<p>We review a number of compartmental models in epidemiology which leads to a nonlinear system of ordinary differential equations. We focus an SIR, SEIR and SIS epidemic models with and without vaccination. A threshold parameter R0 is identified which governs the spread of diseases, and this parameter is known as the basic reproductive number. The models have at least two equilibria, an endemic equilibrium and the disease-free equilibrium. We demonstrate that the disease will die out, if the basic reproductive number R0 &lt / 1. This is the case of a disease-free&nbsp / state, with no infection in the population. Otherwise the disease may become endemic if the basic reproductive number R0 is bigger than unity. Furthermore, stability analysis for both endemic&nbsp / and disease-free steady states are investigated and we also give some numerical simulations. The second part of this dissertation deals with optimal vaccination strategy in epidemiology. We&nbsp / use optimal control technique on vaccination to minimize the impact of the disease. Hereby we mean minimizing the spread of the disease in the population, while also minimizing the effort on&nbsp / vaccination roll-out. We do this optimization for the cases of SIR and SEIR models, and show how optimal strategies can be obtained which minimize the damage caused by the infectious&nbsp / disease. Finally, we describe the numerical simulations using the fourth-order Runge-Kutta method.&nbsp / These are the most useful references: [G. Zaman, Y.H Kang, II. H. Jung. BioSystems 93,&nbsp / (2008), 240 &minus / 249], [K. Hattaf, N. Yousfi. The Journal of Advanced Studies in Biology, Vol. 1(8), (2008), 383 &minus / 390.], [Lenhart, J.T. Workman. Optimal Control and Applied to Biological Models.&nbsp / Chapman and Hall/CRC, (2007).], [P. Van den Driessche, J. Watmough. Math. Biosci., 7,&nbsp / (2005)], and [J. Wu, G. R&uml / ost. Mathematical Biosciences and Engineering, Vol 5(2), (2008), 389 &minus / 391].</p>

>Magister Scientiae - MSc / We review a number of compartmental models in epidemiology which leads to a nonlinear system of ordinary differential equations. We focus an SIR, SEIR and SIS epidemic models with and without vaccination. A threshold parameter R0 is identified which governs the spread of diseases, and this parameter is known as the basic reproductive number. The models have at least two equilibria, an endemic equilibrium and the disease-free equilibrium. We demonstrate that the disease will die out, if the basic reproductive number R0 < 1. This is the case of a disease-free state, with no infection in the population. Otherwise the disease may become endemic if the basic reproductive number R0 is bigger than unity. Furthermore, stability analysis for both endemic and disease-free steady states are investigated and we also give some numerical simulations. The second part of this dissertation deals with optimal vaccination strategy in epidemiology. We use optimal control technique on vaccination to minimize the impact of the disease. Hereby we mean minimizing the spread of the disease in the population, while also minimizing the effort on vaccination roll-out. We do this optimization for the cases of SIR and SEIR models, and show how optimal strategies can be obtained which minimize the damage caused by the infectious disease. Finally, we describe the numerical simulations using the fourth-order Runge-Kutta method.

