On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.This research was supported by the Spanish Institute of Health Carlos III through Grant COV 20/01213 and by the Spanish Government and European Commission through Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE)

Mathematical Modeling of Biological Systems

Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned. The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue includes innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system. Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research. The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms

Bridging between parasite genomic data and population processes: Trypanosome dynamics and the antigenic archive

Antigenic variation processes play a central role in parasite invasion and chronic infectious disease, and are likely to respond to host immune mechanisms and epidemiological characteristics. Whether changes in antigenic variation strategies lead to net positive or negative effects for parasite fitness is unclear. To improve our understanding of pathogen evolution, it is important to investigate the mechanisms by which pathogens regulate antigenic variant expression. This involves consideration of the complex interactions that occur between parasites and their hosts, and top-down and bottom-up factors that might drive changes in the genetic architecture of their antigenic archives. Increasing availability of pathogen genomic data offers new opportunities to understand the fundamental mechanisms of immune evasion and pathogen population dynamics during chronic infection. Motivated by the growing knowledge on the antigenic variation system of the sleeping sickness parasite, the African trypanosome, in this thesis, we present different models that analyze antigenic variation of this parasite at different biological scales, ranging from the within-host level, to between-host transmission, and finally the parasite genetics level. First, we describe mechanistically how the structure of the antigenic archive impacts the parasite population dynamics within a single host, and how it interplays with other within-host processes, such as parasite density-dependent differentiation into transmission life-stages and specific host immune responses. Our analysis focuses first on a single parasitaemia peak and then on the dynamics of multiple peaks that rely on stochastic switching between groups of parasite variants. We show that the interplay between the two types of parasite control within the host: specific and general, depends on the modular structure of the parasite antigenic archive. Our modelling reveals that the degree of synchronization in stochastic variant emergence (antigenic block size) determines the relative dominance of general over specific control within a single peak, and can divide infection scenarios into stationary and oscillatory regimes. A requirement for multiple-peak dynamics is a critical switch rate between blocks of antigenic variants, which depends on host characteristics, such as the immune delay, and implies constraints on variant surface glycoprotein (VSG) archive genetic diversification. Secondly, we study the interactions between the structure and function of the antigenic archive at the transmission level. By using nested modelling, we show that the genetic architecture of the archive has important consequences for pathogen fitness within and between hosts. We find host-dependent optimality criteria for the antigenic archive that arise as a result of typical trade-offs between parasite transmission and virulence. Our analysis suggests that different traits of the host population can select for different aspects of the antigenic archive, reinforcing the importance of host heterogeneity in the evolutionary dynamics of parasites. Variant-specific host immune competence is likely to select for larger antigenic block sizes. Parasite tolerance and host life-span are likely to select for whole archive expansion as more archive blocks provide the parasite with a fitness advantage. Within-host carrying capacity, resulting from density-dependent parasite regulation, is likely to impact the evolution of between-block switch rates in the antigenic archive. Our study illustrates the importance of quantifying the links between parasite genetics and within-host dynamics, and suggests that host body size might play a significant role in the evolution of trypanosomes. In Chapters 4 and 5 we consider the genetics behind trypanosome antigenic variation. Antigen switch rates are thought to depend on a range of genetic features, among which, the genetic identity between the switch-off and switch-on gene. The subfamily structure of the VSG archive is important in providing the conditions for this type of switching to occur. We develop a hidden Markov model to describe and estimate evolutionary processes generating clustered patterns of genetic identity between closely related gene sequences. Analysis of alignment data from high-identity VSG genes in the silent antigen gene archive of the African trypanosome identifies two scales of subfamily diversification: local clustering of sequence mismatches, a putative indicator of gene conversion events with other lower-identity donor genes in the archive, and the sparse scale of isolated mismatches, likely to arise from independent point mutations. In addition to quantifying the respective rates of these two processes, our method yields estimates for the gene conversion tract length distribution and the average diversity contributed locally by conversion events. Model fitting is conducted for a range of models using a Bayesian framework. We find that gene conversion events with lower-identity partners are at least 5 times less common than point mutations for VSG pairs, and the average imported conversion tract is short. However, due to the high frequency of mismatches in converted segments, the two processes have almost equal impact on the rate of sequence diversification between VSG sub-family members. We are able to disentangle the most likely locations of point mutations vs. conversions on each aligned gene pair. Finally we model VSG archive diversification at the global scale, as a result of opposing evolutionary forces: point mutation, which induces diversification, and gene conversion, which promotes global homogenization. By adopting stochastic simulation and theoretical approaches such as population genetics and the diffusion approximation, we find how the stationary identity configuration of the archive depends on mutation and conversion parameters. By fitting the theoretical form of the distribution to the current VSG archive configuration, we estimate the global rates of gene conversion and point mutation. The relative dominance of mutation as an evolutionary force quantifies the high divergence propensity of VSG genes in response to host immune pressures. The success of our models in describing realistic infection patterns and making predictions about the fitness consequences of the parasite antigenic archive illustrates the advantage of using integrative approaches that bridge between different biological scales. Even though quantifying the genetic signatures of antigenic variation remains a challenging task, cross-disciplinary analyses and mechanistic modelling of parasite genomic data can help in this direction, to better understand parasite evolution

