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Pharmaceuticals play a critical role in the eradication of infectious diseases. Effective pharmaceutical inventory management is important for controlling epidemics since medical resources such as pharmaceuticals, medical staff, and hospitals are limited. In this study, a novel epidemiological model is proposed to evaluate the resource requirements for pharmaceuticals and is applied to analyze different pharmaceutical inventory management strategies. We formulate the relationship between the number of infected individuals and the risk of infection to account for virus mutation. Evolutionary game theory is integrated into an epidemiological model to represent human behavioral choices. The proposed model can be developed to forecast the demand for pharmaceuticals and analyze how human behavior affects the demand of pharmaceuticals. This study found that making people aware of the risk of disease has a positive impact on both reducing the number of infections and managing the pharmaceutical inventory. The main contribution of this study is to enhance areas of research in pharmaceutical inventory management. This study revealed that the correct recognition of the risk of disease leads to appropriate pharmaceutical management. There are a few studies on the application of infectious disease models to inventory control problems. This study provides clues toward proper pharmaceutical management.

The applications of differential equations are many. However, many differential equations modelling real-world scenarios are very complex and it can be of great difficulty to find an exact solution if one even exists. Thus, it is of importance to be able to approximate solutions of differential equations. Here, a method using neural networks is explored and its performance is compared to that of a numerical method. To illustrate the method, two first order, two second order and two first order systems of ordinary differential equations are explored. The systems are the Lotka-Volterra system and the SEIR (Susceptible, Exposed, Infected, Removed) epidemiological model. The first four examples have exact solutions to compare to and the observations are then used as a basis when discussing the results of the systems. The results of the thesis show that while the neural network method takes longer to deliver an approximation, it continuously gives better approximations than the implicit Euler method used for comparison. The main contribution of this thesis is the comparison done of the performances of the neural network method and the implicit Euler method., Det finns många användningsområden för differentialekvationer. Däremot är många differentialekvationer som modellerar verkligheten komplexa och det kan vara svårt, om inte omöjligt, att hitta en exakt lösning. På grund av detta är det viktigt att ha metoder som kan approximera lösningar till differentialekvationer. Därför undersöks här en metod som använder sig av neurala nätverk. Dess resultat blir sedan jämförda med en numerisk metod. För att illustrera metoden presenteras två ekvationer av första ordningen, två ekvationer av andra ordningen och två system av differentialekvationer. Systemen som undersöks är Lotka-Volterra ekvationerna samt SEIR (Susceptible, Exposed, Infected, Removed) modellen. De första fyra exemplen som undersöks har exakta lösningar att jämföra med och dessa observationer används sedan vid diskussionerna gällande systemen. Resultaten visar att medan metoden som använder neurala nätverkar tar längre tid att exekvera, så ger metoden bättre approximationer än den implicita Euler metoden som användes som jämförelse. Det huvudsakliga bidraget med det här examensarbetet är jämförelsen av hur de två metoderna presterar.

The applications of differential equations are many. However, many differential equations modelling real-world scenarios are very complex and it can be of great difficulty to find an exact solution if one even exists. Thus, it is of importance to be able to approximate solutions of differential equations. Here, a method using neural networks is explored and its performance is compared to that of a numerical method. To illustrate the method, two first order, two second order and two first order systems of ordinary differential equations are explored. The systems are the Lotka-Volterra system and the SEIR (Susceptible, Exposed, Infected, Removed) epidemiological model. The first four examples have exact solutions to compare to and the observations are then used as a basis when discussing the results of the systems. The results of the thesis show that while the neural network method takes longer to deliver an approximation, it continuously gives better approximations than the implicit Euler method used for comparison. The main contribution of this thesis is the comparison done of the performances of the neural network method and the implicit Euler method., Det finns många användningsområden för differentialekvationer. Däremot är många differentialekvationer som modellerar verkligheten komplexa och det kan vara svårt, om inte omöjligt, att hitta en exakt lösning. På grund av detta är det viktigt att ha metoder som kan approximera lösningar till differentialekvationer. Därför undersöks här en metod som använder sig av neurala nätverk. Dess resultat blir sedan jämförda med en numerisk metod. För att illustrera metoden presenteras två ekvationer av första ordningen, två ekvationer av andra ordningen och två system av differentialekvationer. Systemen som undersöks är Lotka-Volterra ekvationerna samt SEIR (Susceptible, Exposed, Infected, Removed) modellen. De första fyra exemplen som undersöks har exakta lösningar att jämföra med och dessa observationer används sedan vid diskussionerna gällande systemen. Resultaten visar att medan metoden som använder neurala nätverkar tar längre tid att exekvera, så ger metoden bättre approximationer än den implicita Euler metoden som användes som jämförelse. Det huvudsakliga bidraget med det här examensarbetet är jämförelsen av hur de två metoderna presterar.