<p><b>In this dissertation we present the quantitative response of an epidemic of the so-called SIR-type, in a population consisting of a local component and a migrant component. Each component can be divided into three classes, the susceptible individuals, usually denoted by S, who are uninfected but may contract the disease, infected individuals (I) who are infected and can spread the disease to the susceptible individuals and the class (R) of recovered individuals. If a susceptible individual becomes infected, it moves into the infected class. An infected individual, at recovery, moves to the class R. Firstly we develop a model describing two interacting populations with vaccination. Assuming the vaccination rate in both groups or components are constant, we calculate a threshold parameter and we call it a vaccination reproductive number. This invariant determines whether the disease will die out or becomes endemic on the (in particular, local) population. Then we present the stability analysis of equilibrium points and the effect of vaccination. Our primary finding is that the behaviour of the disease free equilibrium depend on the vaccination rates of the combined population. We show that the disease free equilibrium is locally asymptotically stable if the vaccination reproductive number is less than one. Also our stability analysis show that the global stability of the disease free equilibrium depends on the basic reproduction number, not the vaccination reproductive number. If the vaccination reproductive number is greater than one, then the disease free equilibrium is unstable and there exists three endemic equilibrium points in our model. Two of these three endemic equilibria are so-called boundary equilibrium points, which means that the infection is only in one group of the population. The third one which we focus on is the general endemic point for the whole system. We derive a threshold condition that determines whether the endemic equilibria is locally asymptotically stable or not. Secondly, by assuming that the rate of vaccination in the migrant population is constant, we apply optimal control theory to find an optimal vaccination strategy in the local population. Our numerical simulation shows the effectiveness of the control strategy. This model is suitable for modeling the real life situation to control many communicable diseases. Models similar to the model used in the main contribution of our dissertation do exist in the literature. In fact, our model can be regarded as being in-between those of [Jia et al., Theoretical Population Biology 73 (2008) 437-448] and [Piccolo and Billings, Mathematical and Computer Modeling 42 (2005) 291-299]. Nevertheless our stability analysis is original, and furthermore we perform an optimal control study whereas the two cited papers do not. The essence of chapter 5 and 6 of this dissertation is being prepared for publication.</b></p>

<p><b>In this dissertation we present the quantitative response of an epidemic of the so-called SIR-type, in a population consisting of a local component and a migrant component. Each component can be divided into three classes, the susceptible individuals, usually denoted by S, who are uninfected but may contract the disease, infected individuals (I) who are infected and can spread the disease to the susceptible individuals and the class (R) of recovered individuals. If a susceptible individual becomes infected, it moves into the infected class. An infected individual, at recovery, moves to the class R. Firstly we develop a model describing two interacting populations with vaccination. Assuming the vaccination rate in both groups or components are constant, we calculate a threshold parameter and we call it a vaccination reproductive number. This invariant determines whether the disease will die out or becomes endemic on the (in particular, local) population. Then we present the stability analysis of equilibrium points and the effect of vaccination. Our primary finding is that the behaviour of the disease free equilibrium depend on the vaccination rates of the combined population. We show that the disease free equilibrium is locally asymptotically stable if the vaccination reproductive number is less than one. Also our stability analysis show that the global stability of the disease free equilibrium depends on the basic reproduction number, not the vaccination reproductive number. If the vaccination reproductive number is greater than one, then the disease free equilibrium is unstable and there exists three endemic equilibrium points in our model. Two of these three endemic equilibria are so-called boundary equilibrium points, which means that the infection is only in one group of the population. The third one which we focus on is the general endemic point for the whole system. We derive a threshold condition that determines whether the endemic equilibria is locally asymptotically stable or not. Secondly, by assuming that the rate of vaccination in the migrant population is constant, we apply optimal control theory to find an optimal vaccination strategy in the local population. Our numerical simulation shows the effectiveness of the control strategy. This model is suitable for modeling the real life situation to control many communicable diseases. Models similar to the model used in the main contribution of our dissertation do exist in the literature. In fact, our model can be regarded as being in-between those of [Jia et al., Theoretical Population Biology 73 (2008) 437-448] and [Piccolo and Billings, Mathematical and Computer Modeling 42 (2005) 291-299]. Nevertheless our stability analysis is original, and furthermore we perform an optimal control study whereas the two cited papers do not. The essence of chapter 5 and 6 of this dissertation is being prepared for publication.</b></p>

