Journal of Mathematical Biology最新文献

We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class of associated McKean-Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology.

In this work, we study the model of a fish species growing logistically exploited by a fishing fleet in a heterogeneous environment. The environment is made up of a network of fishing patches connected by fish migrations taking place on a fast time scale. We are interested in the maximum economic yield (MEY) which corresponds to the maximum profit made by the fishing fleet. We show that the total MEY profit of the fishery made up of all the connected fishing patches can be greater than the sum of the MEY profits of isolated patches in the absence of migration. We study the general case with any number of connected patches then focus on the case of a system composed of two patches. In the latter case, we show that asymmetry in fish migration plays an important role in increasing the total profit at the MEY by connecting patches. We illustrate our results with numerical simulations allowing us to compare the MEY fishery system with connected patches compared to the system with isolated patches.

The epidemiological behavior of Plasmodium vivax malaria occurs across spatial scales including within-host, population, and metapopulation levels. On the within-host scale, P. vivax sporozoites inoculated in a host may form latent hypnozoites, the activation of which drives secondary infections and accounts for a large proportion of P. vivax illness; on the metapopulation level, the coupled human-vector dynamics characteristic of the population level are further complicated by the migration of human populations across patches with different malaria forces of (re-)infection. To explore the interplay of all three scales in a single two-patch model of Plasmodium vivax dynamics, we construct and study a system of eight integro-differential equations with periodic forcing (arising from the single-frequency sinusoidal movement of a human sub-population). Under the numerically-informed ansatz that the limiting solutions to the system are closely bounded by sinusoidal ones for certain regions of parameter space, we derive a single nonlinear equation from which all approximate limiting solutions may be drawn, and devise necessary and sufficient conditions for the equation to have only a disease-free solution. Our results illustrate the impact of movement on P. vivax transmission and suggest a need to focus vector control efforts on forest mosquito populations. The three-scale model introduced here provides a more comprehensive framework for studying the clinical, behavioral, and geographical factors underlying P. vivax malaria endemicity.

In addition to non-pharmaceutical interventions, antiviral drugs and vaccination are considered as the optimal solutions to control and eliminate the COVID-19 pandemic. It is necessary to couple within-host and between-host models to investigate the impact of treatment and vaccination. Hence, we propose an age-structured model, where the infection age is used to link the within-host viral dynamics and the disease dynamics at the population level. We conduct a detailed analysis of the local and global dynamics of the model, and the threshold dynamics are completely determined by the basic reproduction number $R_0$ . Thus, the disease-free equilibrium is globally asymptotically stable and the disease eventually dies out when $R_0<1$ ; the disease-free equilibrium is globally attractive when $R_0=1$ ; the disease is uniformly persistent, and the unique endemic equilibrium is globally asymptotically stable when $R_0>1$ . The numerical simulation quantitatively studies the impact of the within-host viral dynamics on between-host transmission dynamics. The results show that the combination of antiviral drugs and vaccines can play a key role in mitigating the spread of COVID-19, but it is challenging to eliminate COVID-19 alone. To achieve the complete elimination of COVID-19, we need highly effective antiviral drugs and near-universal vaccine coverage.

We investigate sub-leading orders of the classic SEIR-model using contact matrices from modeling of the Omicron and Delta variants of COVID-19 in Denmark. The goal of this is to illustrate when the growth rate, and by extension the infection transmission potential (basic or initial reproduction number), can be estimated in a new outbreak, e.g. after introduction of a new variant of a virus. In particular, we look at the time scale on which this happens in a realistic outbreak to guide future data collection. We find that as long as susceptible depletion is a minor effect, the transients are gone within around 3 weeks corresponding to about 4-5 times the incubation time. We also argue that this result generalizes to other airborne diseases in a fully mixed population.

