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Sanja Teodorović University of Novi Sad Faculty of Science Mathematical modeling Standard (medical) studies are not enough Abstraction sufficient for further assessment of the solution; makes it possible to perceive a real problem in a simplified manner Description of real problems using various mathematical tools; used for the analysis, design and optimization Correctly set if: the solution of the initial problem exists the solution of the initial problem is unique the solution of the initial problem continuously depend on the initial conditions Mathematical modeling Mathematical 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 History Daniel Bernoulli, 1760 – smallpox; the first model True development William Hamer, 1906 – measles number of new cases depends on the concentration of sensitive and infectious Sir Ronald Ross, 1922 – malaria model of differential equations Kermack and McKendrick, 1926 condition for the occurrence of epidemic – number of sensitive is greater then a finite number middle of 20th century – accelerate d development History Passive 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) Classification of epidemiology models Five basic groups (classes): M class: class of people who have passive immunity S class: susceptible class E class: exposed class; infected but not infectious I class: infectious class R klasa: recovered class (removed) classes M and E are often neglected MSEIR, MSEIRS, SEIR, SEIRS, SIR, SEI, SEIS, SI, SIS SIR basic model Classification of epidemiology models births with passive immunity M death births without passive immunity S latent period E adequate contact death infectious period I death R death death Basic quantities in epidemiological models Time and age component Number of people in classes and the fractions in the classes passively immune fraction susceptible fraction exposed fraction infectious fraction recovered fraction Basic quantities in epidemiological models Transfer rates: M, E, I; - number of adequate contacts in the unit of time Threshold 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 (σ) Replacement number – average number of secondary infections; infectious person infects during its infectious period (R) equality holds in the initial point The classic SIR epidemic model 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 constant population is homogeneously mixed an infectious person can only become a recovered person and cannot become a susceptible person The classic SIR epidemic model Special case of the MSEIR model Vital dynamics are neglected Meets basic assumptions s(t), i(t) and r(t) converge number of susceptible must be greater so the epidemic could spread throughout the population; the basic reproduction number average number of adequate contacts of the infectious person during its infectious period; the contact number the 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 model susceptible fraction infectious fraction The solution of the classic epidemic model (SIR) The classic SIR endemic model Modification of the classic epidemic model Vital 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 MSEIR model Constant size of population is not realistic; different rates of birth and death, significant data about the dead Black Plague (14th century) – 25% of the population; AIDS Size of population additional variable additional differential equation All five basic groups More accurate, more realistic and more difficult to solve MSEIR model where the size of the population changes Directly transmitted illnesses Lasting immunity b – rate of birth, d – rate of death population growth q = b – d has impact on the size of population HIV virus HIV (Human immunodeficiency virus) Basic condition for the appearance of AIDS Gradual decrease of the immune system T cells, macrophage and dendritic cells; CD4+ T cells Stages of the infection: acute phase – 2 to 4 weeks; symptoms of the flu; decrease number of virus cells, increase number of CD4+ T cells seroconversion phase – the immune system gets activated and the number of virus cells decreases asymptotic phase of the infection – absence of symptoms, up to 10 years, number of virus cells varies AIDS – immune system is unable to fight any infection The simplest HIV dynamic model P – unknown function that describes the production of the virus c – clearance rate constant V – virus concentration assumption that the drug completely blocks the virus easy to calculate incorrect, imprecise and incomplete A model that incorporates viral production T – concentration of uninfected T cells s – production rate of new uninfected T cells p – average reproduction rate of T cells Tmax – maximal concentration of T cells when the reproduction stops dT – death rate of the uninfected T cells k – infection rate T* – concentration of infected T cells – death rate of the infected T cells N – virus particulates c – clearance rate constant A model that incorporates viral production 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 precise Analysis of the model that incorporates viral production Concentration of the virus is constant before the treatment dV 0 dt dT * 0 V, N, , c are const. concentration of infected cells is const. dt quasi – steady state Analysis of the model 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 zero Analysis of the model that incorporates viral production the concentration of virus increases indefinitely 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 values Models of drug therapy RT inhibitors RT inhibitors reduce the appearance of the infected target cells. Perfect inhibitor k 0 T does not depend on the concentration of the virus Naive and nonrealistic, demands modifications Models of drug therapy RT inhibitors RT – efficiency of the RT inhibitor inhibitor is 100% efficient inhibitor has no influence Models of drug therapy RT inhibitors Unreal assumption, since it is known that with the decrease of virus cells, CD4+ T cells increase RT should be much greater in order to eliminate the virus from the human body Models of drug therapy Protease inhibitors Protease inhibitors enable the production of infectious virus cells VI – virus cells created before the treatment; infectious VNI – virus cells created after the treatment; not infectious 100% inhibitor Models of drug therapy Protease inhibitors Before the usage of the inhibitor, all virus cell are infectious more parameters unreal Models of drug therapy Imperfect protease inhibitors PI – efficiency of protease inhibitor Models of drug therapy Combination therapy The attack on the virus in two independent points 100% efficient inhibitors Models of drug therapy Combination therapy Assumption of 100% efficiency model is simpler and easier to solve Exponentially decreasing values – medical researchers goal Not real Viral generation time Time 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 cell Conclusion A way to represent the appearances in nature and especially medicine, by using different mathematical tools and rules A 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 modified