Distributing scarce resources such as COVID-19 vaccines is often a highly time-sensitive and mission-critical operation. Our research was prompted by a significant obstacle that the United States and other nations encountered during the early months of the COVID-19 vaccination campaign: Most COVID-19 vaccines require two doses given 3 or 4 weeks apart. Given the severely limited supply and mounting pressure on many countries to reduce hospitalizations and mortality, how to effectively roll out two-dose vaccines was a critical policy decision. In this paper, we first model and analyze inventory dynamics of the rollout process under three rollout strategies: (1) holding back second doses, (2) releasing second doses, and (3) stretching the lead time between doses. Then we develop an SEIR (susceptible, exposed, infectious, recovered) model that incorporates COVID-19 asymptomatic and symptomatic infections to evaluate these strategies in terms of infections, hospitalizations, and mortality. Among our findings, we show releasing second doses reduces infections but creates uneven vaccination patterns. In addition, to ensure second doses are given on time without holding back inventory, strictly less than half of the supply can be allocated to first-dose appointments. Stretching the between-dose lead time flattens the infection curve and reduces both hospitalizations and mortality compared with the strategy of releasing second doses. We also consider an alternative single-dose vaccine with lower efficacy and show that the vaccine can be more effective than its two-dose counterparts in reducing infections and mortality. We conduct extensive sensitivity analyses related to age composition, risk-based prioritization, supply disruptions, and disease transmissibility. Our paper provides important implications for policymakers to develop effective vaccine rollout strategies in developed and developing countries alike. More broadly, our paper sheds light on how to develop effectiv

Is it possible that the origin of Mycobacterium tuberculosis (Mtb) infection was around 70,000 years before the common era? At that time Homo sapiens was just another primate species with discrete growth and a very low-density geographic occupation. Therefore, it is difficult to understand the origin of a highly virulent obligate human pathogen. We have designed a new SEIR model (TBSpectr) that allows the differentiation of smear-positive and -negative tuberculosis. The model reconciles currently accepted growth rates for the Middle Paleolithic (0.003%/year) and Neolithic (0.1%/year). The obtained data link the origin of Mtb infection in the Middle Paleolithic to the induction of smear-negative TB, and reveal that its persistence required interrelations among hunter–gatherer groups, while the risk of human extinction was negligible. It also highlights the number of people infected per case and the fast progression to disease for Mtb infection maintenance, as well as the link between poor health in the Neolithic with the increased incidence of more severe forms of TB (smear-positive). In conclusion, our data support the origin of TB as a well-tolerated, highly persistent disease, even in low-density populations, showing the difficulty of its eradication and highlighting the necessity for providing better health conditions to humans to reduce its severity., Peer Reviewed, Objectius de Desenvolupament Sostenible::3 - Salut i Benestar, Postprint (published version)

Many infectious diseases like the Spanish flu (1918-1919) or currently SARS-CoV-2, also known as COVID19, exhibit a certain wave-like behaviour. An epidemic wave can be defined as the time-distance between two consecutive peaks of infectious population. In many cases there is a seasonal reason for a wave-like behaviour. In others, like in COVID19, this is not so clear and it could be related to several circumstances such as mobility restrictions imposed by governments, general lockdown, social contact constraints, school year, holidays... The goal of this Bachelor's degree thesis is to study epidemic waves from a dynamical systems approach, considering a simple SEIR model. We start by exploring theoretical results associated to a simple SEIR model as the basic reproduction number, Hethcote's theorem or variational equations. Then, we study how the SEIR parameter infection rate can lead to epidemic waves. Finally, we propose an original numerical method to apply our simple SEIR model to Catalonia's data. This method helps us analyse the importance of restrictions and to examine how small variations in the infection rate can lead to higher or smaller epidemic waves.