>Magister Scientiae - MSc / In this dissertation we present the quantitative response of an epidemic of the so-called SIR-type, in a population consisting of a local component and a migrant component. Each component can be divided into three classes, the susceptible individuals, usually denoted by S, who are uninfected but may contract the disease, infected individuals (I) who are infected and can spread the disease to the susceptible individuals and the class (R) of recovered individuals. If a susceptible individual becomes infected, it moves into the infected class. An infected individual, at recovery, moves to the class R. Firstly we develop a model describing two interacting populations with vaccination. Assuming the vaccination rate in both groups or components are constant, we calculate a threshold parameter and we call it a vaccination reproductive number. This invariant determines whether the disease will die out or becomes endemic on the (in particular, local) population. Then we present the stability analysis of equilibrium points and the effect of vaccination. Our primary finding is that the behaviour of the disease free equilibrium depend on the vaccination rates of the combined population. We show that the disease free equilibrium is locally asymptotically stable if the vaccination reproductive number is less than one. Also our stability analysis show that the global stability of the disease free equilibrium depends on the basic reproduction number, not the vaccination reproductive number. If the vaccination reproductive number is greater than one, then the disease free equilibrium is unstable and there exists three endemic equilibrium points in our model. Two of these three endemic equilibria are so-called boundary equilibrium points, which means that the infection is only in one group of the population. The third one which we focus on is the general endemic point for the whole system. We derive a threshold condition that determines whether the endemic equilibria is locally asymptotically stable or not. Secondly, by assuming that the rate of vaccination in the migrant population is constant, we apply optimal control theory to find an optimal vaccination strategy in the local population. Our numerical simulation shows the effectiveness of the control strategy. This model is suitable for modeling the real life situation to control many communicable diseases. Models similar to the model used in the main contribution of our dissertation do exist in the literature. In fact, our model can be regarded as being in-between those of [Jia et al., Theoretical Population Biology 73 (2008) 437-448] and [Piccolo and Billings, Mathematical and Computer Modeling 42 (2005) 291-299]. Nevertheless our stability analysis is original, and furthermore we perform an optimal control study whereas the two cited papers do not. The essence of chapter 5 and 6 of this dissertation is being prepared for publication. / South Africa

Comparatively, HIV like most viruses is very minute, unadorned organism which cannot reproduce unaided. It remains the most deadly disease which has ever hit the planet since the last three decades. The spread of HIV has been very explosive and mercilessly on human population. It has tainted over 60 million people, with almost half of the human population suffering from AIDS related illnesses and death finally. Recent theoretical and computational breakthroughs in delay differential equations declare that, delay differential equations are proficient in yielding rich and plausible dynamics with reasonable parametric estimates. This paper seeks to unveil the niche of delay differential equation in harmonizing low HIV viral haul and thereby articulating the adopted model, to delve into structured treatment interruptions. Therefore, an ordinary differential equation is schemed to consist of three components such as untainted CD4+ T-cells, tainted CD4+ T-cells (HIV) and CTL. A discrete time delay is ushered to the formulated model in order to account for vital components, such as intracellular delay and HIV latency which were missing in previous works, but have been advocated for future research. It was divested that when the reproductive number was less than unity, the disease free equilibrium of the model was asymptotically stable. Hence the adopted model with or without the delay component articulates less production of virions, as per the decline rate. Therefore CD4+ T-cells in the blood remains constant at 𝛿1/𝛿3, hence declining the virions level in the blood. As per the adopted model, the best STI practice is intimated for compliance. / Mathematical Sciences / Ph.D. (Applied Mathematics)

Tato práce je přehledem zpožděných diferenciálních rovnic v dynamických systémech. Počínaje obecným přehledem zpožděných diferenciálních rovnic představujeme koncept zpožděných diferenciálů a použití jeho modelů, od biologie a populační dynamiky po fyziku a inženýrství. Poskytneme také přehled Dynamické systémy a diferenciální rovnice zpoždění v dynamických systémech. Oblastí pro modelování s rovnicemi zpožďovacích diferenciálů je Epidemiologie. Důraz je kladen na vývoj epidemiologického modelu Susceptible-Infected-Removed (SIR) bez časového zpoždění. Analyzujeme naše dva modely v rovnováze a lokální stabilitě pomocí předpokládaných dat COVID -19. Výsledky by byly porovnány mezi modelem bez zpoždění a modelem se zpožděním.

