Mathematical modeling with measures

Joshua Kiddy K.

mathematical modeling with measures

Vaccination and treatment are the most effective ways of controlling the transmission of most infectious diseases. In this paper, a nonlinear deterministic model with time-dependent controls has been proposed to describe the dynamics of bacterial meningitis in a population. The model is shown to exhibit a unique globally asymptotically stable disease-free equilibriumwhen the effective reproduction numberand a globally asymptotically stable endemic equilibriumwhen ; and it exhibits a transcritical bifurcation at.

Carriers have been shown by Tornado plot to have a higher chance of spreading the infection than those with clinical symptoms who will sometimes be bound to bed during the acute phase of the infection.

What is Statistical Modeling For Data Analysis?

In order to find the best strategy for minimizing the number of carriers and ill individuals and the cost of control implementation, an optimal control problem is set up by defining a Lagrangian function to be minimized subject to the proposed model.

Numerical simulation of the optimal problem demonstrates that the best strategy to control bacterial meningitis is to combine vaccination with other interventions such as treatment and public health education. Meningitis is an inflammation of the meninges which are membranes that surround the spinal cord and the brain [ 1 ]. It is often caused by viruses, bacteria, and protozoa. Bacterial meningitis is common in children and young adults. Bacterial meningitis is generally caused by germs such as Listeria monocytogenesStreptococcus pneumoniaeGroup B StreptococcusNeisseria meningitidisand Haemophilus influenzaewhich spreads from one person to another [ 3 ].

This infection varies by age groups: Group B StreptococcusStreptococcus pneumoniaeListeria monocytogenesand Escherichia coli are mostly found in newborn babies; Streptococcus pneumoniaeNeisseria meningitidisHaemophilus influenzae type b Hiband Group B Streptococcus are common in babies and children; Neisseria meningitidis and Streptococcus pneumoniae are predominant in teens and young adults; and Streptococcus pneumoniaeNeisseria meningitidisHaemophilus influenzae type b HibGroup B Streptococcusand Listeria monocytogenes are commonly found in older adults [ 3 ].

Bacterial meningitis is characterized by intense headache and fever, vomiting, sensitivity to light, and stiff neck, which result in convulsion, delirium, and death. It is estimated that meningococcal meningitis causes over 10, deaths annually in Sub-Saharan Africa [ 4 ]. About 4, cases of bacterial meningitis occurred between and in the United States [ 35 ]. Infections from bacterial meningitis can cause permanent disabilities such as brain damage, hearing loss, and learning disabilities [ 3 ].

The illness of bacterial meningitis becomes worse when symptoms are not detected early enough; even with proper treatment, the individual could die [ 2 ]. Vaccination is the most effective way of protecting children against certain types of bacterial meningitis [ 3 ]. Vaccines that can prevent meningitis include Haemophilus influenza type B Hibpneumococcal conjugate, and meningococcal vaccine [ 67 ]. The conjugate meningitis A vaccine, MenAfrivac, is recommended to protect people in Sub-Saharan Africa against the most common type, serotype A [ 8 ].

In the United States, the primary means of preventing meningococcal meningitis is antimicrobial chemoprophylaxis [ 9 ]. Empirical therapy includes ceftriaxone or cefotaxime and vancomycin for Streptococcus pneumoniae [ 2 ]. Trotter and Ramsay [ 10 ] outlined some recommendations on the use of conjugate vaccines in Europe based on the experience with meningococcal C conjugate MCC vaccines.

In areas with limited health infrastructure and resources, there are a number of antibiotics including penicillin, ampicillin, and chloramphenicol that can be used to treat the infection meningitis.

Mathematical models have been shown to help increase the understanding of the spread and control of infectious diseases. Broutin et al. The research work in [ 13 ] gives a detailed description of the use of antibiotics for the prevention and treatment of meningitis infection. Irving et al. They demonstrated that the complex and irregular timing of epidemics could be caused by the interaction of temporary immunity conferred by carriage of the bacteria together with seasonal changes in the transmissibility of infection.

Actually, there have been a significant number of studies of various types of Meningitis in Africa and Europe without the use of optimal control analysis see [ 15 — 28 ]. It is obvious that mathematical modelling has become crucial in investigating the epidemiological behaviour of meningitis. Furthermore, mathematical modelling helps to identify the risk factors for diseases, so as to find out why everyone does not have the same infection uniformly [ 29 ].Last Updated: April 24, References.

To create this article, 10 people, some anonymous, worked to edit and improve it over time. This article has been viewedtimes. Learn more A mathematical model is a description of a system using mathematical language. Mathematical models are used not only in the natural sciences and engineering disciplines but they are also used in biology, economics and sociology.

Mathematical models can range from simple to complex. Mathematical models help to represent a system using mathematical language, and you can make your own to predict outcomes and solve problems.

Mathematical Modelling of Bacterial Meningitis Transmission Dynamics with Control Measures

First, figure out what information you already know and what information you need to solve. If you don't know what formula you need, try looking it up online. You can also draw a diagram to use as a map when you make your model. For a more advanced model, you can even design it using computer software before you build it physically.