Against the COVID-19 pandemic, non-pharmaceutical interventions have been widely applied and vaccinations have taken off. The upcoming question is how the interplay between vaccinations and social measures will shape infections and hospitalizations. Hence, we extend the Susceptible-Exposed-Infectious-Removed (SEIR) model including these elements. We calibrate it to data of Luxembourg, Austria and Sweden until 15 December 2020. Sweden results having the highest fraction of undetected, Luxembourg of infected and all three being far from herd immunity in December. We quantify the level of social interaction, showing that a level around 1/3 of before the pandemic was still required in December to keep the effective reproduction number Refft below 1, for all three countries. Aiming to vaccinate the whole population within 1 year at constant rate would require on average 1,700 fully vaccinated people/day in Luxembourg, 24,000 in Austria and 28,000 in Sweden, and could lead to herd immunity only by mid summer. Herd immunity might not be reached in 2021 if too slow vaccines rollout speeds are employed. The model thus estimates which vaccination rates are too low to allow reaching herd immunity in 2021, depending on social interactions. Vaccination will considerably, but not immediately, help to curb the infection; thus limiting social interactions remains crucial for the months to come.

Against the COVID-19 pandemic, non-pharmaceutical interventions have been widely applied and vaccinations have taken off. The upcoming question is how the interplay between vaccinations and social measures will shape infections and hospitalizations. Hence, we extend the Susceptible-Exposed-Infectious-Removed (SEIR) model including these elements. We calibrate it to data of Luxembourg, Austria and Sweden until 15 December 2020. Sweden results having the highest fraction of undetected, Luxembourg of infected and all three being far from herd immunity in December. We quantify the level of social interaction, showing that a level around 1/3 of before the pandemic was still required in December to keep the effective reproduction number Refft below 1, for all three countries. Aiming to vaccinate the whole population within 1 year at constant rate would require on average 1,700 fully vaccinated people/day in Luxembourg, 24,000 in Austria and 28,000 in Sweden, and could lead to herd immunity only by mid summer. Herd immunity might not be reached in 2021 if too slow vaccines rollout speeds are employed. The model thus estimates which vaccination rates are too low to allow reaching herd immunity in 2021, depending on social interactions. Vaccination will considerably, but not immediately, help to curb the infection; thus limiting social interactions remains crucial for the months to come.

BACKGROUND: Hand, foot, and mouth disease (HFMD) is a common illness in young children. A monovalent vaccine has been developed in China protecting against enterovirus-71, bivalent vaccines preventing HFMD caused by two viruses are under development. OBJECTIVE: To predict and compare the incidence of HFMD under different vaccination scenarios in China. METHODS: We developed a compartmental model to capture enterovirus transmission and the natural history of HFMD in children aged 0-5, and calibrated to reported cases in the same age-group from 2015 to 2018. We compared the following vaccination scenarios: different combinations of monovalent and bivalent vaccine; a program of constant vaccination to that of pulse vaccination prior to seasonal outbreaks. RESULTS: We estimate 1,982,819, 2,258,846, 1,948,522 and 2,398,566 cases from 2015 to 2018. Increased coverage of monovalent vaccine from 0 to 80% is predicted to decrease the cases by 797,262 (49.1%). Use of bivalent vaccine at an 80% coverage level would decrease the cases by 828,560. Use of a 2.0× pulse vaccination for the bivalent vaccine in addition to 80% coverage would reduce cases by over one million. The estimated R0 for HFMD in 2015-2018 was 1.08, 1.10, 1.35 and 1.17. CONCLUSIONS: Our results point to the benefit of bivalent vaccine and using a pulse vaccination in specific months over routine vaccination. Other ways to control HFMD include isolation of patients in the early stage of dissemination, more frequent hand-washing and ventilation, and better treatment options for patients.

