Salmonella is one of the major sources of toxi-infection in humans in France and UnitedStates. The incidence of human salmonellosis has considerably increased over the past 20 years and this can be largely attributed to epidemics of S. enteritidis phage type 4 in poultry in numerous countries. In this article, we formulate and analyse a model in which the transmission of the disease is determined by contact between hens and Salmonella in the environment.

Simple epidemiological models with information dependent vaccination functions cangenerate sustained oscillations via Hopf bifurcation of the endemic state. The onset of these oscillations depend on the shape of the vaccination function. A “global” approach is used to characterize the instability condition and identify classes of functions that always lead to stability/instability. The analysis allows the identification of an analytically determined “threshold vaccination function” having a simple interpretation: coverage functions lying always above the threshold always lead to oscillations, whereas coverage functions always below never lead to instability.

The aim of this paper is to study the steady states of the mathematical models with delaykernels which describe pathogen-immune dynamics of infectious diseases. In the study of mathematicalmodels of infectious diseases it is important to predict whether the infection disappearsor the pathogens persist. The delay kernel is described by the memory function that reflects theinfluence of the past density of pathogen in the blood and it is given by a nonnegative bounded andnormated function k defined on [ 0, ∞ ). By using the coefficient of the kernel k, as a bifurcationparameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of bifurcation periodic solutions are obtained by applying the normal form theory and the center manifold theorems. Some numerical simulation examples for justifying the theoretical results are also given.

We give a survey of results on global stability for deterministic compartmental epidemiologicalmodels. Using Lyapunov techniques we revisit a classical result, and give a simple proof.By the same methods we also give a new result on differential susceptibility and infectivity modelswith mass action and an arbitrary number of compartments. These models encompass the so-calleddifferential infectivity and staged progression models. In the two cases we prove that if the basicreproduction ratio $\mathcal{R}_0$ ≤ 1, then the disease free equilibrium is globally asymptotically stable. If $\mathcal{R}_0$ > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.

Previous work has shown that intracellular delay needs to be taken into account toaccurately determine the half-life of free virus from drug perturbation experiments [1]. The delayalso effects the estimated value for the infected T-cell loss rate when we assume that the drug isnot completely effective [19]. Models of virus infection that include intracellular delay are moreaccurate representations of the biological data.
We analyze a non-linear model of the human immunodeficiency virus (HIV) infection thatconsiders the interaction between a replicating virus, CD4+ T-cell and the cytotoxic-lymphocytes(CTL).We then investigate the intracellular delay effect on the stability of the endemically infectedsteady state. Criteria are given to ensure that the infected steady state is asymptotically stable forall delays. Model analysis also allows the prediction of a critical delay $\tau_c$ below which the effectorCTL can play a significant role in the immune control mechanism even when the basic reproductionnumber is high.

This paper presents mathematical models for tuberculosis and its dynamics under theimplementation of the direct observation therapy strategy (DOTS) in Nigeria. The models establishconditions for the eradication of tuberculosis in Nigeria based on the fraction of detectedinfectious individuals placed under DOTS for treatment. Both numerical and qualitative analysisof the models were carried out and the effect of the fraction of detected cases of active TB on thevarious epidemiological classes is investigated. The results showed that, provided that the fractionof detected infectious individuals exceeded a critical value, there exists a globally stable diseasefree equilibrium. However, if this critical detection level is not reached, the disease-free equilibriumwill be unstable even with the very high probability successful treatment under DOTS. Theresults showed that DOTS expansion in Nigeria must include significant increase in the detectionrate of infectious individuals; otherwise the effect in reducing the incidence in Nigeria will notbe achieved disregarding the tremendous efforts in any other direction, and the huge number ofundetected cases will make DOTS insignificant with respect to tuberculosis control.

The emergence of multidrug resistance among gram-negative bacilli is complex. Numerousfactors need to be considered, including the biological fitness cost of resistance, fitnesscompensatorymutations and frequency and type of antibiotic exposure. A mathematical modelevaluating these complex relationships was developed in an individual colonized with strains ofpan-susceptible, single-, two- and multidrug-resistant (MDR) gram-negative bacilli (GN). The effectof bacterial fitness, compensatory mutations and the frequency of three-antimicrobial regimenexposure to predominance of multidrug-resistant strains were quantified. The model predicts thatinitially, in the absence of antibiotic exposure, the biologically fitter pan-susceptible strain predominatesover the resistant strains. Over time, the fitness of the MDR strains increases fasterwith repeated antimicrobial exposure, through compensatory-fitness mutations. Increasing thefrequency of exposure to the three-antimicrobial regimen or, increasing the initial fitness of theresistant strains, substantially decreases the time to MDR-GN predominance. The model impliesthat when MDR-GN strains evolve into strains that are fitter than susceptible strains, a reductionin antimicrobial exposure may not result in a decrease of MDR-GN, since the absence of selectiveantimicrobial pressure would no longer favor susceptible strains. The model also implies thatantimicrobial cycling may promote the emergence of MDR-GN.