[ES] Actualmente existe una clara concienciación de la población por la sostenibilidad, el cuidado del medio ambiente y el bienestar animal. Pero, además los consumidores exigen alimentos seguros lo que implica a toda la cadena productiva empezando por la producción primaria. Un adecuado control de las enfermedades transmisibles a este nivel es uno de los pilares fundamentales de la seguridad alimentaria junto con el control en el momento del sacrificio, procesado y distribución. En esta tesis se plantea la utilización de herramientas matemáticas que permitan optimizar el uso de las medidas de bioseguridad, de implantación general en granjas de aves, como son la vacunación, la limpieza y desinfección, y la detección y eliminación de animales infectados. Esto con la fi nalidad de lograr una producción libre de infección y por lo tanto evitar el sacrificio temprano de los animales. De esta forma, se puede contribuir a la sostenibilidad de las granjas. Además, de garantizar la inocuidad de los alimentos a nivel de la producción primaria. Así se ha estudiado el comportamiento de un modelo matemático estructurado que incorpora la contaminación del medio ambiente como un modo indirecto de transmisión de la enfermedad, centrándose en el análisis de un brote de Salmonella en una granja de pollos. Las variables consideradas han sido: individuos susceptibles e infectados y la cantidad de bacterias acumuladas en el recinto (sistema (SIB), y se considera la reposición de los individuos muertos de forma que el tamaño de la población es el mismo en cualquier etapa. El sistema se considera dinámico y no lineal, en tiempo discreto y por ello su modelización se basa en ecuaciones en diferencias. Se ha analizado el comportamiento del sistema alrededor de los puntos de equilibrio a) libre de enfermedad y b) endémico. Tras el análisis del proceso se ha obtenido el número reproductivo básico R0. Este número indica el comportamiento de la enfermedad, ya que si R0 es menor que la unidad, la enfermedad tiende a desaparecer pero en caso contrario la enfermedad permanece endémica o puede llegar a crecer. El resultado obtenido del modelo indica que R0 es menor que uno, si y sólo si, la población se mantiene por debajo de cierto valor umbral, lo que permite tener la enfermedad controlada hacia su desaparición. También, se han estudiado tres modelos para conseguir redirigir la evolución de la enfermedad hacia su desaparición considerando las siguientes medidas: a) vacunación, b) limpieza y desinfección periódica y c) análisis y eliminación periódica de individuos infectados. Los objetivos a alcanzar con el modelo propuesto fueron que la vacunación redujese la incidencia de la enfermedad entre los sujetos susceptibles y determinar su impacto sobre la incidencia. Respecto a la desinfección del recinto y la eliminación de infectados, el objetivo ha sido construir, en cada caso, un nuevo sistema dinámico con coefi cientes periódicos que representase matemáticamente la estrategia de acción periódica elegida. La fi nalidad ha sido optimizar el número de etapas que se puede estar sin actuar sobre el proceso y manteniéndose este estable, es decir con el número reproductivo básico menor que la unidad. Y, por último, se han comparado ambas estrategias, en base a sus periodos máximos. Los resultados obtenidos indican que, respecto a la efectividad de la vacunación, el nuevo número reproductivo básico es función de la tasa de vacunación y de la tasa de efectividad de la vacuna. Si el proceso transcurre con un número reproductivo básico muy alto se requiere vacunar a un mayor número de individuos. Además, cuanto más efectiva sea la vacuna la tasa de vacunación se puede reducir. Para el modelo de impacto la vacunación, se ha indicado que la tasa de vacunación en los programas de vacunación se reduce si el impacto de ésta es positivo reduciendo la tasa de contagios entre los vacunados respecto a la de los susceptibles. / [CA] Actualment existeix una clara conscienciacio de la poblacio per la sostenibilitat, la cura del medi ambient i el benestar animal. Pero, a mes els consumidors exigeixen aliments segurs el que implica a tota la cadena productiva comencant per la produccio primaria. Un adequat control de les malalties transmissibles a aquest nivell es un dels pilars fonamentals de la seguretat alimentaria juntament amb el control en el moment del sacri ci, processament i distribucio. En aquesta tesi es planteja la utilitzacio d'eines matematiques que permeten optimitzar l'us de les mesures de bioseguretat, d'implantacio general en granges d'ocells, com son la vacunacio, la neteja i desinfeccio, i la deteccio i eliminaci o d'animals infectats. Aixo amb la nalitat d'aconseguir una produccio lliure