Van Den Driessche Research Articles

New Results and Open Questions for SIR-PH Epidemic Models with Linear Birth Rate, Loss of Immunity, Vaccination, and Disease and Vaccination Fatalities

Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, and states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the “R0 alternative” of Van den Driessche and Watmough, which states that the local stability condition of the disease-free equilibrium may be expressed as R0<1, where R0 is the famous basic reproduction number, which also plays a major role in the theory of branching processes. The literature suggests that it is impossible to find general laws concerning the endemic points. However, it is quite common that 1. When R0>1, there exists a unique fixed endemic point, and 2. the endemic point is locally stable when R0>1. One would like to establish these properties for a large class of realistic epidemic models (and we do not include here epidemics without casualties). We have introduced recently a “simple” but broad class of “SIR-PH models” with varying populations, with the express purpose of establishing for these processes the two properties above. Since that seemed still hard, we have introduced a further class of “SIR-PH-FA” models, which may be interpreted as approximations for the SIR-PH models, and which include simpler models typically studied in the literature (with constant population, without loss of immunity, etc.). For this class, the first “endemic law” above is “almost established”, as explicit formulas for a unique endemic point are available, independently of the number of infectious compartments, and it only remains to check its belonging to the invariant domain. This may yet turn out to be always verified, but we have not been able to establish that. However, the second property, the sufficiency of R0>1 for the local stability of an endemic point, remains open even for SIR-PH-FA models, despite the numerous particular cases in which it was checked to hold (via Routh–Hurwitz time-onerous computations, or Lyapunov functions). The goal of our paper is to draw attention to the two open problems above, for the SIR-PH and SIR-PH-FA, and also for a second, more refined “intermediate approximation” SIR-PH-IA. We illustrate the current status-quo by presenting new results on a generalization of the SAIRS epidemic model.

Brown adipose tissue involution associated with progressive restriction in progenitor competence.

Human brown adipose tissue (BAT) undergoes progressive involution. This involution process is not recapitulated in rodents, and the underlying mechanisms are poorly understood. Here we show that the interscapular BAT (iBAT) of rabbits whitens rapidly during early adulthood. The transcriptomic remodeling and identity switch of mature adipocytes are accompanied by loss of brown adipogenic competence of progenitors. Single-cell RNA sequencing reveals that rabbit and human iBAT progenitors highly express the FSTL1 gene. When iBAT involutes in rabbits, adipocyte progenitors reduce FSTL1 expression and are refractory to brown adipogenic recruitment. Conversely, FSTL1 is constitutively expressed in mouse iBAT to sustain WNT signaling and prevent involution. Progenitor incompetence and iBAT paucity can be induced in mice by genetic deletion of the Fstl1 gene or ablation of Fstl1+ progenitors. Our results highlight the hierarchy and dynamics of the BAT progenitor compartment and implicate the functional incompetence of FSTL1-expressing progenitors in BAT involution.

Global behaviour of a class of discrete epidemiological SI models with constant recruitment of susceptibles

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:10.1007/s12190-021-01503-2], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number We establish the global asymptotic stability of the disease-free equilibrium (GAS) for We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for . For the transcritical bifurcation appears. For we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number .

A Review of Matrix SIR Arino Epidemic Models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”,which may be useful to obtain approximate control policies.

Who Gets Credit for AI-Generated Art?

SummaryThe recent sale of an artificial intelligence (AI)-generated portrait for $432,000 at Christie's art auction has raised questions about how credit and responsibility should be allocated to individuals involved and how the anthropomorphic perception of the AI system contributed to the artwork's success. Here, we identify natural heterogeneity in the extent to which different people perceive AI as anthropomorphic. We find that differences in the perception of AI anthropomorphicity are associated with different allocations of responsibility to the AI system and credit to different stakeholders involved in art production. We then show that perceptions of AI anthropomorphicity can be manipulated by changing the language used to talk about AI—as a tool versus agent—with consequences for artists and AI practitioners. Our findings shed light on what is at stake when we anthropomorphize AI systems and offer an empirical lens to reason about how to allocate credit and responsibility to human stakeholders.

