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چكيده انگليسي :

Global dynamics of an epidemiological model with age of infection and disease relapse Iman Mohammad Rahimi Esfarjani i mohammd@math iut ac ir January 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Rasoul Asheghi r asheghi@cc iut ac irAdvisor Dr Majid Salamat m salamat@cc iut ac ir2000 MSC 37N25 92B05 37B25Keywords Age of infection disease relapse Lyapunov functional LaSalle s invariance principle stabilityAbstract Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic andhelp inform public health interventions Models use some basic assumptions and mathematics to find parametersfor various infectious diseases and use those parameters to calculate the effects of different interventions like massvaccination programmes The modelling can help in deciding which intervention s to avoid and which to trial The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread to predict the future course of an outbreak and to evaluate strategies to control an epidemic The first scientist whosystematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations madeupon the Bills of Mortality in 1662 The bills he studied were listings of numbers and causes of deaths publishedweekly Graunt s analysis of causes of death is considered the beginning of the theory of competing risks whichaccording to Daley and Gani is a theory that is now well established among modern epidemiologists The earliest account of mathematical modelling of spread of disease was carried out in 1766 by Daniel Bernoulli Trained as a physician Bernoulli created a mathematical model to defend the practice of inoculating against small pox The calculations from this model showed that universal inoculation against smallpox would increase the lifeexpectancy from 26 years 7 months to 29 years 9 months Daniel Bernoulli s work preceded the modern understand ing of germ theory In the early 20th century William Hamer and Ronald Ross applied the law of mass action toexplain epidemic behaviour The 1920s saw the emergence of compartmental models The Kermack McKendrickepidemic model 1927 and the Reed Frost epidemic model 1928 both describe the relationship between suscepti ble infected and immune individuals in a population The Kermack McKendrick epidemic model was successful inpredicting the behavior of outbreaks very similar to that observed in many recorded epidemics It is well known that for some diseases recovered individuals may relapse with reactivation of latent infection andrevert back to the infective class This recurrence of disease is an important feature of some animal and human dis eases for example tuberculosis TB including human and bovine and herpes For human TB incomplete treatmentcan lead to relapse but relapse can also occur in patients who took a full course of treatment and were declared cured Herpes is a human disease that is transmitted by close physical or sexual contact Important features of herpes are thatan individual once infected remains infected for life and the virus reactivates regularly with reactivation producinga relapse period of infectiousness In this thesis an epidemiological model with age of infection and disease relapse is investigated The basic repro duction number for the model is identified and it is shown to be a sharp threshold to completely determine the globaldynamics of the model By analysing the corresponding characteristic equations the local stability of a disease freesteady state and an endemic steady state of the model is established By means of suitable Lyapunov functionals andLaSalle s invariance principle it is verified that if the basic reproduction number is less than unity the disease freesteady state is globally asymptotically stable and hence the disease dies out if the basic reproduction

محمدرحيمي اسفرجاني، ايمان

ديناميك سراسري يك مدل اپيدميولوژيك با سن عفونت و عودت بيماري

توابع لياپانوف , عدد بازتوليد پايه , اصل پايايي لاسال , پايداري سراسري