Communication in Biomathematical Sciences

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, Nawel Abdesselam

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

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.

Fuzzy Mathematical Modelling and Numerical Analysis of HIV Transmission Dynamics

Muhammad Bilal, Muhammad Rafiq, Zulfiqar Ali, Shah Zeb, Rizwana Kausar

Human Immunodeficiency Virus (HIV) and Acquired Immunodeficiency Syndrome (AIDS) still are major global healthcare issues, and apparently, their constant toll supports the necessity of stringent quantitative research. Mathematical modeling in the same context offers an orderly method to the study of the mechanisms of transmission, and the evaluation of potential intervention strategies in a more direct manner. Within the current research, a compartmental fuzzy system is built up of a system of linear ordinary differential equations to realize the spread of HIV/AIDS among a specified group of people. Basic reproduction number which is referred to as $\mathcal{R}_c$ is specifically determined in an attempt to describe the natural transmission capability of the pathogen. Analytic study shows that the disease free equilibrium or DFE is locally asymptotically stable at ${\mathcal{R}_c}<1$ implying the infection cannot persist in this case. On the contrary, at values of $\mathcal{R}_c$ greater than unity, the DFE becomes unstable and an endemic equilibrium occurs. This endemic condition seems to be maintained and constant at ${\mathcal{R}_c}>1$, which shows the constant presence of the viruses in the host population. Numerical experimentation of the system is then done by trying to understand the dynamical behavior of the system using two methods of discretization: standard finite difference scheme and non-standard finite difference formulation. The conventional finite difference approach proves conditional convergence only and, depending on the spacing between calculations, can put forward biologically impossible solutions. Contrary to this fact, non standard finite difference scheme proposed does not lose some qualitative properties of the full scale model, such as state variables positivity and equilibrium stability. Stability analysis of the non-standard scheme is also given in detail in order to support these observations. The analytical findings are backed by comparative numerical simulations, which, again, may be more vital, demonstrate improved reliability of the non standard finite difference technique in retrieving the pertinent transmission dynamics of the HIV/AIDS with more fidelity. Furthermore, fuzzy membership functions are utilized to represent uncertainty in model parameters and epidemiological data. This fuzzy representation enhances the flexibility of the model and allows a more realistic description of the variability present in real-world disease transmission.