Communication in Biomathematical Sciences

Mathematical Models of Tumour--Immune Dynamics with Immune-Checkpoint Inhibition and Resistance: A Critical Review and Future Directions

Hafez Elfakie

Cancer remains a leading cause of global mortality. Mathematical modelling has become indispensable for understanding tumour growth, immune surveillance, and therapeutic response. This review critically examines mathematical models of tumour--immune interactions and immune-checkpoint inhibitor therapy from 2000 to 2025, organised thematically: classical growth models, deterministic ODE tumour--immune frameworks, models incorporating PD-1/PD-L1 and CTLA-4 checkpoint pathways, PDE and fractional-order models, and optimal control formulations. For each class, we evaluate mathematical structure, biological fidelity, and suitability for optimal control analysis. A comparative taxonomy reveals that no existing framework simultaneously incorporates structured multi-resistant tumour subpopulations, checkpoint-modulated cytotoxicity for both NK cells and CTLs, and Pontryagin-based optimal control for dual-blockade immunotherapy. We identify nine open challenges and propose a research agenda centred on compartmental ODE models with checkpoint regulation and optimal dual-blockade control.

Optimal Control Strategies for the Transmission Dynamics of Giardiasis

Timothy Ado Shamaki

This work develops and examines a deterministic mathematical model of giardiasis transmission, explicitly including the frequently neglected influence of asymptomatic carriers and the environmental pathogen reservoir. A thorough dynamical systems analysis of the autonomous model is performed, confirming the positivity and boundedness of solutions to ensure epidemiological well-posedness. We calculate the basic reproduction number ($\mathcal{R}_0$) employing the next-generation matrix approach and formally delineate the criteria for the local asymptotic stability of the disease-free equilibrium. We enhance the model by incorporating optimum control theory to meet the essential requirement for economical, time-sensitive intervention tactics. We provide three dynamic control functions that represent the medical treatment of symptomatic persons, the active screening of asymptomatic carriers, and environmental sanitation initiatives. Utilizing Pontryagin's Maximum Principle, we establish the requisite conditions for optimal control and simulate diverse intervention scenarios. Our findings demonstrate that single-intervention methods achieve limited efficacy, but a multifaceted, integrated strategy substantially reduces both the human disease burden and environmental pollution. Moreover, we illustrate that a focused strategy integrating immediate treatment with proactive asymptomatic screening provides a highly cost-effective option for resource-limited environments. This study offers substantial mathematical insights and a quantitative public health framework for the development of effective and sustainable giardiasis control initiatives.

A Modified HIV Model with Recovery Rate: Parameter Estimation and Dynamical Analysis

Mohammed Bassoudi

A mathematical model for the dynamics of the Human Immunodeficiency Virus (HIV) that includes a cure rate is presented in this study. In order to guarantee biological feasibility, we prove the positivity and boundedness of solutions for non-negative beginning circumstances. The next-generation matrix approach is used to determine the fundamental reproduction number, and the existence of endemic and disease-free equilibrium points is investigated. Additionally, to increase the study's practical applicability, model parameters are determined using the epidemiological data that is currently accessible. To demonstrate the theoretical results and investigate the influence of important parameters on the transmission dynamics, numerical simulations are carried out. To show how well the suggested model captures observed HIV dynamics, an applied case study is provided. The findings offer insightful information about how recovery affects HIV prevention.

Integrated Optimal Control of Malaria Transmission Considering Recrudescence and Reinfection within a SEITR-SEI Model

Kabir Idowu

Malaria is a vector-borne infectious disease initiated by Plasmodium parasites and transmitted to humans
through the bites of infected Anopheles mosquitoes, it still remains one of the most serious public health
challenges in sub-Saharan Africa even with decades of control efforts. Persistent transmission is still occurring
because of the biological mechanisms, such as reinfection and recrudescence, which are rarely modeled by
standard epidemiological models. This study fill this gap by developing and analysing an integrated SEITR–SEI
compartmental model incorporate explicitly reinfection and recrudescence dynamics in the context of optimal
control. The model evaluates the combined effect of three important interventions: chemoprevention, diagnosis
and treatment, and vector control. All mathematical modelling formulations, calculations, and analytical
procedures are strictly established using relevant theorems, such as positivity, boundedness, reproduction
number, equilibrium analysis, and stability theorems. The novelty of this research lies in integrating unified
mathematical structure of these biological complexities and control strategies. The results indicate that failure
to consider these biological factors implies the underestimation of disease persistence and control costs.
Consequently, integrated implementation of chemoprevention, treatment, and vector control is recommended
for sustainable malaria control and movement toward eradication in sub-Saharan Africa.