We examine here the effects of recurrent vaccination and waning immunity on the establishment of an endemic equilibrium in a population. An individual-based model that incorporates memory effects for transmission rate during infection and subsequent immunity is introduced, considering stochasticity at the individual level. By letting the population size going to infinity, we derive a set of equations describing the large scale behavior of the epidemic. The analysis of the model's equilibria reveals a criterion for the existence of an endemic equilibrium, which depends on the rate of immunity loss and the distribution of time between booster doses. The outcome of a vaccination policy in this context is influenced by the efficiency of the vaccine in blocking transmissions and the distribution pattern of booster doses within the population. Strategies with evenly spaced booster shots at the individual level prove to be more effective in preventing disease spread compared to irregularly spaced boosters, as longer intervals without vaccination increase susceptibility and facilitate more efficient disease transmission. We provide an expression for the critical fraction of the population required to adhere to the vaccination policy in order to eradicate the disease, that resembles a well-known threshold for preventing an outbreak with an imperfect vaccine. We also investigate the consequences of unequal vaccine access in a population and prove that, under reasonable assumptions, fair vaccine allocation is the optimal strategy to prevent endemicity.

The HPT complex, consisting of the hypothalamus, pituitary and thyroid, functions as a regulated system controlled by the respective hormones. This system maintains an intrinsic equilibrium, called the set point, which is unique to each individual. In order to optimize the treatment of thyroid patients and understand the dynamics of the system, a validated theoretical representation of this set point is required. Therefore, the research field of mathematical modeling of the HPT complex is significant as it provides insights into the interactions between hormones and the determination of this endogenous equilibrium. In literature, two mathematical approaches are presented for the theoretical determination of the set point in addition to a time-dependent model. The two approaches are based on the maximum curvature of the pituitary response function and the optimal gain factor in the representation of the HPT complex as a closed feedback system. This paper demonstrates that all hormone curves described by the model converge to the derived set point with increasing time. This result establishes a clear correlation between the physiological equilibrium described by the set point and the mathematical equilibrium with respect to autonomous systems of differential equations. It thus substantiates the validity of the theoretical set point approaches.

We propose an integral model describing an epidemic of an infectious disease. The model is behavioural in the sense that the force of infection includes the information index that describes the opinion-driven human behavioural changes. The information index contains a memory kernel to mimic how the individuals maintain memory of the past values of the infection. We obtain sufficient conditions for the endemic equilibrium to be locally stable. In particular, we show that when the infectivity function is represented by an exponential distribution, stability is guaranteed by the weak Erlang memory kernel. However, through numerical simulations, we show that oscillations, possibly self-sustained, may arise when the memory is more focused in the disease's past history, as exemplified by the strong Erlang kernel. We also show the model solutions in cases of different infectivity functions taken from studies where specific diseases like Influenza and SARS are considered.

We propose a general framework for the study of the genealogy of neutral discrete-time populations. We remove the standard assumption of exchangeability of offspring distributions appearing in Cannings models, and replace it by a less restrictive condition of non-heritability of reproductive success. We provide a general criterion for the weak convergence of their genealogies to $\Xi$ -coalescents, and apply it to a simple parametrization of our scenario (which, under mild conditions, we also prove to essentially include the general case). We provide examples for such populations, including models with highly-asymmetric offspring distributions and populations undergoing random but recurrent bottlenecks. Finally we study the limit genealogy of a new exponential model which, as previously shown for related models and in spite of its built-in (fitness) inheritance mechanism, can be brought into our setting.

One of the central results of evolutionary matrix games is that a state corresponding to an evolutionarily stable strategy (ESS) is an asymptotically stable equilibrium point of the standard replicator dynamics. This relationship is crucial because it simplifies the analysis of dynamic phenomena through static inequalities. Recently, as an extension of classical evolutionary matrix games, matrix games under time constraints have been introduced (Garay et al. in J Theor Biol 415:1-12, 2017; Křivan and Cressman in J Theor Biol 416:199-207, 2017). In this model, after an interaction, players do not only receive a payoff but must also wait a certain time depending on their strategy before engaging in another interaction. This waiting period can significantly impact evolutionary outcomes. We found that while the aforementioned classical relationship holds for two-dimensional strategies in this model (Varga et al. in J Math Biol 80:743-774, 2020), it generally does not apply for three-dimensional strategies (Varga and Garay in Dyn Games Appl, 2024). To resolve this problem, we propose a generalization of the replicator dynamics that considers only individuals in active state, i.e., those not waiting, can interact and gain resources. We prove that using this generalized dynamics, the classical relationship holds true for matrix games under time constraints in any dimension: a state corresponding to an ESS is asymptotically stable. We believe this generalized replicator dynamics is more naturally aligned with the game theoretical model under time constraints than the classical form. It is important to note that this generalization reduces to the original replicator dynamics for classical matrix games.