BACKGROUND: Hand, foot, and mouth disease (HFMD) is a common illness in young children. A monovalent vaccine has been developed in China protecting against enterovirus-71, bivalent vaccines preventing HFMD caused by two viruses are under development. OBJECTIVE: To predict and compare the incidence of HFMD under different vaccination scenarios in China. METHODS: We developed a compartmental model to capture enterovirus transmission and the natural history of HFMD in children aged 0-5, and calibrated to reported cases in the same age-group from 2015 to 2018. We compared the following vaccination scenarios: different combinations of monovalent and bivalent vaccine; a program of constant vaccination to that of pulse vaccination prior to seasonal outbreaks. RESULTS: We estimate 1,982,819, 2,258,846, 1,948,522 and 2,398,566 cases from 2015 to 2018. Increased coverage of monovalent vaccine from 0 to 80% is predicted to decrease the cases by 797,262 (49.1%). Use of bivalent vaccine at an 80% coverage level would decrease the cases by 828,560. Use of a 2.0× pulse vaccination for the bivalent vaccine in addition to 80% coverage would reduce cases by over one million. The estimated R0 for HFMD in 2015-2018 was 1.08, 1.10, 1.35 and 1.17. CONCLUSIONS: Our results point to the benefit of bivalent vaccine and using a pulse vaccination in specific months over routine vaccination. Other ways to control HFMD include isolation of patients in the early stage of dissemination, more frequent hand-washing and ventilation, and better treatment options for patients.

In this thesis, we present a study to obtain a clear and accurate overview of the progress and behaviour of COVID-19 in the Netherlands. We distinguish two parts for this study. The first part is to estimate the total number of infected people as a function of time by combining data from hospital admissions, daily reported cases and serological data. Using these data sets, we found that our estimation for the number of infected people was comparable to the estimations provided by the RIVM and Sanquin. Furthermore, we found that on average only 39.3% of the total number of cases were detected. 1.2% of the total number of infected people is admitted to the hospital and 18.6% of the hospitalized patients is admitted to the ICU. The second part is to develop a representative model that reproduces the estimated total number of infections using a modified SEIR model. These modifications include modelling the infection rate β(t) as a function of time using a simple linear ODE, a system of ODEs inspired by the Lotka-Volterra equations, the implementation of gamma distributed exposed and infected stages and lastly the incorporation of spatial heterogeneity. We found that our Lotka-Volterra inspired model was able to model multiple consecutive waves, which differs from the standard compartmental models. The other modifications however seemed to have only minor effects on the model and had some difficulties with matching historical data. We conclude that our Lotka-Volterra inspired model should be used to model consecutive waves for a longer period of time. The other modifications can be used to optimize the model., Applied Mathematics

BACKGROUND: Hand, foot, and mouth disease (HFMD) is a common illness in young children. A monovalent vaccine has been developed in China protecting against enterovirus-71, bivalent vaccines preventing HFMD caused by two viruses are under development. OBJECTIVE: To predict and compare the incidence of HFMD under different vaccination scenarios in China. METHODS: We developed a compartmental model to capture enterovirus transmission and the natural history of HFMD in children aged 0-5, and calibrated to reported cases in the same age-group from 2015 to 2018. We compared the following vaccination scenarios: different combinations of monovalent and bivalent vaccine; a program of constant vaccination to that of pulse vaccination prior to seasonal outbreaks. RESULTS: We estimate 1,982,819, 2,258,846, 1,948,522 and 2,398,566 cases from 2015 to 2018. Increased coverage of monovalent vaccine from 0 to 80% is predicted to decrease the cases by 797,262 (49.1%). Use of bivalent vaccine at an 80% coverage level would decrease the cases by 828,560. Use of a 2.0× pulse vaccination for the bivalent vaccine in addition to 80% coverage would reduce cases by over one million. The estimated R0 for HFMD in 2015-2018 was 1.08, 1.10, 1.35 and 1.17. CONCLUSIONS: Our results point to the benefit of bivalent vaccine and using a pulse vaccination in specific months over routine vaccination. Other ways to control HFMD include isolation of patients in the early stage of dissemination, more frequent hand-washing and ventilation, and better treatment options for patients.