Stability analysis of epidemiological models incorporating heterogeneous infectivity

In this paper we analyze general deterministic epidemiological models described by autonomous ordinary differential equations taking into account heterogeneity related to the infectivity and vital dynamics, in which the flow into the compartment of the susceptible individuals is given by a generic function. Our goal is to provide a new tool that facilitates the qualitative analysis of equilibrium points, which represent the disease free population, generalizing the result presented by Leite et al. (Math Med Biol J IMA 17:15–31, 2000) , and population extinction. The epidemiological models exposed are the type SEIRS (Susceptible-Exposed-Infectious-Recovered-Susceptible) and SEIR (Susceptible-Exposed-Infectious-Recovered) with vaccination. Moreover, we computed the basic reproduction number from the models by van den Driessche and Watmough (Math Biosci 180:29–48, 2002) and correlate this threshold parameter with the stability of the equilibrium point representing the disease free population.

A Tribute: Robert John van den Driessche, Tree Physiologist, 1933–2018

Robert (Bob) John van den Driessche passed away on July 19, 2018 leaving a legacy of research in tree physiology and nutrition that will continue to influence forest regeneration in British Columbia for years to come. This tribute outlines Bob’s career in the context of forest history in the province, and the many significant contributions Bob made to our understanding of conifer seedling nutrition and stress tolerance.

Reply to the comment on “Chronostratigraphy and paleoclimatology of the Lodève Basin, France: Evidence for a pan-tropical aridification event across the Carboniferous–Permian boundary” by Michel et al., (2015). Palaeogeography, Palaeoclimatology, Palaeoecology 430, 118–131

Michel et al. (2015) present new radioisotopic age estimates coupled with paleosol data from the Lodève basin, France that indicate a Pennsylvanian–Permian trend toward aridity that is similar to trends seen in other western equatorial basins across tropical Pangaea. In their comment, Pochat and Van Den Driessche (2015) call into question these newly published U-Pb zircon chemical-abrasion isotope dilution thermal ionization mass spectrometry (CA-IDTIMS) age estimates from volcanic ash layers preserved within the Lodève Basin. Furthermore, Pochat and Van Den Driessche (2015) reject our assertion that the rates of sediment accumulation for the Carboniferous–Permian strata in the Lodève basin are in need of recalculation as a result of the new geochronological dates. Finally, Pochat and Van Den Driessche (2015) suggest that the paleosol data presented in Michel et al. (2015) do not provide meaningful proxies of paleoclimate change, and reassert the position presented in Pochat and Van Den Driessche (2011) that sedimentary strata are not strongly affected by paleoclimatic change compared with the influence of tectonic processes of the Lodève basin.

A note on dynamics of an age-of-infection cholera model.

A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if i(a,t)/i*(a)=0 on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.

Minimum ranks of sign patterns via sign vectors and duality

A sign pattern matrix is a matrix whose entries are from the set {+,−,0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is shown in this paper that for any m×n sign pattern A with minimum rank n − 2, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer n ≥ 9, there exists a nonnegative integer m such that there exists an m × n sign pattern matrix with minimum rank n − 3 for which rational realization is not possible. A characterization of m × n sign patterns A with minimum rank n − 1 is given (which solves an open problem in Brualdi et al. [R. Brualdi, S. Fallat, L. Hogben, B. Shader, and P. van den Driessche. Final report: Workshop on Theory and Applications of Matrices Described by Patterns. Banff International Research Station, Jan. 31 – Feb. 5, 2010.]), along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. Several related open problems are stated along the way.

Global dynamics of a general class of multi-group epidemic models with latency and relapse.

