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On the Role of Early Case Detection and Treatment Failure in Controlling Tuberculosis Transmission: A Mathematical Modeling Study Aldila, Dipo; Ramadhan, Derio A.; Chukwu, Chidozie W.; Handari, Bevina D.; Shahzad, Muhammad; Putri Zahra Kamalia
Communication in Biomathematical Sciences Vol. 7 No. 1 (2024)
Publisher : The Indonesian Bio-Mathematical Society

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/cbms.2024.7.1.4

Abstract

Tuberculosis (TB) remains a pressing global health concern, demanding urgent attention to mitigate its spread and impact. In this study, we present a rigorous mathematical model of TB transmission that incorporates early case detection and addresses the critical issue of treatment failure. Through the development of a system of nonlinear ordinary differential equations, we conduct comprehensive analyses to assess the dynamics of TB transmission and the efficacy of intervention strategies. Our findings underscore the urgent need for effective TB control measures. Mathematical analyses reveal that the model exhibits a TB-free equilibrium, which is globally asymptotically stable only if the control reproduction number falls below one. However, we identify a concerning phenomenon: the model demonstrates a forward bifurcation when the control reproduction number equals one, suggesting that the disease-free equilibrium loses its stability, while simultaneously, the stable unique endemic equilibrium begins to emerge. Moreover, sensitivity analysis highlights the complex interplay between case detection rates, treatment failure probabilities, and TB transmission dynamics. Contrary to expectations, increasing case detection rates and minimizing treatment failure probabilities may not consistently reduce the basic reproduction number or the size of the infected population. Instead, there exists a critical threshold for intervention effectiveness, beyond which TB transmission can be significantly curtailed. Biologically, this phenomenon may occur if there is no balance between case detection and treatment efforts. If treatment quality does not improve, then case detection will not have a significant impact, and in the worst case scenario, it can exacerbate the intervention’s negative effects. These findings underscore the urgency of implementing targeted intervention strategies to combat TB transmission effectively. Failure to meet the critical intervention threshold risks undermining TB elimination efforts and exacerbating the global TB burden. Through numerical simulations, we elucidate potential intervention scenarios necessary for achieving TB elimination goals in human populations. In conclusion, our study highlights the urgent imperative for coordinated action to control TB transmission effectively. By elucidating the dynamics of TB spread and intervention efficacy, we provide valuable insights to inform evidence-based policy decisions and accelerate progress towards TB elimination on a global scale.
Implementation of non-standard finite difference on a predator prey model considering cannibalism on predator and harvesting on prey Luis, Prisalo; Kamalia, Putri Zahra; Peter, Olumuyiwa James; Aldila, Dipo
Jambura Journal of Biomathematics (JJBM) Volume 6, Issue 1: March 2025
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/jjbm.v6i1.30550

Abstract

The type of interaction between two different species in the same ecosystem plays an important role in the coexistence between these species. One type of interaction between species is predator-prey interaction. Several important factors are crucial to guarantee the existence of predator and prey in the same ecosystem, such as the carrying capacity of the ecosystem for the survival of prey, the intensity of predation, cannibalism in the predator population, and many other factors. External factors such as human intervention, such as harvesting, increase the complexity of the problem. Here in this article, we discuss a predator-prey model that takes predation and harvesting in prey populations into account. We implement a Non-Standard Finite Difference (NSFD) numerical scheme to solve our model due to it good performance on stability and approximation. Mathematical analysis on the existence and stability of equilibrium points from the discrete model was analyzed in detail. We implement a Nonstandard Finite Difference (NSFD) scheme to ensure numerical stability across various simulation scenarios. It is shown that NSFD has a better numerical stability compared to the standard numerical scheme like Euler or fourth-order Runge-Kutta method. From the sensitivity of autonomous simulation, we have shown that increases of cannibalism in predator populations will reduce predator populations, and as a result, the population of prey will increase due to the lack of number of predators. We also showed that increasing harvesting in prey populations may cause extinction in prey and predator populations. Furthermore, we have shown how periodic harvesting on prey populations may cause a critical condition on the existence of prey populations that takes a longer period to get recovered.
Epidemic Dynamics with Nonlinear Incidence Considering Vaccination Effectiveness Kamalia, Putri Zahra; Aldila, Dipo
Jambura Journal of Biomathematics (JJBM) Volume 6, Issue 3: September 2025
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/jjbm.v6i3.33815

Abstract

This paper presents a mathematical model that examines the effect of nonlinear incidence on disease transmission dynamics.  Furthermore, we also accommodate newborn and adult vaccination strategy as the prevention strategy to prevent rapid spread of the disease due to nonlinear incidence rate. Assuming a constant population  size,  the  system is  reduced  to  a  two-dimensions and  nondimensionalized using  the  average infectious period as the time scale.   Analytical results reveal the existence of both disease-free and endemic equilibria, with the possibility of backward bifurcation when the nonlinear incidence parameter exceeds a critical threshold.   This implies that disease persistence may still occur even when the basic reproduction number is less than one.  Numerical simulations using MATCONT conducted to visualize the occurrence of both forward and backward bifurcations phenomena.    Using COVID-19 parameter values,  a  global sensitivity analysis via Partial Rank Correlation Coefficient - Latin Hypercube Sampling method indicates that newborn vaccination has a stronger impact on reducing the basic reproduction number. These findings provide important insights for designing effective vaccination strategies and understanding the complex dynamics arising from nonlinear transmission and imperfect immunization.