BACKGROUND: Hand, foot, and mouth disease (HFMD) is a common illness in young children. A monovalent vaccine has been developed in China protecting against enterovirus-71, bivalent vaccines preventing HFMD caused by two viruses are under development. OBJECTIVE: To predict and compare the incidence of HFMD under different vaccination scenarios in China. METHODS: We developed a compartmental model to capture enterovirus transmission and the natural history of HFMD in children aged 0-5, and calibrated to reported cases in the same age-group from 2015 to 2018. We compared the following vaccination scenarios: different combinations of monovalent and bivalent vaccine; a program of constant vaccination to that of pulse vaccination prior to seasonal outbreaks. RESULTS: We estimate 1,982,819, 2,258,846, 1,948,522 and 2,398,566 cases from 2015 to 2018. Increased coverage of monovalent vaccine from 0 to 80% is predicted to decrease the cases by 797,262 (49.1%). Use of bivalent vaccine at an 80% coverage level would decrease the cases by 828,560. Use of a 2.0× pulse vaccination for the bivalent vaccine in addition to 80% coverage would reduce cases by over one million. The estimated R0 for HFMD in 2015-2018 was 1.08, 1.10, 1.35 and 1.17. CONCLUSIONS: Our results point to the benefit of bivalent vaccine and using a pulse vaccination in specific months over routine vaccination. Other ways to control HFMD include isolation of patients in the early stage of dissemination, more frequent hand-washing and ventilation, and better treatment options for patients.

BACKGROUND: Hand, foot, and mouth disease (HFMD) is a common illness in young children. A monovalent vaccine has been developed in China protecting against enterovirus-71, bivalent vaccines preventing HFMD caused by two viruses are under development. OBJECTIVE: To predict and compare the incidence of HFMD under different vaccination scenarios in China. METHODS: We developed a compartmental model to capture enterovirus transmission and the natural history of HFMD in children aged 0-5, and calibrated to reported cases in the same age-group from 2015 to 2018. We compared the following vaccination scenarios: different combinations of monovalent and bivalent vaccine; a program of constant vaccination to that of pulse vaccination prior to seasonal outbreaks. RESULTS: We estimate 1,982,819, 2,258,846, 1,948,522 and 2,398,566 cases from 2015 to 2018. Increased coverage of monovalent vaccine from 0 to 80% is predicted to decrease the cases by 797,262 (49.1%). Use of bivalent vaccine at an 80% coverage level would decrease the cases by 828,560. Use of a 2.0× pulse vaccination for the bivalent vaccine in addition to 80% coverage would reduce cases by over one million. The estimated R0 for HFMD in 2015-2018 was 1.08, 1.10, 1.35 and 1.17. CONCLUSIONS: Our results point to the benefit of bivalent vaccine and using a pulse vaccination in specific months over routine vaccination. Other ways to control HFMD include isolation of patients in the early stage of dissemination, more frequent hand-washing and ventilation, and better treatment options for patients.

Classical Continuous System Simulation (CSS) is restricted to modelling continuous flows, and therefore, cannot correctly realise a conceptual model with discrete objects. The development of Full Potential CSS solves this problem by (1) handling discrete quantities as discrete and continuous matter as continuous, (2) preserving the sojourn time distribution of a stage, (3) implementing attributes correctly, and (4) describing different types of uncertainties in a proper way. In order to apply Full Potential CSS a new software, StochSD, has been developed. This thesis evaluates StochSD's ability to model Full Potential CSS, where the points 1-4 above are included. As a test model a well-defined conceptual epidemic model, which includes all aspects of Full Potential CSS, was chosen. The study was performed by starting with a classical SIR model and then stepwise add the different aspects of the Conceptual Model. The effects of each step were demonstrated in terms of size and duration of the epidemic. Finally, the conceptual model was also realised as an Agent Based Model (ABM). The results from 10 000 replications each of the CSS and ABM models were compared and no statistical differences could be confirmed. The conclusion is that StochSD passed the evaluation.

Classical Continuous System Simulation (CSS) is restricted to modelling continuous flows, and therefore, cannot correctly realise a conceptual model with discrete objects. The development of Full Potential CSS solves this problem by (1) handling discrete quantities as discrete and continuous matter as continuous, (2) preserving the sojourn time distribution of a stage, (3) implementing attributes correctly, and (4) describing different types of uncertainties in a proper way. In order to apply Full Potential CSS a new software, StochSD, has been developed. This thesis evaluates StochSD's ability to model Full Potential CSS, where the points 1-4 above are included. As a test model a well-defined conceptual epidemic model, which includes all aspects of Full Potential CSS, was chosen. The study was performed by starting with a classical SIR model and then stepwise add the different aspects of the Conceptual Model. The effects of each step were demonstrated in terms of size and duration of the epidemic. Finally, the conceptual model was also realised as an Agent Based Model (ABM). The results from 10 000 replications each of the CSS and ABM models were compared and no statistical differences could be confirmed. The conclusion is that StochSD passed the evaluation.