d'infeccio i per tant evitar el sacri ci primerenc dels animals. D'aquesta manera, es pot contribuir a la sostenibilitat de les granges. A mes, de garantir la innocu tat dels aliments a nivell de la produccio primaria. Aix, s'ha estudiat el comportament d'un model matematic estructurat que incorpora la contaminacio del medi ambient com una manera indirecta de transmissio de la malaltia, centrant-se en l'analisi d'un brot de Salmonella en una granja de pollastres. Les variables considerades han sigut: individus susceptibles i infectats i la quantitat de bacteris acumulats en el recinte (sistema SIB), i, a mes, es considera reposicio dels individus morts de manera que la grandaria de la poblacio es el mateix en qualsevol etapa. El sistema es considera dinamic i no lineal, en temps discret i per aixo la seua modelitzacio es basa en equacions en diferencies. S'ha analitzat el comportament del sistema al voltant dels punts d'equilibri a) lliure de malaltia i b) endemic. Despres de l'analisi del proces s'ha obtingut el numero reproductiu basic R0. Aquest numero indica el comportament de la malaltia, ja ix que si R0. es menor que la unitat, la malaltia tendeix a desapareixer pero en cas contrari la malaltia roman endemica o pot arribar a creixer. El resultat obtingut del model indica que R0. es menor que un, si i nomes si, la poblacio es mante per davall d'un cert valor llindar, la qual cosa permet tindre la malaltia controlada cap a la seua desaparicio. Tambe, s'han estudiat tres models per a aconseguir redirigir l'evolucio de la malaltia cap a la seua desaparicio considerant les seg uents mesures: a) vacunacio, b) neteja i desinfeccio periodica i c) analisi i eliminacio periodica d'individus infectats. Els objectius a aconseguir amb el model proposat van ser que la vacunacio redu ra la incidencia de la malaltia entre els subjectes susceptibles i determinar el seu impacte sobre la incidencia. Respecte a la desinfeccio del recinte i l'eliminacio d'infectats, l'objectiu ha sigut construir, en cada cas, un nou sistema dinamic amb coe cients periodics que representara matematicament l'estrategia d'accio periodica triada. La nalitat ha sigut optimitzar el nombre d'etapes que es pot estar sense actuar sobre el proces i mantenint-se aquest estable, es a dir amb el numero reproductiu basic menor que la unitat. I, nalment, s'han comparat totes dues estrategies, sobre la base dels seus perodes maxims. Els resultats obtinguts indiquen que, respecte a l'efectivitat de la vacunacio, el nou numero reproductiu basic es funcio de la taxa de vacunacio i de la taxa d'efectivitat de la vacuna. Si el proces transcorre amb un numero reproductiu basic molt alt es requereix vacunar a un major nombre d'individus. A mes, com mes efectiva siga la vacuna la taxa de vacunacio es pot reduir. Per al model d'impacte la vacunacio, s'ha indicat que la taxa de vacunacio en els programes de vacunacio es redueix si l'impacte d'aquesta es positiu reduint la taxa de contagis entre els vacunats respecte a la dels susceptibles. / [EN] There is currently a clear awareness of the population for sustainability, care for the environment and animal welfare. But in addition, consumers demand safe food, which involves the entire production chain, starting with primary production. Adequate control of communicable diseases at this level is one of the fundamental pillars of food security along with control at the time of slaughter, processing and distribution. This thesis proposes the use of mathematical tools to optimize the use of biosafety measures, general implementation in bird farms, such as vaccination, cleaning and disinfection, and the detection and elimination of infected animals. This in order to achieve an infection free production and therefore avoid early slaughter of animals. In this way, it can contribute to the sustainability of farms. In addition, to ensure food safety at the level of primary production. Thus, the behavior of a structured mathematical model that incorporates environmental pollution as an indirect mode of disease transmission has been studied, focusing on the analysis of a Salmonella outbreak on a farm. chickens. The variables considered were susceptible and infected individuals and the amount of bacteria accumulated in the enclosure (SIB system), and, in addition, replacement of dead individuals is considered so that the size of the population is the same at any stage. The system is considered dynamic and nonlinear, in discrete time and therefore its modeling is based on equations in dierences. The behavior of the system around the equilibrium points a) free of disease and b) endemic has been analyzed. After the analysis of the process the basic reproductive number R0. was