We consider the dynamics of a virus spreading through a population that produces a mutant strain with the ability to infect individuals that were infected with the established strain. Temporary cross-immunity is included using a time delay, but is found to be a harmless delay. We provide some sufficient conditions that guarantee local and global asymptotic stability of the disease-free equilibrium and the two boundary equilibria when the two strains outcompete one another. It is shown that, due to the immune evasion of the emerging strain, the reproduction number of the emerging strain must be significantly lower than that of the established strain for the local stability of the established-strain-only boundary equilibrium. To analyze the unique coexistence equilibrium we apply a quasi steady-state argument to reduce the full model to a two-dimensional one that exhibits a global asymptotically stable established-strain-only equilibrium or global asymptotically stable coexistence equilibrium. Our results indicate that the basic reproduction numbers of both strains govern the overall dynamics, but in nontrivial ways due to the inclusion of cross-immunity. The model is applied to study the emergence of the SARS-CoV-2 Delta variant in the presence of the Alpha variant using wastewater surveillance data from the Deer Island Treatment Plant in Massachusetts, USA.

We introduce a distributed-delay differential equation disease spread model for COVID-19 spread. The model explicitly incorporates the population's time-dependent vaccine uptake and incorporates a gamma-distributed temporary immunity period for both vaccination and previous infection. We validate the model on COVID-19 cases and deaths data from the state of Michigan and use the calibrated model to forecast the spread and impact of the disease under a variety of realistic booster vaccine strategies. The model suggests that the mean immunity duration for individuals after vaccination is $ 350 $ days and after a prior infection is $ 242 $ days. Simulations suggest that both high population-wide adherence to vaccination mandates and a more-than-annually frequency of booster doses will be required to contain outbreaks in the future.

We introduce a novel n-stage vaccination model and corresponding system of differential equations that stratify a population according to their vaccination status. The model is an extension of the classical SIR-type models commonly used for time-course simulations of infectious disease spread and allows for the mitigation effects of vaccination to be uncoupled from other factors, such as changes in social behavior and the prevalence of virus variants. We fit the model to the Virginia Department of Health data on new COVID-19 cases, hospitalizations, and deaths broken down by vaccination status. The model suggests that, from 23 January through 11 September, fully vaccinated individuals were 89.8% less likely to become infected with COVID-19 and that the B.1.617.2 (Delta) variant is 2.08 times more transmissible than previously circulating strains of COVID-19. We project the model trajectories into the future to predict the impact of booster shots.

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than $ 1 $ ($ R_0 < 1 $), the disease-free steady state is asymptotically stable. When $ R_0>1 $, the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratio-dependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Background: A major challenge in cancer treatment is the ability of tumor cells to adapt to immunotherapy through immune escape, often mediated by the PD-1/PD-L1 pathway. To investigate this, we adapted an ordinary differential equation model of combination therapy, incorporating the dynamics of the immune checkpoint inhibitor Avelumab and the immunostimulant NHS-muIL12. Methods: Using literature-derived parameter values, we refitted a single parameter across therapies, which showed that PD-L1 expression increased with immunotherapy, while Avelumab blocked its functional signaling, preventing PD-L1 from suppressing T-cell activity. Incorporating therapy-dependent, dynamically regulated PD-L1 expression enabled a biologically grounded mechanism to reproduce experimental observations, leading us to formulate PD-L1 tumor expression as a dynamic variable ($ϵ$) and providing a mechanistic basis for both therapeutic synergy and treatment failure. Results: We validated this mechanistic framework by showing that the distinct outcomes observed in two independent cancer datasets (EMT-6 and MC38) can be captured by the same model structure, differing only in the parameterization of tumor-specific parameters and PD-L1 regulatory dynamics. Our results indicate that tumor resistance is linked to dose-dependent upregulation of PD-L1 following NHS-muIL12 treatment, explaining treatment failure, while PD-1/PD-L1 blockade in combination therapy enables effective antitumor immune responses. Conclusions: This work provides a validated mechanistic framework for adaptive resistance in combination immunotherapy. Quantified parameter differences between responder and non-responder phenotypes enable clearer biological interpretation and support the development of predictive tools for optimizing treatment strategies.