We provide, calibrate and test a realistic model of the spread of SARS-Cov-2 in an economy where the population has different age groups and sectors. The model takes into account factors that have proved to be essential in explaining features of the effect of the epidemic, in particular the constraint in the number of Intensive Care Units available in a region and the different response to the epidemic of individuals of different ages. We characterize the policies of containment of the epidemic that are efficient with respect to important outcomes such as number of fatalities and GDP loss. Our main finding is that prudent policies of gradual return to work even in the short period, may save many lives with limited economic costs, as long as a threshold is not reached. Further attempts to reduce fatalities beyond this threshold cause GDP losses that become extremely large. The policies that allow this safe return to productive activity are a combination of selection criteria of individual allowed to return to work on the basis of age and risk of the sector in which they are employed.

We provide, calibrate and test a realistic model of the spread of SARS-Cov-2 in an economy where the population has different age groups and sectors. The model takes into account factors that have proved to be essential in explaining features of the effect of the epidemic, in particular the constraint in the number of Intensive Care Units available in a region and the different response to the epidemic of individuals of different ages. We characterize the policies of containment of the epidemic that are efficient with respect to important outcomes such as number of fatalities and GDP loss. Our main finding is that prudent policies of gradual return to work even in the short period, may save many lives with limited economic costs, as long as a threshold is not reached. Further attempts to reduce fatalities beyond this threshold cause GDP losses that become extremely large. The policies that allow this safe return to productive activity are a combination of selection criteria of individual allowed to return to work on the basis of age and risk of the sector in which they are employed.

Pontianak is one of the cities in the globe hit by the Covid-19 pandemic. Therefore, an effort to assess the hazard of the pandemic on a local scale is critical. This study implements the SEIR model to evaluate the dynamics of the case in the city and determine the reproduction rate that must be achieved by the local government so that the NHS will not be collapse. The modeling uses two scenarios: without and with maximum intervention. The simulation results show that the death cases in the city due to Covid-19 could reach 21,588 persons without an intervention. On the contrary, the death cases can be reduced to only 20 persons with a maximum intervention. Based on the capacity of the isolation room available in the city's hospitals, the intervention process must reduce the transmission rate to below 1.05 so that all critical cases can be handled.

Here we used the recursive point process and SEIR model to describe the transmission dynamics of pertussis. The recursive point process is a refined version of the Hawkes point process which defines productivity as a function of conditional intensity. The SEIR model is a variant of the basic compartmental model, SIR, which additionally takes into account a period when an individual is infected but not infectious yet. We attempted to fit the models to the reported cases of pertussis in Nevada since 1940. The evaluation of the model fit and prediction errors suggests that the recursive point process performs better in modeling the spread of pertussis and predicting the weekly incidence. However, we still recommend using it along with the SEIR model considering that the compartmental model has been more commonly used for an epidemic study.

Here we used the recursive point process and SEIR model to describe the transmission dynamics of pertussis. The recursive point process is a refined version of the Hawkes point process which defines productivity as a function of conditional intensity. The SEIR model is a variant of the basic compartmental model, SIR, which additionally takes into account a period when an individual is infected but not infectious yet. We attempted to fit the models to the reported cases of pertussis in Nevada since 1940. The evaluation of the model fit and prediction errors suggests that the recursive point process performs better in modeling the spread of pertussis and predicting the weekly incidence. However, we still recommend using it along with the SEIR model considering that the compartmental model has been more commonly used for an epidemic study.

