We introduce a set of ordinary differential equations (ODE) that qualitatively reproduces delayed responses observed in immune checkpoint blockade therapy, such as anti-CTLA-4 Ipilimumab. Immune checkpoint inhibitors have revolutionised cancer treatment, offering durable responses in some patients, but their efficacy is often hindered by delayed anti-tumour immune responses. Our ODE system can be considered as two modular submodels: one focusing on dendritic cell (DC) dynamics within the tumour microenvironment and the other on T cell activation within lymph nodes. Together, these submodels provide a mechanistic framework to explore the interplay between immune activation and tumour-driven suppression. By calibrating key parameters, our models naturally reproduce clinically observed phenomena that range from no response to delayed anti-tumour activity. Simulations suggest that insufficient co-stimulatory signals by DCs may play a key role in delayed responses. Furthermore, we offer a framework on how individual immune profiles shape treatment outcomes in anti-CTLA-4 and anti-PD-1 combination therapies. Our model also suggests potential insights into long-term dynamics, including tumour dormancy, oscillatory behaviour and the conditions necessary for durable tumour eradication. We hope this exploratory study will inspire further research into the intricate and nuanced dynamics of tumour-immune interactions.

Malaria and typhoid fever are major infectious diseases that pose significant public health challenges in many parts of the world, particularly in sub-Saharan Africa. This study develops a mathematical model to investigate the dynamics of malaria–typhoid co-infection, incorporating both vector and non-vector malaria transmission routes and environmental transmission for typhoid. Model parameters were drawn from the literature, and simulations were conducted in MATLAB. The results show that vector-borne transmission accounts for over 80% of malaria infections, while typhoid transmission sustains a persistent infection level. Co-infected individuals constitute approximately 25–35% of the total infected population at peak conditions, underscoring the substantial burden of simultaneous infection. Sensitivity analysis identifies malaria and typhoid transmission rates as key drivers of co-infection prevalence. A backward bifurcation in the malaria subsystem indicates that malaria may persist even when its reproduction number is below one, thereby continually seeding co-infection and indirectly maintaining typhoid transmission through co-infected individuals. These findings highlight the need for integrated and sustained control strategies, including vector control, typhoid vaccination, and improved sanitation, to effectively reduce the dual disease burden. Overall, the model provides a useful analytical framework to support public health planning and evidence-based resource allocation in regions where both infections remain endemic.

This study extends the model proposed by Lin et al. (J Math Biol, 84:39, 2022) by incorporating an age-dependent competition within species into the logistic equation. The proposed model is a differential equation with an integral delay, reflecting variations in competition. In the single-species model, sustained oscillations occur due to Hopf bifurcations related to the immature period or competition variation index. A threshold dynamics can be determined, either global convergence to a trivial equilibrium or the uniform persistence of positive solutions. Additionally, we can demonstrate the dynamics of convergence towards a positive equilibrium under another criterion by using the convergence property of a class of linear functional differential equations. In the two-species model, we certify that the population dynamics converge toward the dominant single-species equilibria when one species can survive in the environment without competitors while the other cannot. If a positive equilibrium exists with strong competition between the two species, we numerically observe bistable dynamics. On the other hand, when competition between the two species is low, we numerically observe multiple stability switches in the coexistence state. Additionally, we examine how the maturation time affects the dynamics of invader populations under steady or oscillatory conditions of the resident species.

We conduct sensitivity analysis of an optimal control problem applied to agricultural irrigation. The aim is to minimize the square of the amount of irrigation water while ensuring the health grow of the crop. A crucial parameter of our model is the percentage of water loss due to deep percolation, β, a parameter hard to estimate and subject to perturbations. Thus, our problem is a parametric state constrained optimal control problem with a L2 cost. Our goal is to study how perturbations of β affect the optimal solutions of our problem using sensitivity analysis of our problem using sensitivity analysis. To solve numerically our optimal control problem we use the direct method, transcribing the problem into a non-linear programming problem. We show how sensitivity analysis applied to the non-linear programming problem provides information on the variation of optimal solutions of the original problem in terms of β. Valid approximations of optimal solution and cost, provided by sensitivity analysis, are computed for values of β within of a certain neighbourhood. Remarkably, we show that for all β in such neighbourhood, the irrigation period is kept constant. Only the flow rate of irrigation water changes.

