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International audience; The coffee berry borer (CBB) Hypothenemus hampei (Coleoptera: Scolytidae) is the most important insect pest affecting coffee production worldwide and generating huge economic losses. As most of its life cycle occurs inside the coffee berry, its control is extremely difficult. To tackle this issue, we solve an optimal control problem based on a berry age-structured dynamical model that describes the infestation dynamics of coffee berries by CBB during a cropping season. This problem consists in applying a bio-insecticide at discrete times in order to maximise the economic profit of healthy coffee berries while minimising the CBB population for the next cropping season. We derive analytically the first-order necessary optimality conditions of the control problem. Numerical simulations are provided to illustrate the effectiveness of the optimal control strategy.

International audience; One of the biggest challenges for the farmer is to increase the incomes generated by his farm. This is not always easy because of pests and the scarcity of nutrients in the soil. In this work, we model, analyze and simulate a self-financing nonlinear mathematical model for cabbage crops. The model takes into account the young biomass, the adult biomass, the main cabbage pest (Plutella Xylostella) in its most devastating developmental state (larvae), the concentration of biopesticides in the culture, the external efforts and the incomes generated by the crop. We assume that the budget is allocated for purchasing new plants, biopesticides, and increasing the growth rate of crops (external efforts). This budget increases only through the sale of adult biomass. We study several variants of the main model. In these different cases, the condition in which the budget must be spent to keep the system alive is obtained. The model analysis reveals that prophylaxis can eradicate larvae in a cabbage crop. In addition, the impact of the delay in the supply of fertilizers and biopesticides to the culture is studied.

Bacteriophages or phages (viruses of bacteria) play numerous roles in shaping the diversity of bacterial communities within the human gut. Either a phage-infected bacterial cell immediately starts a lysis mechanisms (virulent/lytic infection), or it enters a stable state within the host as a prophage (lysogeny), until a trigger event, called Spontaneous Prophage Induction (SPI), allows the lysis phenomenon. We develop an approach to address the role of SPI within phage-bacteria interactions and the influence of lysogeny on the success of phage therapy. This is based on a model structure in terms of time since infection. Our analysis suggests that SPI allows lysogen bacteria to have a competitive advantage over their non-lysogenic counterparts. Moreover, the model exhibits the bistability phenomenon such that, depending on the initial conditions, the dynamic can either converge towards a purely temperate state (only lysogens and temperate phages persist) or a purely lytic state (only nonlysogens and virulent phages persist). We also highlight the existence of SPI's critical values leading to the coexistence of all states through periodic oscillations. Finally, the model suggests that efficient bacteria control through phage therapy can be achieved by minimizing lysogens' offspring.

We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms ...

International audience; The coffee berry borer (CBB), Hypothenemus hampei (Coleoptera: Scolytidae) is the most devastating insect pest of coffee throughout the world, causing a lot of economic damage in all regions where commercial coffee is grown. In this work, we formulate and study a model describing multi-seasonal infestation dynamics of coffee berries by CBB in a rainy/dry season regime, with the aim of identifying how sanitation measures can be applied to ensure CBB eradication. We employ a semi-discrete formalism to describe these dynamics as the epidemiological dynamics are best represented by a continuous model in ODE, while the remaining dynamics, that connect one season to the next, are better or more simply described by discrete-time events: berry harvest and plantation sanitation. During the rainy season, the infestation dynamics of coffee berries by CBB are described as proposed in Fotso Fotso et al (2021) with the coffee berry production rate depending on time. During the dry season, the dynamics are very similar , except that no berry growth takes place and mortality rates are higher as the environment is less favorable. The model is then completed by the discrete events that take place at the end of the rainy season and at the end of the dry season, that correspond to harvesting and plantation cleaning respectively. Based on that model, we first obtain a periodic CBB-free solution, that will be the desirable behaviour of the system. Linearising the dynamics around that solution and applying Floquet theory, we obtain stability conditions for this solution, that give threshold values for the harvesting and cleaning parameters below which asymptotic stability is ensured, hence CBB eradication. Note however that, even with perfect harvesting and cleaning, CBB could still persist through the survival of the colonising forms over the dry season.Finally, with perfect harvesting of the healthy berries at the end of the cropping season, we obtain conditions on harvesting and cleaning of the infested berries, that ensures CBB eradication at the minimum cost, for two versions of the cost.