Optimization and control of virus-host systems

Optimization and control are powerful tools to design a system that works as effectively as possible. In this thesis, we focus on applications of model-based optimization and control in complex virus-host systems at multiple scales. Viruses that infect bacteria, i.e., bacteriophage or ‘phage’, are increasingly considered as treatment options for the control and clearance of bacterial infections, particularly as compassionate use therapy for multi-drug resistant infections. Here, we evaluate principles underlying why careful application of multiple phage (i.e., a ‘cocktail’) might lead to therapeutic success in contrast to the failure of single-strain phage therapy to control an infection. We combine dynamical modeling of phage, bacteria, and host immune cell populations with control-theoretic principles (via optimal control theory) to devise phage cocktails and delivery schedules to control the bacterial populations. However, a risk in using cocktails of different phage is that bacteria could simultaneously develop resistance to all injected phage (i.e., selecting for multi-phage resistant). The next step is to understand how to pre-select phage that have adapted via co-evolution with bacterial strains and then to efficiently use these ‘future’ phage to clear the infection early on. In doing so, we develop the evolutionarily robust phage therapy in immunodeficient hosts given the infection networks that was identified in co-evolutionary training. Optimization and control not only can be applied to bacteria-phage-immune systems (i.e., at the microbial level) to help design phage therapy, but also can be applied to epidemiological systems (i.e., at the large-scale population level) to guide the development and deployment of efficient interventions. Lockdowns and stay-at-home orders have reduced the transmission of SARS-CoV-2 but have come with significant social and economic costs. Here, we describe a control theory framework combining population-scale viral and serological testing as part of an individualized approach to control COVID-19 spread. The aim is to develop policies for modulating individualize contact rates depending on both personalized disease status and the status of the epidemic at the population scale. Altogether in this thesis, we apply control strategies to alleviate the burden or spread of disease at multiple scales.Ph.D

Robust Sliding Control of SEIR Epidemic Models

This paper is aimed at designing a robust vaccination strategy capable of eradicating an infectious disease from a population regardless of the potential uncertainty in the parameters defining the disease. For this purpose, a control theoretic approach based on a sliding-mode control law is used. Initially, the controller is designed assuming certain knowledge of an upper-bound of the uncertainty signal. Afterwards, this condition is removed while an adaptive sliding control system is designed. The closed-loop properties are proved mathematically in the nonadaptive and adaptive cases. Furthermore, the usual sign function appearing in the sliding-mode control is substituted by the saturation function in order to prevent chattering. In addition, the properties achieved by the closed-loop system under this variation are also stated and proved analytically. The closed-loop system is able to attain the control objective regardless of the parametric uncertainties of the model and the lack of a priori knowledge on the system.This work was partially supported by the Spanish Ministry of Economy and Competitiveness through Grant no. DPI2012-30651, the Basque Government (Gobierno Vasco) through Grant no. IE378-10, and by the University of the Basque Country through Grant no. UFI 11/07

Dynamic analysis and optimal control of a novel fractional-order 2I2SR rumor spreading model

In this paper, a novel fractional-order 2I2SR rumor spreading model is investigated. Firstly, the boundedness and uniqueness of solutions are proved. Then the next-generation matrix method is used to calculate the threshold. Furthermore, the stability of rumor-free/spreading equilibrium is discussed based on fractional-order Routh–Hurwitz stability criterion, Lyapunov function method, and invariance principle. Next, the necessary conditions for fractional optimal control are obtained. Finally, some numerical simulations are given to verify the results