Use your equation, diagram, and any other data you have to make your model. Now you can test your model and compare it with your predicted results. For more tips, including how to find inspiration for your mathematical model, read on! Did this summary help you? Yes No. Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker.

Log in Facebook. No account yet? Create an account.Industry Advice Analytics. Those who are pursuing a career in data analytics or data science are likely familiar with the many relevant skills needed to be successful in this demanding field. By making sense of data, you are translating it into fact, drawing conclusions, and using those conclusions to create and tell stories. Luckily, those who take the time to understand the role that statistical modeling plays in data analytics—and the ways in which different modeling techniques can be used to analyze and manipulate data—will have the context needed to do just that.

Statistical modeling is the process of applying statistical analysis to a dataset.

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A statistical model is a mathematical representation or mathematical model of observed data. When data analysts apply various statistical models to the data they are investigating, they are able to understand and interpret the information more strategically. Rather than sifting through the raw data, this practice allows them to identify relationships between variables, make predictions about future sets of data, and visualize that data so that non-analysts and stakeholders can consume and leverage it.

While data scientists are most often tasked with building models and writing algorithms, analysts also interact with statistical models in their work on occasion.

For this reason, analysts who are looking to excel should aim to obtain a solid understanding of what makes these models successful. Below are some of the benefits that come from having a thorough understanding of statistical modeling.

There are many different types of statistical models, and an effective data analyst needs to have a comprehensive understanding of them all. Data is rarely ready for analysis in its raw form. To ensure your analysis is accurate and viable, the data must first be cleaned up. Once you know how various statistical models work and how they leverage data, it will become easier for you to determine what data is most relevant to the question you are trying to answer, as well.

In most organizations, data analysts are required to communicate their findings with two different audiences. The second audience consists of those who are interested in the more granular details; this group will want both the list of broad conclusions and an explanation of how you reached them.

Having a thorough understanding of statistical modeling can help you better communicate with both of these audiences, as you will be better equipped to reach conclusions and therefore generate better data visualizations, which are helpful in communicating complex ideas to non-analysts.

Simultaneously, a complex understanding of how these models work on the backend will allow you to generate and explain those more granular details when necessary.

Before any statistical model can be created, an analyst needs to collect or fetch the data housed on a database, clouds, social media, or within a plain excel file. To do this, analysts must also have a solid grasp of data structure and management, including how and where data is stored, fetched, and maintained. Those working in this field should thus share a passion for facts and data, and understand the basics of data manipulation, as well.

Once it comes time to analyze the data, there are an array of statistical models analysts may choose to utilize. According to Mello, most common techniques will fall into the following two groups:.

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Data analysts use regression models to examine relationships between variables. Regression models are often used by organizations to determine which independent variables hold the most influence over dependent variables—information that can be leveraged to make essential business decisions.

Other examples of regression models can include stepwise regression, ridge regression, lasso regression, and elastic net regression. Classification is a process in which an algorithm is used to analyze an existing data set of known points. The understanding achieved through that analysis is then leveraged as a means of appropriately classifying the data.

Classification is a form of machine learning that can be particularly helpful in analyzing very large, complex sets of data to help make more accurate predictions. There are also the neural networking models that are more used in AI. Digging In Deeper: The unknown process that takes place with this model can be compared to putting raw dough into one side of a black box and getting freshly baked bread out the other side.

Because you understand the inputs dough and outputs bread you can make certain assumptions about what happened inside the box—the dough was cooked—but the exact mechanism of how this happened cannot be known.As COVID spreads worldwide, leaders are relying on mathematical models to make public health and economic decisions.

A new model developed by Princeton and Carnegie Mellon researchers improves tracking of epidemics by accounting for mutations in diseases.

Now, the researchers are working to apply their model to allow leaders to evaluate the effects of countermeasures to epidemics before they deploy them. Vincent Poor, one of the researchers on this study and Princeton's interim dean of engineering. The models currently used to track epidemics use data from doctors and health workers to make predictions about a disease's progression. Poor, the Michael Henry Strater University Professor of Electrical Engineering, said the model most widely used today is not designed to account for changes in the disease being tracked.

This inability to account for changes in the disease can make it more difficult for leaders to counter a disease's spread. Knowing how a mutation could affect transmission or virulence could help leaders decide when to institute isolation orders or dispatch additional resources to an area.

Mathematical model

If the researchers can correctly account for measures to counter the spread of disease, they could give leaders critical insights into the best steps they could take in the face of pandemics. The researchers are building on work published March 17 in the Proceedings of the National Academy of Sciences. In that article, they describe how their model is able to track changes in epidemic spread caused by mutation of a disease organism.

mathematical modeling with measures

The researchers are now working to adapt the model to account for public health measures taken to stem an epidemic as well. The researchers' work stems from their examination of the movement of information through social networks, which has remarkable similarities to the spread of biological infections.