A mathematical model presented in Berge T, Lubuma JM-S, Moremedi GM, Morris N Shava RK, A simple mathematical model for Ebola in Africa, J Biol Dyn 11(1): 42–74 (2016) for the transmission dynamics of Ebola virus is extended to incorporate vaccination and change of behavior for self-protection of susceptible individuals. In the new setting, it is shown that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number [Formula: see text] is less than or equal to unity and unstable when [Formula: see text]. In the latter case, the model system admits at least one endemic equilibrium point, which is locally asymptotically stable. Using the parameters relevant to the transmission dynamics of the Ebola virus disease, we give sensitivity analysis of the model. We show that the number of infectious individuals is much smaller than that obtained in the absence of any intervention. In the case of the mass action formulation with vaccination and education, we establish that the number of infectious individuals decreases as the intervention efforts increase. In the new formulation, apart from supporting the theory, numerical simulations of a nonstandard finite difference scheme that we have constructed suggests that the results on the decrease of the number of infectious individuals is valid.

We deal with the following question: Can the consumption of contaminated bush meat, the funeral practices and the environmental contamination explain the recurrence and persistence of Ebola virus disease outbreaks in Africa? We develop an SIR-type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses. We prove that the full model has one (endemic) equilibrium which is locally asymptotically stable whereas, it is globally asymptotically stable in the absence of the Ebola virus shedding in the environment. For the sub-model without the provision of Ebola viruses, the disease dies out or stabilizes globally at an endemic equilibrium. At the endemic level, the number of infectious is larger for the full model than for the sub-model without provision of Ebola viruses. We design a nonstandard finite difference scheme, which preserves the dynamics of the model. Numerical simulations are provided.

We propose a new deterministic mathematical model for the transmission dynamics of Ebola Virus Disease (EVD) in a complex Ebola virus life ecology. Our model captures as much as possible the features and patterns of the disease evolution as a three cycle transmission process in the two ways below. Firstly it involves the synergy between the epizootic phase (during which the disease circulates periodically amongst non-human primates populations and decimates them), the enzootic phase (during which the disease always remains in fruit bats population) and the epidemic phase (during which the EVD threatens and decimates human populations). Secondly it takes into account the well-known, the probable/suspected and the hypothetical transmission mechanisms (including direct and indirect routes of contamination) between and within the three different types of populations consisting of humans, animals and fruit bats. The reproduction number R0 for the full model with the environmental contamination is derived and the global asymptotic stability of the disease free equilibrium is established when R0andlt;1. It is conjectured that there exists a unique globally asymptotically stable endemic equilibrium for the full model when R0andgt;1. The role of a contaminated environment is assessed by comparing the human infected component for the sub-model without the environment with that of the full model. Similarly, the sub-model without animals on the one hand and the sub-model without bats on the other hand are studied. It is shown that bats influence more the dynamics of EVD than the animals. Global sensitivity analysis shows that the effective contact rate between humans and fruit bats and the mortality rate for bats are the most influential parameters on the latent and infected human individuals. Numerical simulations, apart from supporting the theoretical results and the existence of a unique globally asymptotically stable endemic equilibrium for the full model, suggest further that: (1) fruit bats are more important in the transmission processes and the endemicity level of EVD than animals. This is in line with biological findings which identified bats as reservoir of Ebola viruses; (2) the indirect environmental contamination is detrimental to human beings, while it is almost insignificant for the transmission in bats.

More than 20 outbreaks of Ebola virus disease have occurred in Africa since 1976, and yet no adequate treatment is available. Hence, prevention, control measures and supportive treatment remain the only means to avoid the disease. Among these measures, contact tracing occupies a prominent place. In this paper, we propose a simple mathematical model that incorporates imperfect contact tracing, quarantine and hospitalization (or isolation). The control reproduction number [Formula: see text] of each sub-model and for the full model are computed. Theoretically, we prove that when [Formula: see text] is less than one, the corresponding model has a unique globally asymptotically stable disease-free equilibrium. Conversely, when [Formula: see text] is greater than one, the disease-free equilibrium becomes unstable and a unique globally asymptotically stable endemic equilibrium arises. Furthermore, we numerically support the analytical results and assess the efficiency of different control strategies. Our main observation is that, to eradicate EVD, the combination of high contact tracing (up to 90%) and effective isolation is better than all other control measures, namely: (1) perfect contact tracing, (2) effective isolation or full hospitalization, (3) combination of medium contact tracing and medium isolation.