On the interaction between the immune system and an exponentially replicating pathogen
Alberto d'Onofrio
Mathematical Biosciences & Engineering 2010, 7(3): 579-602 doi: 10.3934/mbe.2010.7.579
In this work, we generalize the Pugliese-Gandolfi Model [A. Pugliese and A. Gandolfi, Math Biosc, 214,73 (2008)] of interaction between an exponentially replicating pathogen and the immune system. After the generalization, we study the properties of boundedness and unboundedness of the solutions, and we also give a condition for the global eradication as well as for the onset of sustained oscillations. Then, we study the condition for the uniqueness of the arising limit cycle, with numerical applications to the Pugliese-Gandolfi model. By means of simulations, we also show some alternative ways to reaching the elimination of the pathogen and interesting effects linked to variations in aspecific immune response. After shortly studying some pathological cases of interest, we include in our model distributed and constant delays and we show that also delays may unstabilize the equilibria.
keywords: Immunology Delay Differential Equations. Limit cycles
Global stability of infectious disease models with contact rate as a function of prevalence index
Cruz Vargas-De-León Alberto d'Onofrio
Mathematical Biosciences & Engineering 2017, 14(4): 1019-1033 doi: 10.3934/mbe.2017053

In this paper, we consider a SEIR epidemiological model with information-related changes in contact patterns. One of the main features of the model is that it includes an information variable, a negative feedback on the behavior of susceptible subjects, and a function that describes the role played by the infectious size in the information dynamics. Here we focus in the case of delayed information. By using suitable assumptions, we analyze the global stability of the endemic equilibrium point and disease-free equilibrium point. Our approach is applicable to global stability of the endemic equilibrium of the previously defined SIR and SIS models with feedback on behavior of susceptible subjects.

keywords: Behavioral epidemiology information variable negative feedback Lyapunov function global stability
Modelling cell populations with spatial structure: Steady state and treatment-induced evolution
Alessandro Bertuzzi Alberto d'Onofrio Antonio Fasano Alberto Gandolfi
Discrete & Continuous Dynamical Systems - B 2004, 4(1): 161-186 doi: 10.3934/dcdsb.2004.4.161
Tumour cells growing around blood vessels form structures called tumour cords. We review some mathematical models that have been proposed to describe the stationary state of the cord and the cord evolution after single-dose cell killing treatment. Whereas the cell population has been represented with age or maturity structure to describe the cord stationary state, for the response to treatment a simpler approach was followed, by representing the cell population by means of the cell volume fractions. In this latter model, where transport of oxygen is included and its concentration is critical for cell viability, some constraints to be imposed on the interface separating the tumour from the necrotic region have a crucial role. An analysis of experimental data from untreated tumour cords, which involves modelling by cell age and by volume fractions, and some results about the cord response to impulsive cell killing, are also presented.
keywords: tumour cord nonlinear systems of differential and integral equations Cell population free boundary problems.
Effect of seasonality on the dynamics of an imitation-based vaccination model with public health intervention
Bruno Buonomo Giuseppe Carbone Alberto d'Onofrio
Mathematical Biosciences & Engineering 2018, 15(1): 299-321 doi: 10.3934/mbe.2018013

We extend here the game-theoretic investigation made by d'Onofrio et al (2012) on the interplay between private vaccination choices and actions of the public health system (PHS) to favor vaccine propensity in SIR-type diseases. We focus here on three important features. First, we consider a SEIR-type disease. Second, we focus on the role of seasonal fluctuations of the transmission rate. Third, by a simple population-biology approach we derive -with a didactic aim -the game theoretic equation ruling the dynamics of vaccine propensity, without employing 'economy-related' concepts such as the payoff. By means of analytical and analytical-approximate methods, we investigate the global stability of the of disease-free equilibria. We show that in the general case the stability critically depends on the 'shape' of the periodically varying transmission rate. In other words, the knowledge of the average transmission rate (ATR) is not enough to make inferences on the stability of the elimination equilibria, due to the presence of the class of latent subjects. In particular, we obtain that the amplitude of the oscillations favors the possible elimination of the disease by the action of the PHS, through a threshold condition. Indeed, for a given average value of the transmission rate, in absence of oscillations as well as for moderate oscillations, there is no disease elimination. On the contrary, if the amplitude exceeds a threshold value, the elimination of the disease is induced. We heuristically explain this apparently paradoxical phenomenon as a beneficial effect of the phase when the transmission rate is under its average value: the reduction of transmission rate (for example during holidays) under its annual average over-compensates its increase during periods of intense contacts. We also investigate the conditions for the persistence of the disease. Numerical simulations support the theoretical predictions. Finally, we briefly investigate the qualitative behavior of the non-autonomous system for SIR-type disease, by showing that the stability of the elimination equilibria are, in such a case, determined by the ATR.

keywords: Seasonality vaccination behavior public health systems game theory imitation game global stability Floquet persistence
Special issue on Erice ‘MathCompEpi 2015’ Proceedings
Alberto D'Onofrio Paola Cerrai Piero Manfredi
Mathematical Biosciences & Engineering 2018, 14(1): i-iv doi: 10.3934/mbe.201801i
keywords: Mathematical epidemiology computational epidemiology public health vaccine prevention, control eradication

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