Communication in Biomathematical Sciences

Optimal Control of Caputo Fractional-Order SVEITR Epidemic Model for H5N1

Fatima OUCHRAA

In this work, we formulate and analyze a Caputo fractional-order compartmental epidemic model for In
fluenza A virus subtype H5N1 incorporating susceptible, vaccinated, exposed, infectious, treatment/quarantine,
and recovered (SV EITR) populations. The model embeds a behavioral feedback mechanism through an
exponential incidence modulation that mechanistically captures the dampening effect of media coverage and
public awareness on transmission. Under a hypothetical pandemic scenario with sustained human-to-human
transmissibility, the basic reproduction number R0 is derived using the next-generation matrix method, while
the global asymptotic stability of the disease-free and endemic equilibria is established through Lyapunov
functionals and LaSalle’s invariance principle. A Pontryagin optimal control framework with four intervention
strategies(transmission reduction, vaccination promotion, and targeted screening of exposed and infectious indi
viduals)is developed and solved numerically using the forward-backward sweep method. Parameter values are
informed by WHO surveillance data and published epidemiological estimates for H5N1. Numerical simulations
with α = 0.95 and R0 = 1.986 show that the combined strategy (u1+u2+u3+u4) reduces the peak number
of infectious individuals by 37% compared with the no-control scenario. These results highlight the potential
of Pontryagin optimal control as a mathematically rigorous framework for H5N1 epidemic management.

Stochastic and Optimal Control of Mpox Dynamics: A Nonstandard Finite Difference Approach

Patience Pokuaa Gambrah

Mpox (monkeypox) has emerged as a critical zoonotic public health threat, with the Democratic Republic of Congo (DRC) recording over 86,798 confirmed cases between 2000 and 2024, including a dramatic surge to 16,088 cases in 2024 alone. Our paper develops and analyzed a novel nine compartmental stochastic epidemic model for Mpox transmission dynamics incorporating human-animal interaction, vaccination, and disease control. A nonstandard finite difference (NSFD) scheme is constructed to preserve the positivity, boundedness, and dynamical consistency of the continuous system. We derive the basic reproduction number, establish conditions for disease free and endemic equilibrium stability, and prove the existence of a unique distribution under stochastic perturbation. An optimal control problem is formulated with four time-dependent control variables which were vaccination, treatment, environmental decontamination, and animal reservoir culling intensity. Pontryagin’s Maximum Principle was applied to characterize optimal strategies. Numerical simulations calibrated against weekly DRC epidemiological data (2000–2024) demonstrate that vaccination and combined strategy reduces cumulative infections by up to 93\% relative to no intervention scenarios. We also found out that, the NSFD schemes outperform Euler and RK4 methods, preserving positivity and boundedness. The stochastic differential equation (SDE) extension captures realistic epidemic uncertainty through multiplicative noise, with widening confidence intervals reflecting genuine epidemiological unpredictability. Moreover, the analysis of the basic reproduction number revealed that, interventions targeting the animal reservoir (rodent population management, habitat separation, market biosafety protocols) are globally as important as human vaccination. This provides quantitative justification for a One Health approach to Mpox control.

A MULTI-STRAIN DENGUE FEVER MODEL WITH REACTIVE SERO-POSITIVE VACCINATION

Taurai Mademutsa

The neglected tropical disease dengue fever remains a major public health challenge in tropical and sub-tropical regions where the disease is endemic. Its dynamics are shaped jointly by temperature-driven vector activity and the complex interaction between co-circulating serotypes, yet existing models typically treat these mechanisms separately and represent vaccination as a fixed-rate intervention, neglecting the behavioural feedback through which uptake responds to perceived risk. We develop a multi-strain deterministic compartmental model of dengue transmission incorporating seasonal temperature forcing, quasi-stationary vector dynamics, temporary cross-immunity, and an incidence-triggered reactive sero-positive vaccination strategy. The vaccination rate $\alpha(I)$ responds dynamically to secondary incidence through a Hill-type function with a psychological threshold $K$ and a responsiveness parameter $n$, capturing how public perception and vaccine uptake track secondary infections. We establish positivity, boundedness, the strain-specific reproduction numbers, and existence conditions for the disease-free, single-strain, and coexistence equilibria, and show that because $\alpha(0)=0$, reactive vaccination leaves the invasion threshold unchanged and therefore reshapes outbreaks rather than preventing them. Phase-plane analysis reveals closed orbits corresponding to recurrent, seasonally driven outbreaks, with reactive vaccination reducing these orbits to ones of smaller diameter and so reducing both overall incidence and severe disease burden. Global sensitivity analysis using PRCC and variance decomposition identifies the vaccination capacity $\alpha_{\max}$, the threshold $K$, and the responsiveness $n$ as dominant drivers of peak infections and cumulative severe burden, indicating that lowering the psychological threshold affords greater control than simply expanding health system capacity. Simulations show alternating serotype dominance and multi-year spacing between severe outbreaks, attributable to cross-immunity and to asymmetry arising from niche partitioning, and sero-positive vaccination is found to protect against severe disease while simultaneously reducing primary infections. The model further reproduces essential qualitative features of weekly dengue case data from several endemic countries, confirming that temperature forcing and multi-strain interaction are together sufficient to generate characteristic dengue dynamics. Public health campaigns should therefore aim at lowering the activation threshold by encouraging early sero-positive vaccination uptake.

Interplay of Fear, Hunting Cooperation, and Group Defence in Stage-Structured Predator-Prey Dynamics

Ankur Jyoti Kashyap

In this study, we propose and analyze a novel predator–prey model incorporating intra-specific
competition, cooperative hunting among predators, and fear-induced effects on prey. The novelty
of the work lies in the simultaneous integration of cooperation-driven predation with dual fear
effects and density-dependent regulation within a unified dynamical system. Unlike classical preydependent
or predator-interference functional responses, this formulation captures more realistic
collective and behavioral interactions. A detailed dynamical analysis is carried out with respect to
key parameters, including cooperation rate, fear intensities, intra-specific competition, maximum
per capita growth rate, and predator mortality. The system exhibits rich dynamics such as stable
coexistence and sustained oscillations arising through Hopf, generalized Hopf, saddle–node, and
Bogdanov–Takens bifurcations. In particular, intra-specific competition and its interaction with
other parameters drive transitions between equilibrium and oscillatory regimes. The maximum
per capita growth rate induces saddle–node bifurcation and may lead to predator-free equilibrium,
while increased predator mortality gives rise to the Hydra effect. These findings emphasize the
crucial roles of cooperation, fear, and density dependence in shaping ecosystem dynamics.