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseases in heterogeneous populations. In each group, the population is divided into susceptible, exposed, infectious, and recovered subclasses. A general nonlinear incidence rate is used in the model. The results show that the global dynamics are completely determined by the basic reproduction number R0. In particular, a matrix-theoretic method is used to prove the global stability of the disease-free equilibrium when R0 ≤ 1, while a new combinatorial identity (Theorem 3.3 in Shuai and van den Driessche) in graph theory is applied to prove the global stability of the endemic equilibrium when R0 > 1. We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted into a graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.

The basic reproduction number for cellular SIR networks.

The basic reproduction number R0 is the average number of new infections produced by a typical infective individual in the early stage of an infectious disease, following the introduction of few infective individuals in a completely susceptible population. If R0 <1, then the disease dies, whereas for R0 >1 the infection can invade the host population and persist. This threshold quantity is well studied for SIR compartmental or mean field models based on ordinary differential equations, and a general method for its computation has been proposed by van den Driessche and Watmough. We concentrate here on SIR epidemiological models that take into account the contact network N underlying the transmission of the disease. In this context, it is generally admitted that R0 can be approximated by the average number R2,3 of infective individuals of generation three produced by an infective of generation two. We give here a simple analytic formula of R2,3 for SIR cellular networks. Simulations on two-dimensional cellular networks with von Neumann and Moore neighborhoods show that R2;3 can be used to capture a threshold phenomenon related the dynamics of SIR cellular network and confirm the good quality of the simple approach proposed recently by Aparicio and Pascual for the particular case of Moore neighborhood.

On a reaction–diffusion model for sterile insect release method on a bounded domain

We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method (SIRM). Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci.5 (1992) 221–247] where the habitat is assumed to be the one-dimensional whole space ℝ, we consider this system in a bounded one-dimensional domain (interval). Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that: (i) when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; (ii) when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.

TWO TYPES OF CONDITION FOR THE GLOBAL STABILITY OF DELAYED SIS EPIDEMIC MODELS WITH NONLINEAR BIRTH RATE AND DISEASE INDUCED DEATH RATE

We study global asymptotic stability for an SIS epidemic model with maturation delay proposed by K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol.39(4) (1999) 332–352. It is assumed that the population has a nonlinear birth term and disease causes death of infective individuals. By using a monotone iterative method, we establish sufficient conditions for the global stability of an endemic equilibrium when it exists dependently on the monotone property of the birth rate function. Based on the analysis, we further study the model with two specific birth rate functions B1(N) = b e -aN and B3(N) = A/N + c, where N denotes the total population. For each model, we obtain the disease induced death rate which guarantees the global stability of the endemic equilibrium and this gives a positive answer for an open problem by X. Q. Zhao and X. Zou, Threshold dynamics in a delayed SIS epidemic model, J. Math. Anal. Appl.257(2) (2001) 282–291.

Global dynamics of delay epidemic models with nonlinear incidence rate and relapse

A mathematical model for a disease with a general exposed distribution, the possibility of relapse and nonlinear incidence rate is proposed. By the method of Lyapunov functionals, it is shown that the disease dies out if ℜ 0 ≤ 1 and that the disease becomes endemic if ℜ 0 > 1 . Applications are also made to the special case with a discrete delay and the result confirms that the endemic equilibrium is globally asymptotically stable as suggested in van den Driessche et al. (2007) [4].

Political Accountability in Europe: Which Way Forward? A Traditional Concept of Parliamentary Democracy in an EU Context – Edited by Luc Verhey, Hansko Broeksteeg and Ilse Van den Driessche

Political Accountability in Europe: Which Way Forward? A Traditional Concept of Parliamentary Democracy in an EU Context – Edited by Luc Verhey, Hansko Broeksteeg and Ilse Van den Driessche

Sparse inertially arbitrary patterns

An n -by- n sign pattern A is a matrix with entries in { + , - , 0 } . An n -by- n nonzero pattern A is a matrix with entries in { ∗ , 0 } where ∗ represents a nonzero entry. A pattern A is inertially arbitrary if for every set of nonnegative integers n 1 , n 2 , n 3 with n 1 + n 2 + n 3 = n there is a real matrix with pattern A having inertia ( n 1 , n 2 , n 3 ) . We explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials. This information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary. Britz et al. [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] conjectured that irreducible spectrally arbitrary patterns must have at least 2 n nonzero entries; we demonstrate that irreducible inertially arbitrary patterns can have less than 2 n nonzero entries.