Mathematical analysis of the physical principles governing phase morphology evolution is important for the fundamental understanding of the microstructure formation in materials science. Since the physical laws describing the full field evolution of phases can be written as partial differential equations (PDEs), it is of interest to examine the general principles of fields evolution as defined by PDEs. Previously, the connection between first order time partial differential equations and the movement of level-sets ( i.e. iso-contours or iso-surfaces) of a general field was established, and demonstrations of the initial calculation with a lagrangian mesh based on moving level-sets was made [A. Pohjonen, J. Phys. Conf. Ser. 2675 (2023) 012031]. Although the first order approximation is capable of estimating field evolution, the stability of numerical methods could be improved by considering second order approximations in time. Furthermore, the theoretical connection between second order time derivative of the field, the differential geometry of the field, velocity and acceleration of level-sets, and their shape evolution provide insights in to the dynamical evolution of the field geometry, as defined by partial differential equations in general.

Acquired resistance is one of the major causes of failure of standard therapies in cancer patients. Chemotherapeutic agents are still widely used and the understanding of the mechanisms leading to secondary resistance to these molecules are still puzzling. Recently, the role of the tumor immune microenvironment has been recognized. Among the cells potentially involved, macrophages seem to be the perfect culprits. In a previous work, we have shown that hypoxic macrophages are able to provide strong protection against 5-fluorouracil, a first-line chemotherapeutic agent in digestive cancers. In the present work, we use mathematical modeling to explore the spatiotemporal aspects of the treatment-induced organization of the tumor environment. Based on analytical and numerical analysis, we propose that macrophage-driven protection against chemotherapy under treatment does not rely solely on biochemical degradation, but is enhanced by the emergence of spatially structured chemotherapeutic protective niches. This work paves the way for the development of new therapeutic strategies that rely on targeting the spatial organization of tumors as a way to control treatment resistance.

A mathematical model for coregistered time-dependent electroencephalography (EEG) and diffusive optical tomography (DOT) is developed and analysed. Evolution with time is introduced by considering time-dependent dipolar sources in the EEG model and time-dependent optical parameters for DOT. Dimensional analysis shows that time-derivatives can be neglected. A non-linear system of differential equations from literature is used to model the postsynaptic current and hemodynamic parameters at the neuron level. A key point of the full model is to explain how these quantities provide, at the level of the whole head, the moment of the dipolar source term of the EEG problem and the behaviour in time of the optical parameters of the DOT model. The well-posedness of the timedependent EEG problem is proved by the subtraction approach for moments with L2-regularity in time and continuous source trajectories. For the time-dependent DOT model with continuous optical parameters in time, standard results of functional analysis apply. We explain the full pipeline from the stimulation current up to the simulated signals recorded at the electroptodes. Numerical results for a three-dimensional realistic head model illustrate the capacity of simultaneous EEG/DOT measurements to attest neurovascular coupling between the neural activity and changes in the hemodynamic parameters.

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to beyorzindependent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.

We consider a one-dimensional model of hemodynamics for a tube defined by a Koiter shell, and investigate traveling wave solutions. The system of partial differential equations is reduced to a fourth-order ordinary differential equation for such solutions. We find a single equilibrium point and determine the condition for the changing type of the equilibria for the corresponding system of first-order differential equations. The conditions for changing the type of the equilibrium point are formulated. Then we introduce the classification of blood flow regimes relative to the type of the equilibrium point. Numerical experiments are carried out to confirm and analyze the obtained results, various regimes of blood flow are considered.

This paper investigates a three-strategy (cooperators, toxin producers, and cheaters) N-player division of labor game in bacterial populations. We construct the replicator equation to discuss the evolution of the frequency of the three strategies. Firstly, we prove that the interior equilibrium is always unstable, the three strategies cannot coexist. Secondly, according to Sotomayor’s theorem, the system undergoes transcritical bifurcation. Furthermore, the sensitivity of the two-dimensional evolutionary state diagrams to the third parameter (toxin rate, absorption rate, toxin quantity, etc) is analyzed. In summary, high toxicity rates, high levels of toxins, and low levels of competition tend to promote cooperation. All players choose to perform the task, and the cheater disappears. When the absorption rate of cooperators is high enough, only cooperators exist in the population over time. When the absorption rate of the cooperator is low, and the absorption rate of the toxin producer is greater than the threshold, the cooperator and the toxin producer coexist. All players perform the task. Finally, the triangle diagrams and three-dimensional diagrams are presented, which show the initial conditions of the three strategies also affect the dynamic results. As the amount of toxin increases, the range of players who choose to perform tasks widens.