Neke diferencijalne jednaine sa primenama u biologiji

Analysis of mathematical models in medicine and scienceSanja TeodoroviUniversity of Novi SadFaculty of ScienceMathematical modelingStandard (medical) studies are not enough

Abstraction sufficient for further assessment of the solution; makes it possible to perceive a real problem in a simplified mannerDescription of real problems using various mathematical tools; used for the analysis, design and optimization

Correctly set if:the solution of the initial problem existsthe solution of the initial problem is uniquethe solution of the initial problem continuously depend on the initial conditions Mathematical modelingMathematical models:linear and nonlinear;deterministic and stochastic predetermined by previous values and impossible to predict the probability of change of the certain values;statistical and dynamic constant and dependent on time;discrete and continuous in certain points of time and continuously in time;deductive, inductive and floating theoretical and experimental reasoning and an estimate of the expected relationship between variables.

Formulation: simplified, real, precise

Computer simulations and experimental testing of theoretical conclusions HistoryDaniel Bernoulli, 1760 smallpox; the first modelTrue developmentWilliam Hamer, 1906 measlesnumber of new cases depends on the concentration of sensitive and infectiousSir Ronald Ross, 1922 malariamodel of differential equationsKermack and McKendrick, 1926condition for the occurrence of epidemic number of sensitive is greater then a finite numbermiddle of 20th century accelerate d developmentHistoryPassive immunity, a gradual loss of immunity, social mixing between groups, vaccination, quarantine, different medicines etc.

Smallpox, measles, diphtheria, malaria, rabies, gonorrhea, herpes, syphilis and HIV and AIDS

Epidemiological (for sudden and rapid outbreaks) and endemic (for infections that extend over a longer period of time)Classificationof epidemiology modelsFive basic groups (classes):M class: class of people who have passive immunityS class: susceptible classE class: exposed class; infected but not infectiousI class: infectious classR klasa: recovered class (removed)classes M and E are often neglected

MSEIR, MSEIRS, SEIR, SEIRS, SIR, SEI, SEIS, SI, SISSIR basic modelMSEIRbirths withpassiveimmunitybirths withoutpassiveimmunitydeathdeathdeathdeathdeathClassificationof epidemiology modelsadequatecontactlatentperiodinfectiousperiodTime and age componentNumber of people in classes and the fractions in the classesBasic quantitiesin epidemiological models

passively immune fraction susceptible fractionexposed fractioninfectious fractionrecovered fractionTransfer rates: M, E, I; - number of adequate contacts in the unit of timeThreshold quantities:The basic reproduction number average number of infected people after the invasion of the disease; quantitative threshold (R0)Contact number average number of adequate contacts of any infectious person during its infectious period ()Replacement number average number of secondary infections; infectious person infects during its infectious period (R)

equality holds in the initial pointBasic quantitiesin epidemiological models

Classic, most primitive and the simplest model

S susceptibles, I infectives and R removed

Significant insight into the dynamics of infectious diseases

Basic assumptions:total population size is constantpopulation is homogeneously mixedan infectious person can only become a recovered person and cannot become a susceptible person

The classic SIR epidemic model

The classic SIR epidemic modelSpecial case of the MSEIR modelVital dynamics are neglectedMeets basic assumptionss(t), i(t) and r(t) converge

number of susceptible must be greater so the epidemic could spread throughout the population; the basic reproduction numberaverage number of adequate contacts of the infectious person during its infectious period; the contact numberthe replacement number

The classic SIR epidemic model

at each next moment less people become infectious and so the epidemic loses its strength

number of infectious increases until it reaches the maximum number of infectious The epidemic begins to lose its strength at the moment

The contact number can be experimentally calculated

The classic SIR epidemic modelsusceptible fractioninfectious fractionThe solution of the classic epidemic model (SIR)

The classic SIR endemic model

Modification of the classic epidemic modelVital dynamic (rates of birth and death) the average life expectancy

the basic reproduction number

the contact number is equal to the basic reproduction number, at any given moment, since there is an assumption that after the invasion of the disease there are no new cases of susceptible or infectious