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Notably, the spread of information is affected by slight changes in the information itself. If something becomes slightly more exciting to recipients, for example, they might be more likely to pass it along or to pass it along to a wider group of people. By modeling such variations, one can see how changes in the message change its target audience.

Our model allows us to consider changes to information as it spreads through the network and how those changes affect the spread. Obtaining accurate information is extremely difficult during an ongoing pandemic when circumstances shift daily, as we have seen with the COVID virus.

You can't always wait until you collect data to make decisions -- having a model can help fill this void," Poor said. Materials provided by Princeton University, Engineering School. Original written by John Sullivan. Note: Content may be edited for style and length. Science News. Vincent Poor. The effects of evolutionary adaptations on spreading processes in complex networks.

ScienceDaily, 25 March Princeton University, Engineering School. New mathematical model can more effectively track epidemics.A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences such as physicsbiologyearth sciencechemistry and engineering disciplines such as computer scienceelectrical engineeringas well as in non-physical systems such as the social sciences such as economicspsychologysociologypolitical science.

Mathematical models are also used in music [1]linguistics [2] and philosophy for example, intensively in analytic philosophy. A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.

Mathematical models can take many forms, including dynamical systemsstatistical modelsdifferential equationsor game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

In the physical sciencesa traditional mathematical model contains most of the following elements:. Mathematical models are usually composed of relationships and variables. Relationships can be described by operatorssuch as algebraic operators, functions, differential operators, etc.

Variables are abstractions of system parameters of interest, that can be quantified. Several classification criteria can be used for mathematical models according to their structure:. In business and engineeringmathematical models may be used to maximize a certain output. The system under consideration will require certain inputs.

The system relating inputs to outputs depends on other variables too: decision variablesstate variablesexogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants.

The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system represented by the state variables. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user.

Depending on the context, an objective function is also known as an index of performanceas it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases.

For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.

mathematical modeling with measures

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available.

A black-box model is a system of which there is no a priori information available. A white-box model also called glass box or clear box is a system where all necessary information is available.A difference of 2 goals. Our scoreline of 2-1 Getafe win was correct too. Said, Sociedad wouldn't win. It remained 1-0 till the 90th minute. Hoffeinheim equalized at the edge of the game. Match ended 1-1 as we predicted. Gladbach scored 3 goals quickly.

It stayed 3-1 for long.

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Game ended, 4-2, Gladbach win. WI scored 356, became 0. Also said, despite starting favs at home, don't see Dortmund having an edge at all. Match ended 3-0 PSG. Chelsea won in the last moments of the last minute of the game. RomaUEFA Champions League 2017-18 Chelsea vs A. Roma, 18-10-2017Said, no team looks outstanding for this game. Indeed, it ended up in a 3-3 Draw.

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Asked to lay Pak early(0. Can be better than that. Simply can't ignore SL as the game becomes tricky towards the end. A chance of the game going closer than 35-27, nearing a Draw. It ended 25-24 NZ win. Match ended 37-20 Australia. Trinbago Knight Riders won the CPL17 trophy. After a Liverpool Red Card, City won 5-0. HalepFrench Open 2017 Women's Final, 10-06-2017Yes, at 20, unseeded, Jelena Ostapenko won the French Open 2017.

Our prediction was correct. We'll be back with new and exciting features.Certain Zacks Rank stocks for which no month-end price was available, pricing information was not collected, or for certain other reasons have been excluded from these return calculations.

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The MATH of Epidemics - Intro to the SIR Model

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NASDAQ data is at least 15 minutes delayed. Space weather refers to changes in the space environment, particularly the region between the Earth and Sun. The "solar wind" from the Sun stream past the Earth and is mostly deflected by the Earth's magnetic field, but variations in the solar wind cause changes in the Earth's magnetic field.

Occasionally, a huge release of magnetic energy, called a solar flare, occurs on the Sun. Flares can produce large quantities of x-rays which affect the Earth's atmosphere. They can also accelerate atomic particles (mostly protons) to very high speeds (a substantial fraction of the speed of light.

These high energy particles are dangerous to man and can reach the stratosphere where jetliners fly.

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Most aspects of space weather affect us to some extent. The more our society becomes dependent on technology and the more we utilise space, the more we are affected by space weather.

Some aspects of space weather are benevolent, and allow activities not otherwise possible such as long range radio communications. Some aspects are benign but fascinating such as the Aurora, and some are malevolent. Like terrestrial weather, it sometimes depends on the situation and the event. The image below is an artists impression of the solar wind interacting with the Earth's magnetic field. Home Forums Movies Oscars 2018 Nominations: Part 6 (Predict Today. The Shape Of Water Three Billboards Outside Ebbing, MissouriDIRECTOR Darren Aronofsky, mother.

Frances McDormand, Three Billboards Outside Ebbing, MissouriSupporting Actor: Sam Rockwell, Three Billboards Outside Ebbing, Missouri Alt. Michael Stuhlbarg, Call Me By Your NameWith the past few years of voting patterns, and seeming openness of this year, it seems pretty ripe for a very spread out field of winners.


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