A Susceptible, Exposed, Infectious, and Recovered (SEIR) type control model describing the Ebola epidemic in a population of constant size is considered over a fixed time interval. This model is an extension of the well-known SEIR model and is more suitable to the study of the control mechanism of Ebola epidemics. Along with the traditional SEIR compartments, this model contains an isolated infectious compartment representing the number of infected and exposed individuals that have been isolated from the susceptible individuals. The model has two intervention controls reflecting efforts to protect susceptible individuals from infected and exposed individuals. Additionally,there are two control functions that define efforts for the detection and isolation of infected and exposed individuals. The minimization problem of the sum of total fractions of infected and exposed individuals and total weighted costs of control constraints over a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. Accordingly, the controls are bang-bang functions determined by the corresponding switching functions. In order to estimate the number of zeros of the switching functions, a new approach is proposed based on the analys is of the Cauchy problems for the derivatives of these functions. It is found that theoptimal controls of the original problem have at most one switching. This allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of three variables. Results of the numerical solution to this problem and their analysis are provided., Se considera un modelo de tipo Susceptible, Expuesto, Infeccioso y Recuperado (SEIR) que describe la epidemia del ébola en una población de tamaño constante sobre un intervalo de tiempo fijo. Este modelo es una extensión del bien conocido modelo SEIR y es más adecuado para el estudio del mecanismo de control de la epidemia del ébola. Además de los compartimientos tradicionales del SEIR, este modelo contiene un compartimientoaisladoinfecciosoquerepresentaelnúmerodeindividuos infectados y expuestos que han sido aislados de los individuos susceptibles. El modelo tiene dos controles de intervención que reflejan los esfuerzos para proteger a los individuos susceptibles de los individuos infectados y expuestos. Adicionalmente, hay dos funciones de control que definen los esfuerzos para la detección y aislamiento de individuos infectados y expuestos. Se plantea el problema de minimización de la suma del total de cocientes de individuos infectados y expuestos y el total de costos ponderados de restricciones de control sobre un intervalo de tiempo. Para el análisis de los correspondientes controles óptimos, se usa el principio del máximo de Pontryanguin. En consecuencia, los controles son funciones bang-bang determinadas por las correspondientes funciones de cambio. Con el fin de estimar el número de ceros de las funciones de cambio, se propone un nuevo enfoque basado en el análisis de los problemas de Cauchy para las derivadas de estas funciones. Se encontró que los controles óptimos del problema original tienen a lo sumo un cambio. Esto permite la reducción del complejo problema original de control óptimo a resolver un problema mucho más simple de minimización condicional de una función de tres variables. Se presentan los resultados y análisis de la solución numérica a este problema.

A Susceptible, Exposed, Infectious, and Recovered (SEIR) type control model describing the Ebola epidemic in a population of constant size is considered over a fixed time interval. This model is an extension of the well-known SEIR model and is more suitable to the study of the control mechanism of Ebola epidemics. Along with the traditional SEIR compartments, this model contains an isolated infectious compartment representing the number of infected and exposed individuals that have been isolated from the susceptible individuals. The model has two intervention controls reflecting efforts to protect susceptible individuals from infected and exposed individuals. Additionally,there are two control functions that define efforts for the detection and isolation of infected and exposed individuals. The minimization problem of the sum of total fractions of infected and exposed individuals and total weighted costs of control constraints over a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. Accordingly, the controls are bang-bang functions determined by the corresponding switching functions. In order to estimate the number of zeros of the switching functions, a new approach is proposed based on the analys is of the Cauchy problems for the derivatives of these functions. It is found that theoptimal controls of the original problem have at most one switching. This allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of three variables. Results of the numerical solution to this problem and their analysis are provided., Se considera un modelo de tipo Susceptible, Expuesto, Infeccioso y Recuperado (SEIR) que describe la epidemia del ébola en una población de tamaño constante sobre un intervalo de tiempo fijo. Este modelo es una extensión del bien conocido modelo SEIR y es más adecuado para el estudio del mecanismo de control de la epidemia del ébola. Además de los compartimientos tradicionales del SEIR, este modelo contiene un compartimientoaisladoinfecciosoquerepresentaelnúmerodeindividuos infectados y expuestos que han sido aislados de los individuos susceptibles. El modelo tiene dos controles de intervención que reflejan los esfuerzos para proteger a los individuos susceptibles de los individuos infectados y expuestos. Adicionalmente, hay dos funciones de control que definen los esfuerzos para la detección y aislamiento de individuos infectados y expuestos. Se plantea el problema de minimización de la suma del total de cocientes de individuos infectados y expuestos y el total de costos ponderados de restricciones de control sobre un intervalo de tiempo. Para el análisis de los correspondientes controles óptimos, se usa el principio del máximo de Pontryanguin. En consecuencia, los controles son funciones bang-bang determinadas por las correspondientes funciones de cambio. Con el fin de estimar el número de ceros de las funciones de cambio, se propone un nuevo enfoque basado en el análisis de los problemas de Cauchy para las derivadas de estas funciones. Se encontró que los controles óptimos del problema original tienen a lo sumo un cambio. Esto permite la reducción del complejo problema original de control óptimo a resolver un problema mucho más simple de minimización condicional de una función de tres variables. Se presentan los resultados y análisis de la solución numérica a este problema.

