has cost over one million lives. The pandemic has caused a global lockdown that in turn caused a huge economic burden. Releasing the lockdown needs to be under controlled conditions, being extremely careful. Currently, most countries are in the ‘second wave’ [32,33] and we are still far from resolving this pandemic. The elderly population are the most susceptible to this virus, and therefore many suggestions have been made of releasing the general population, while keeping the elderly quarantined. However, social interactions are known to be extremely important in the elderly population [34,35], and such conditions may backfire leading to both mental and emotional deterioration in this population.

In this study, we tested several states that limit the interactions between people in different age groups, and we showed that all these scenarios reduce mortality rate. In the first two states, we kept the interactions of family members within the same household, and eliminated connections outside of the age group, yet keeping the network connectivity (by adding more connections within the same age group). This drastically reduces the overall death toll by 62%. The reasoning for the improvement in the overall mortality is that younger people have a higher probability of being asymptomatic (or weekly symptomatic). Asymptomatic people usually keep on their social interactions, infecting many people for a very long time. Older people have a much lower probability of being asymptomatic. This means that they are usually infectious for only approximately 5–6 days (the asymptomatic period is assumed to have a Weibull distribution with an average of 6.4 days). At this point, they usually know that they are sick and are isolated. Therefore, an older individual is less likely to spread the disease.

In the other two states, we reduced the interactions (state 4) or completely diminished the interactions (state 5) between the different age groups. This may be a reasonable assumption, since not all connections between age groups will be replaced by connections within the age group (such as connections with distant family). In state 5, all the connections outside the age group were eliminated, which reduced mortality rate by 93% in a population with an Israeli distribution (99% in a population with an Italian distribution). A real scenario will probably be some compromise between state 3 and state 5, since some connections will indeed be replaced (for example, seating in a restaurant) and some will be lost (such as meeting distant family members). It is important to keep in mind that our model is a stochastic model that depends on the initial conditions. To better understand how variable the results are with respect to different realizations or sample paths, we ran the model 10 times and plotted in figure 6 the confidence interval around the mean total infected and deceased individuals for each of the conditions.

It is important to note that although age separation is difficult to implement generally, it is relatively easy to find microenvironments in which age separation is possible and will enable the older population to maintain social connections while reducing infection and death rates in these populations. Such microenvironments may include grocery shops, theatres or airplane rides, and would allow important social interactions for the elderly population. Overall, our study shows that age separation is extremely beneficial and can be imposed in an important intermediate period until resuming normal life when finding a cure or a vaccine to COVID-19.

Watch the video: Μαθηματική Μοντελοποίηση: Επιδημιολογικά μοντέλα (July 2022).


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