Hopf bifurcation for neutral functional differential equations

In this paper, we extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al. [N.D. Kazarinoff, P. van den Driessche, Y.H. Wan, Hopf bifurcation and stability of periodic solutions of differential–difference and integro-differential equations, J. Inst. Math. Appl. 21 (1978) 461–477] to a kind of neutral functional differential equation (NFDE). As an example, a neutral delay logistic differential equation is considered, and the explicit formulas for determining the direction of bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out to support the analytic results.

Survey of recent results of multi-compartments intra-host models of malaria and HIV

We present the recent results obtained for the within-host models of malaria and HIV. We briefly recall the Anderson-May-Gupter model. We also recall the Van Den Driessche method of computation for the basic reproduction ratio R0 ; here, a very simple formula is given for a new class of models. The global analysis of these models can be founded in [1, 2, 3, 5]. The results we recall here are for a model of one strain of parasites and many classes of age, a general model of n strains of parasites and k classes of age, a S E1 E2 · · ·En I S model with one linear chain of compartments and finally a general S Ei1 Ei2 · · ·Ein I S model with k linear chains of compartments. When R0 &lt;=1, the authors prove that there is a trivial equilibria calling disease free equilibrium (DFE) which is globally asymptotically stable (GAS) on the non-negative orthant , and when R0 &gt; 1, they prove the existence of a unique endemic equilibrium in the non-negative orthant and give an explicit formula. They provided a weak condition for the global stability of endemic equilibrium Le travail que nous présentons ici est un résumé de quelques résultats récents obtenus dans [1, 2, 3, 5] concernant les modèles intra-hôtes multi-compartimentaux. Il s’agit d’une analyse mathématique et globale des modèles intra-hôtes de paludisme et de V.I.H . Mais avant de présenter ces résultats, nous rappelons d’abord la méthode de calcul développée par Van Den Driessche[71] concernant le taux de reproduction de base R0 car c’est cette méthode qu’utilisent les auteurs dans leur analyse. Ces modèles sont dits de Anderson-May-Gupter dont le modèle original est considéré comme précurseur. Une formule simple est donnée ici pour le calcul de R0 dans les modèles étudiés. Les résultats que nous rappelons ici sont obtenus pour un modèle du paludisme à un génotype de parasites et k classes d’âge, le modèle général à n génotypes de parasites et k classes d’âge, un modèle S E1 E2 · · ·En I S avec une chaîne linéaire de parasites et enfin le modèle général S Ei1 Ei2 · · ·Ein I S avec k chaîne linéaire de parasites. Lorsque R0 1, les auteurs montrent qu’il existe un point d’équilibre évident, le DFE (Disease Free Equilibrium) qui est GAS (globalement asymptotiquement stable) sur l’orthant positif. Lorsque R0 &gt; 1, ils montrent l’existence d’un unique équilibre endémique dans l'orthant positif et moyennant une petite condition ils montrent que cet équilibre est globalement asymptotiquement stable.

A note on global stability of an SEI epidemic model with acute and chronic stages

In this paper, we have investigated the dynamic behavior of an SEI model with acute and chronic stages. Firstly by using Theorem 4.1 of Busenberg and van den Driessche [S. Busenberg, P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28 (1990), 257–270], we have shown that if the basic reproduction number R 0 > 1, the unique endemic equilibrium is global asymptotically stable. The results obtained here improved the main results of Yuan and Yang [J. Yuan, Z. Yang, Global dynamics of an SEI model with acute and chronic stages, J. Comput. Appl. Math. (2007), doi: 10.1016/j.cam.2007.01.042]. Then the criterion for robustness of the system is established under stochastic perturbations. Finally, numerical simulations are performed to illustrate the analytical results.