The 2014 Ebola outbreak in Sierra Leone is analyzed using a susceptible-exposed-infectious-removed (SEIR) epidemic compartmental model. The discrete time-stochastic model for the epidemic evolution is coupled to a set of ordinary differential equations describing the dynamics of the expected proportions of subjects in each epidemic state. The unknown parameters are estimated in a Bayesian framework by combining data on the number of new (laboratory confirmed) Ebola cases reported by the Ministry of Health and prior distributions for the transition rates elicited using information collected by the WHO during the follow-up of specific Ebola cases. The time-varying disease transmission rate is modeled in a flexible way using penalized B-splines. Our framework represents a valuable stochastic tool for the study of an epidemic dynamic even when only irregularly observed and possibly aggregated data are available. Simulations and the analysis of the 2014 Sierra Leone Ebola data highlight the merits of the proposed methodology. In particular, the flexible modeling of the disease transmission rate makes the estimation of the effective reproduction number robust to the misspecification of the initial epidemic states and to underreporting of the infectious cases.

Tato práce se zabývá spojitými epidemiologickými deterministickými kompartmentovými modely a specifickými rysy modelování pandemie COVID-19. Numericky je studována rozdílnost různých pravděpodobnostních rozdělení setrvání jedinců v kompartmentech na základní reprodukční číslo, respektive konečnou velikost epidemie. Je navržen nový model pro retrospektivní analýzu epidemiologických dat ze severní Itálie první poloviny roku 2020. Odhad parametrů modelu je proveden minimalizací sumy váhovaných čtvercových reziduí a prohledávaní prostoru parametrů implementací algoritmu BFGS., This thesis deals with the continuous epidemiological deterministic compartmental models and the COVID-19 pandemic modeling distinctive features. The effect of different probability distributions of individuals stay in compartments is studied numerically in relation to basic reproductive number and the final size of the epidemic, respectively. New model for a retrospective analysis of the first half of 2020 northern Italy epidemiological data is proposed. The model parameters estimation is performed using minimisation of weighted sum of squared residuals and the search through parameter space with BFGS algorithm implementation.

Tato práce se zabývá spojitými epidemiologickými deterministickými kompartmentovými modely a specifickými rysy modelování pandemie COVID-19. Numericky je studována rozdílnost různých pravděpodobnostních rozdělení setrvání jedinců v kompartmentech na základní reprodukční číslo, respektive konečnou velikost epidemie. Je navržen nový model pro retrospektivní analýzu epidemiologických dat ze severní Itálie první poloviny roku 2020. Odhad parametrů modelu je proveden minimalizací sumy váhovaných čtvercových reziduí a prohledávaní prostoru parametrů implementací algoritmu BFGS., This thesis deals with the continuous epidemiological deterministic compartmental models and the COVID-19 pandemic modeling distinctive features. The effect of different probability distributions of individuals stay in compartments is studied numerically in relation to basic reproductive number and the final size of the epidemic, respectively. New model for a retrospective analysis of the first half of 2020 northern Italy epidemiological data is proposed. The model parameters estimation is performed using minimisation of weighted sum of squared residuals and the search through parameter space with BFGS algorithm implementation.

Tato práce se zabývá spojitými epidemiologickými deterministickými kompartmentovými modely a specifickými rysy modelování pandemie COVID-19. Numericky je studována rozdílnost různých pravděpodobnostních rozdělení setrvání jedinců v kompartmentech na základní reprodukční číslo, respektive konečnou velikost epidemie. Je navržen nový model pro retrospektivní analýzu epidemiologických dat ze severní Itálie první poloviny roku 2020. Odhad parametrů modelu je proveden minimalizací sumy váhovaných čtvercových reziduí a prohledávaní prostoru parametrů implementací algoritmu BFGS., This thesis deals with the continuous epidemiological deterministic compartmental models and the COVID-19 pandemic modeling distinctive features. The effect of different probability distributions of individuals stay in compartments is studied numerically in relation to basic reproductive number and the final size of the epidemic, respectively. New model for a retrospective analysis of the first half of 2020 northern Italy epidemiological data is proposed. The model parameters estimation is performed using minimisation of weighted sum of squared residuals and the search through parameter space with BFGS algorithm implementation.