obtained. This number indicates the behavior of the disease, as if R0 is less than unity, the disease tends to disappear but otherwise the disease remains xiii endemic or may grow. The result obtained from the model indicates that R0 is less than one, if and only if, the population remains below a certain threshold value, which allows to have the disease controlled towards its disappearance. Also, three models have been studied to redirect the evolution of the disease towards its disappearance considering the following measures: a) vaccination, b) periodic cleaning and disinfection and c) periodic analysis and elimination of infected individuals. The objectives to be achieved with the proposed model were that vaccination would reduce the incidence of the disease among susceptible subjects and determine its impact on the incidence. Regarding the disinfection of the enclosure and the elimination of infected, the aim has been to build, in each case, a new dynamic system with periodic coecients that will mathematically represent the chosen periodic action strategy. The aim has been to optimize the number of stages that can be left without acting on the process and keeping it stable, ie with the basic reproductive number less than the unit. And nally, both strategies have been compared, based on their maximum periods. The results obtained indicate that, with respect to the eectiveness of vaccination, the new basic reproductive number is a function of the vaccination rate and the vaccine eectiveness rate. If the process proceeds with a very high basic reproductive number it is required to vaccinate a larger number of individuals. In addition, the more eective the vaccine, the lower the vaccination rate. For the vaccination impact model, it has been indicated that the vaccination rate in vaccination programs is reduced if the impact of this is positive by reducing the rate of transmission among vaccinated compared to those susceptible. / Poveda Giner, JJ. (2022). Análisis de procesos epidemiológicos mediante modelos matemáticos: aplicación a la seguridad alimentaria [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/182456 / TESIS

Master of Science / Department of Electrical and Computer Engineering / Faryad Darabi Sahneh / Recent experience of Ebola outbreak of 2014 highlighted the importance of immediate response to impede Ebola transmission at its very early stage. To this aim, efficient and effective allocation of limited resources is crucial. Among standard interventions is the practice of following up with physical contacts of individuals diagnosed with Ebola virus disease -- known as contact tracing. In an effort to objectively understand the effect of possible contact tracing protocols, we explicitly develop a model of Ebola transmission incorporating contact tracing. Our modeling framework has several features to suit early–stage Ebola transmission: 1) the network model is patient–centric because when number of infected cases are small only the myopic networks of infected individuals matter and the rest of possible social contacts are irrelevant, 2) the Ebola disease model is individual–based and stochastic because at the early stages of spread, random fluctuations are significant and must be captured appropriately, 3) the contact tracing model is parameterizable to analyze the impact of critical aspects of contact tracing protocols. Notably, we propose an activity driven network approach to contact tracing, and develop a Monte-Carlo method to compute the basic reproductive number of the disease spread in different scenarios. Exhaustive simulation experiments suggest that while contact tracing is important in stopping the Ebola spread, it does not need to be done too urgently. This result is due to rather long incubation period of Ebola disease infection. However, immediate hospitalization of infected cases is crucial and requires the most attention and resource allocation. Moreover, to investigate the impact of mitigation strategies in the 2014 Ebola outbreak, we consider reported data in Guinea, one the three West Africa countries that had experienced the Ebola virus disease outbreak. We formulate a multivariate sequential Monte Carlo filter that utilizes mechanistic models for Ebola virus propagation to simultaneously estimate the disease progression states and the model parameters according to reported incidence data streams. This method has the advantage of performing the inference online as the new data becomes available and estimating the evolution of the basic reproductive ratio R₀(t) throughout the Ebola outbreak. Our analysis identifies a peak in the basic reproductive ratio close to the time of Ebola cases reports in Europe and the USA.