Constant size of population is not realistic; different rates of birth and death, significant data about the deadBlack Plague (14th century) 25% of the population; AIDS

Size of population additional variable additional differential equationAll five basic groups

More accurate, more realistic and more difficult to solve

MSEIR model

MSEIR model where the size of the population changesDirectly transmitted illnessesLasting immunity

b rate of birth, d rate of death

population growth q = b d has impact on the size of population

HIV (Human immunodeficiency virus)Basic condition for the appearance of AIDSGradual decrease of the immune systemT cells, macrophage and dendritic cells; CD4+ T cellsStages of the infection:acute phase 2 to 4 weeks; symptoms of the flu; decrease number of virus cells, increase number of CD4+ T cellsseroconversion phase the immune system gets activated and the number of virus cells decreasesasymptotic phase of the infection absence of symptoms, up to 10 years, number of virus cells variesAIDS immune system is unable to fight any infectionHIV virus

P unknown function that describes the production of the virusc clearance rate constantV virus concentrationThe simplest HIVdynamic model

assumption that the drug completely blocks the virus easy to calculate incorrect, imprecise and incomplete

A model that incorporates viral productionT concentration of uninfected T cellss production rate of new uninfected T cellsp average reproduction rate of T cellsTmax maximal concentration of T cells when the reproduction stopsdT death rate of the uninfected T cellsk infection rateT* concentration of infected T cells death rate of the infected T cellsN virus particulatesc clearance rate constant

The probability of contact between T cells and HIV cells is proportional to the product of their concentrations

The system provides significant data about the concentration of the virus and the behavior of cells that the virus had impact on

The system is more accurate; can be modified in order to get results that are more preciseA model that incorporates viral productionAnalysis of themodel that incorporates viral productionConcentration of the virus is constant before the treatment

V, N, , c are const. concentration of infected cells is const.

quasi steady stateAnalysis of themodel that incorporates viral production

the virus clearance rate is greater than the speed of virus production so therefore after a finite amount of time the concentration of virus decreases to zeroAnalysis of themodel that incorporates viral production

no single point is stable, but the entire line is a set of possible equilibrium points; the multiplicity of equilibrium point provides the ability to maintain the parameters V and T* on some positive and finite valuesthe concentration of virus increases indefinitelyModels of drug therapyRT inhibitorsRT inhibitors reduce the appearance of the infected target cells.

Perfect inhibitor

T does not depend on the concentration of the virusNaive and nonrealistic, demands modificationsModels of drug therapyRT inhibitorsRT efficiency of the RT inhibitor

inhibitor is 100% efficientinhibitor has no influence

Models of drug therapyRT inhibitorsUnreal assumption, since it is known that with the decrease of virus cells, CD4+ T cells increaseRT should be much greater in order to eliminate the virus from the human body

Models of drug therapyProtease inhibitorsProtease inhibitors enable the production of infectious virus cells

VI virus cells created before the treatment; infectiousVNI virus cells created after the treatment; not infectious

100% inhibitor

Before the usage of the inhibitor, all virus cell are infectiousModels of drug therapyProtease inhibitors

more parametersunreal

Models of drug therapyImperfect protease inhibitors

PI efficiency of protease inhibitorModels of drug therapyCombination therapyThe attack on the virus in two independent points

100% efficientinhibitors

Models of drug therapyCombination therapy

Assumption of 100% efficiency model is simpler and easier to solveExponentially decreasing values medical researchers goal Not realTime the virus needs to infect cells and reproduce

V0 of virus cells infects a patient with T0 uninfected target cells

The sum of an average lifetime of a free cell and an average lifespan of an infected cellViral generation time

A way to represent the appearances in nature and especially medicine, by using different mathematical tools and rulesA way to study a behavior of the disease in a real situations

The mathematical modeling of epidemics and HIV can significantly contribute to solving these problems

Given mathematical models haven certain limitations and they can be additionally modifiedConclusionThank you for your attention!