Transcript T - Math & Computer

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