Amira Rachah

Mathematical Modelling, Simulation, and Optimal Control of the 2014 Ebola Outbreak in West Africa

The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.

Dynamics and Optimal Control of Ebola Transmission

A major Ebola outbreak occurs in West Africa since March 2014, being the deadliest epidemic in history. As an infectious disease epidemiology, Ebola is the most lethal and is moving faster than in previous outbreaks. On 8 August 2014, the World Health Organization (WHO) declared the outbreak a public health emergency of international concern. Last update on 7 July 2015 by WHO reports 27,609 cases of Ebola with a total of 11,261 deaths. In this work, we present a mathematical description of the spread of Ebola virus based on the SEIR (Susceptible–Exposed–Infective–Recovered) model and optimal strategies for Ebola control. In order to control the propagation of the virus and to predict the impact of vaccine programmes, we investigate several strategies of optimal control of the spread of Ebola: control infection by vaccination of susceptible; minimize exposed and infected; reduce Ebola infection by vaccination and education.

Predicting and controlling the Ebola infection

We present a comparison between two different mathematical models used in the description of the Ebola virus propagation currently occurring in West Africa. In order to improve the prediction and the control of the propagation of the virus, numerical simulations and optimal control of the two models for Ebola are investigated. In particular, we study when the two models generate similar results.

Transmission dynamics of intramammary infections caused by Corynebacterium species

The development of reliable models for transmission of intramammary infections (IMI) is the subject of extensive research. Such models are useful to enhance the identification and understanding of factors that affect pathogen-specific IMI dynamics. Longitudinal transmission models are valuable for predicting infection outbreak risks, quantifying the effectiveness of response tactics, and performing response planning. In this work, we focused on modeling Corynebacterium spp. by using a compartmental model. Previous investigations have considered modeling the transmission dynamics of several bacterial pathogens, but not Corynebacterium spp. We established a Corynebacterium spp. Susceptible-Infectious-Susceptible (SIS) model. We simulated the model numerically by using parameters that we estimated by a generalized linear model approach, using month of study as the time variable. The data, from which the parameters of the model were estimated, were obtained in a field trial conducted in 2 US dairy herds. Altogether, 786 cows were sampled at least once during the 13-mo study period. The total number of quarter milk cultures and cases of IMI caused by Corynebacterium spp. were 11,744 and 556, respectively, in farm A; the corresponding figures for farm B were 11,804 and 179. Our modeling study included only transmission from persistent IMI caused by Corynebacterium spp. within the lactation pens. The rate of new infections was significantly related to preexisting IMI in both farms, underscoring the importance of preexisting Corynebacterium spp. IMI for the transmission of Corynebacterium spp. within lactation pens. The estimated basic reproduction numbers (R0) in the 2 farms were 1.18 and 0.98, respectively. The nonsignificant disparity in R0 was associated with significant differences in cure rates between farms.

Milk-flow data collected routinely in an automatic milking system: an alternative to milking-time testing in the management of teat-end condition?

BackgroundHaving a poor teat-end condition is associated with increased mastitis risk, hence avoiding milking machine settings that have a negative effect on teat-end condition is important for successful dairy production. Milking-time testing (MTT) can be used in the evaluation of vacuum conditions during milking, but the method is less suited for herds using automatic milking systems (AMS) and relationships with teat end condition is poorly described. This study aimed to increase knowledge on interpretation of MTT in AMS and to assess whether milk-flow data obtained routinely by an AMS can be useful for the management of teat-end health. A cross-sectional study, including 251 teats of 79 Norwegian Red cows milked by AMS was performed in the research herd of the Norwegian University of Life Sciences. The following MTT variables were obtained at teat level: Average vacuum level in the short milk tube during main milking (MTVAC), average vacuum in the mouthpiece chamber during main milking and overmilking, teat compression intensity (COMPR) and overmilking time. Average and peak milk flow rates were obtained at quarter level from the AMS software. Teat-end callosity thickness and roughness was registered, and teat dimensions; length, and width at apex and base, were measured. Interrelationships among variables obtained by MTT, quarter milk flow variables, and teat dimensions were described. Associations between these variables and teat-end callosity thickness and roughness, were investigated.ResultsPrincipal component analysis showed clusters of strongly related variables. There was a strong negative relationship between MTVAC and average milk flow rate. The variables MTVAC, COMPR and average and peak milk flow rate were associated with both thickness and roughness of the callosity ring.ConclusionsQuarter milk flow rate obtained directly from the AMS software was useful in assessing associations between milking machine function and teat-end condition; low average milk flow rates were associated with a higher likelihood of the teat having a thickened or roughened teat-end callosity ring. Since information on milk flow rate is readily available from the herd management system, this information might be used when evaluating causes for impaired teat-end condition in AMS.

Modelling and dynamics of intramammary infections caused by Corynebacterium species

We present a deterministic mathematical model that describes the transmission dynamics of intramammary infections (IMI) caused by Corynebacterium spp. (Corynebacterium species) in lactating dairy herds. Longitudinal, quantitative, dynamic models are likely to be valuable for predicting infections outbreak risk, quantify the effectiveness of response tactics and performing response planning. Previous investigations on IMI have considered the transmission of several bacterial pathogens, but not Corynebacterium spp., in modelling investigations. We have established a new Corynebacterium spp. SnISw compartmental model subject on appropriate unknown parameters that we estimated by using a deterministic approach, based on real data fitting procedure. The real data, from which the parameters of the model are estimated, were obtained in a field trial conducted in a New York dairy herd (US).

Mathematical Analysis of a Continuous Crystallization Process

In this paper we discuss a mathematical model of crystallization of KCl in continuous operational mode with fines dissolution and classified product removal. We prove the global existence and the uniqueness of solutions of the model under realistic hypotheses.

Modeling and control of a semi-batch cooling seeded crystallizer

We present a mathematical model of semi-batch cooling seeded crystallization of α-lactose monohydrate. The crystallization dynamics are described by population, mass and energy balance equations including the phenomena of nucleation (crystal birth) and growth. After developing the model we present two optimal control studies, where control of temperature and feed rate are used to increase the number of particles in a given narrow target size range. We compare our results obtained by optimal control with those obtained by heuristic strategies as often recommended in the literature.

Optimal control of crystallization of alpha-lactose monohydrate

We present a mathematical model for solvated crystallization of a-lactose monohydrate in semi-batch mode. The process dynamics are governed by conservation laws including population, molar and energy balance equations. We present and discuss the model and then control the process with the goal to privilege the production of small particles in the range between 10-5 and 10-4μm. We compare several specific and unspecific cost functions leading to optimal strategies with significantly different effects on product quality. Control inputs are temperature, feed rate, and the choice of an appropriate crystal seed.