We present a non-autonomous delayed mathematical model of tumor-immune interactions incorporating immunotherapy and targeted therapy. Recent clinical trials have demonstrated the significant efficacy of this combination, even in advanced stages of cancer. We introduce a time-dependent function representing the concentration of the active substance of administered targeted therapy and analyse how the mathematical framework reflects key biological mechanisms. We establish the existence of tumor-free equilibrium solutions and provide theoretical analysis of their stability in both autonomous and non-autonomous settings. Numerical simulations complement the theoretical results, illustrating the complex effects of combined therapies. Finally, a parameter estimation is performed, using recent clinical data on tumor growth, validating the model’s relevance and predictive capabilities.
Trait evolution can be both a population’s rescue and its ruin. This raises a central eco–evolutionary question: under a strong Allee effect, can the evolution of a quantitative trait help a sparse population escape extinction, or can it instead destabilize an otherwise viable population and increase extinction risk? To address this, we study a discrete-time Darwinian extension of the Ricker model coupling population density to the evolution of a mean trait: a low-density reproductive limitation term generates the strong Allee effect, and the trait evolves along the fitness gradient. The model is a two-dimensional map with eco–evolutionary feedback: density shapes selection, while the evolving trait alters reproduction and competition. After determining the equilibrium structure and the threshold separating extinction from positive-density survival, we analyze how the upper positive equilibrium loses stability. Period-doubling (PD) and Neimark–Sacker (NS) bifurcations replace steady survival by alternating-generation and oscillatory dynamics. Codimension-two phenomena—the 1:2 point where the PD and NS curves meet, together with the 1:3 and 1:4 strong resonances—organize transitions toward two-, three-, and four-generation cycles, with quasiperiodic and more complex fluctuations. These mechanisms arise from the non-monotone Ricker-type recruitment term and are absent from the monotone one-dimensional Beverton–Holt recruitment map. We show that evolutionary response and trait-mediated competition can either favor persistence or promote destabilization, depending on parameters. Basin computations reveal coexistence between positive-density attractors and extinction, so long-term survival depends on both parameters and initial conditions. Overall, the model clarifies how Darwinian trait evolution and low-density demographic limitation jointly shape extinction thresholds, oscillatory survival, and population fate.
Crimean Congo hemorrhagic fever (CCHF) is a viral disease in animals and humans. Ticks transmit the virus that causes this disease to susceptible hosts by feeding on them, and ticks can acquire the virus by feeding on an infected host. The tick species most commonly associated with CCHF feed on different hosts at different stages in their life cycle: larvae and nymphs feed on small mammals, while adults feed on large mammals, like humans and cattle. An additional transmission pathway, co-feeding, occurs when infected and susceptible ticks feed in close proximity on the same host, allowing viral transmission between ticks even if the host remains uninfected. In this work, we developed the first system of ordinary differential equations capturing CCHF transmission that includes co-feeding dynamics, seasonal variation in tick populations, and all three feeding life stages of ticks with their subsequent hosts. We adapted the model to Uganda’s Cattle Corridor, a region with frequent human–cattle interactions and reported CCHF activity. Simulations explored both baseline and immigration event scenarios, capturing realistic ecological and epidemiological dynamics. By varying the co-feeding transmission, we found notable effects on the hosts and ticks, including a shift in timing of seasonal infection peaks and changes to the number of human infections from ticks or cattle. The inclusion of additional tick life stages demonstrates how co-feeding strengthens viral circulation within ticks and increases host infectivity. These results highlight the importance of including ecological complexity in models of CCHF transmission and to illustrate the effects of co-feeding.
Intensive care unit (ICU) control of methicillin-resistant Staphylococcus aureus (MRSA) occurs over short horizons, whereas R_0 summarizes average transmission at the importation-free disease-free state without encoding timing. We couple within-host bacterial and immune dynamics to an ICU model structured by colonization age, generating transmission and progression kernels that vary with colonization age. A two-stage framework uses a branching process to quantify finite-threshold crossing and a deterministic ward model to quantify conditional burden after the threshold is reached. Calibration fixes η =0.10 and distributes colonized admissions across pre-admission colonization ages. Of 1, 000, 000 parameter sets, 45, 250 satisfy cumulative R^2≥ 0.90 . The representative calculation is subcritical, with R_0=0.907957 ; across accepted sets, the median and 95 [0.814593, 1.032975] , with 92.460% below one. At Z_crit=5 , exact threshold-reaching probabilities are 0.114659 for age-dependent transmission and 0.117074 for matched constant transmission. Their accepted-set difference range crosses zero, so this contrast has no robust direction. By contrast, age-dependent transmission reaches 50 98.378% of accepted sets. In 200-day conditional simulations, delaying τ _crit from 5 to 15 days increases the age-dependent peak total MRSA burden from 5.629 to 7.298 patients and cumulative total burden from 683.8 to 1398.0 patient-days. At the 9-day baseline and equal R_0 , age dependence gives a 7.372% higher peak but a 9.304% lower cumulative burden than constant transmission. Increasing κ _max from 0.6 to 1.4 times baseline reduces R_0 by 4.133% . Thus, matching R_0 does not preserve timing-sensitive outcomes, but their magnitude and direction depend on the endpoint; over the examined ranges, clearance timing has greater leverage than immune-strength intensification.
