Math Models Nihar

Mathematical Modeling in Health Introduction

Dr Nihar Ranjan Ray Bhubaneswar, India

Mathematics is the language to formulate biological phenomena occurring in nature…..Application of this principle in biology for simplification understanding is the sole aim of Mathematical Modeling. Biology is a vast science ..a lot of departments… !ecology, epidemiology, immunology, virology, physiology, evolutionary biology, neurobiology, cell biology, biochemistry, genetics, …and so many. It is very difficult to understand the problem and its complexity designing a mechanism to tackle the problem Mathematical Biology: conceptual framework to identify fundamental variables and their influences on each other

Historical Precedents ... • • • • •

Modelling of infectious diseases of humans initiated with Bernoulli’s model for the transmission dynamics of smallpox in 1760 Foundation of modern mathematical modeling of epidemiology between 1900-1935 In 1909, Sir Rossintroduced a disease control factor (R0): “The Basic Reproductive Number” Interest in mathematical immunology emerged in 1970:G.I. Bell, Journal of Theoretical Biologyand Nature A new theoretical field in biology:advent ofHIVin the 1980s

Contributers to the Mathematical Modeling: Hamer WH, Epidemic disease in England. Lancet 1906; 1: 733–739 Ronald Ross , The prevention of malaria( His works in India),London, England, John Murray, 1909 McKendrick AG, Applications of mathematics to medical problems. Proc Edin Math Soc 1926; 14: 98-130 Kermak WO, McKendrick AG, A contribution to the mathematical theory of epidemics. Proc R Soc Lond B 1927; 115: 700-721 Kermak WO, McKendrick AG, Contributions to the mathematical theory of epidemics. Proc R Soc Lond B 1931; 138: 55-83

Utility of Mathematical Models • • •

Models are essential tools for understanding the transmission dynamics of diseases designing proper and cost-effective control strategies To understand the Complexity in the newer generation of epidemic concepts modes of transmission, incubation period, duration of infectiousness, treatment and preventive strategies, age and social behavior ,socioeconomic demographics

Mathematical models are essential tools for: • • • •

determining principles of interaction between pathogens and the immune system evaluating the ability of the immune system to control rapidly evolving pathogens understanding the relationship between disease characteristics and control measures predicting disease dynamics and consequences of various control strategies

Control Strategies: Prevention and Treatment Traditional methods: • reducing number of contacts • therapeutic treatments • quarantine and isolation Modern methods: • Prevention, immunization • ecological intervention . Integration of mathematical models with: • mechanisms of diseases pathogenesis and epidemiology • evolutionary biology of viral pathogens • intervention strategies at individual and population levels • empirical data (in vivoas well as in the population) So there is need for uniting expertise: mathematics, computer science, statistics, immunology, virology, epidemiology, evolutionary biology, public health to master in mathematical biology. There is need of emergence of new science against new infections Mathematical Tools: Theory and Applications • Dynamical Systems (ODEs, PDEs): stability, bifurcation, perturbation, singularity, normal form, and Index theories,… • Statistical Analysis: experimental design, data classification and analysis, sensitivity and uncertainty analysis,… • Computational biology: stochastic modeling, simulations, finitedifference methods, numerical software,… Examples of Popular Mathematical Models The Concept of Mass Action The state variables in an epidemiological model correctly refer to population density rather than population size. The contact rate is often a function of

population density, reflecting the fact that contacts take time and saturation occurs. The Size of an Epidemic Consider an epidemic that occurs on a timescale that is much shorter than that of the population, in other words regard the population as having a constant size and ignore births and deaths. Assume that upon recovery the individual remains immune. At any time the population consists of S susceptible individuals, I infected and R immune (removed). A simple model that describes the changes in these numbers with time is Compartmental Models A compartmental model is one for which the individuals in a population are classified into compartments depending on their status with regard to the infection under study. The Basic Reproduction Ratio Definition: the average number of new infections produced by one infected individual introduced into a wholly susceptible population during the course of infection The basic reproduction ratio of an infectious disease is a pivotal concept in epidemiology. It is defined as the expected number of secondary cases that would arise from the introduction of a single primary case into a fully susceptible population. Clearly, when R0 < 1 each successive ‘infection generation’ is smaller than its predecessor, and the infection cannot persist. Conversely, when R0 > 1 successive ‘infection generations’ are larger than their predecessors Why is R0so important? Tells us how easy or difficult it is to eradicate an infection assesses the effectiveness of a disease control measure 1

1 Models for Vector-Born Infections: A simple model that captures the essential elements of malaria epidemiology and other vector borne diseases. Models for Parasite Populations : model for the dynamics of a population of parasitic helminths in a host population Models with Structure: the possible demographic impact of HIV in developing countries. will require models formulated in an age-structured setting.

Ref: ©Encyclopedia of Life Support Systems (EOLSS) National Research Council Canada