1. Cuscuta species are rootless, leafless and branchless stems without meaningful photosynthesis. Hence, sink-free and source-free mechanistic growth models are possible yet remain unexplored. Furthermore, phytohormone expression has not yet been studied in the absence of hosts. 2. We use mass conservation in second-order differential equations to build mechanistic models for Cuscuta growth dynamics and UPLC-MS/MS to identify and mathematically score phytohormone expression. 3. We identified four sequential stages of growth – exponential, linear, parabolic deceleration and terminal stages and, in Cuscuta chinensis, the phases are discernable by eye. 4. Analytical solutions to the differential equations fit the growth data well and the model also predicts faster growth in Cuscuta species with smaller seeds, in agreement with the fact that Cuscuta chinensis attains terminal stage faster than Cuscuta japonica. 5. We found evidence for stage-specific phytohormone expression and for the existence of stem-wide phytohormone gradients, especially for the Auxin components MEIAA and ICAId in Cuscuta japonica. 6. We have built models for the roles of phytohormones in Cuscuta growth dynamics, and performed ad hoc calculations which suggest a continuously-increasing Cytokinin/Auxin ratio in growing Cuscuta seedlings, thereby implying a maximum value beyond-which growth slow-down begins. 7. Synthesis: We have created the first-ever source-free and sink-free plant growth models using mass conservation in second-order differential equations and in so-doing, uncovered four growth stages in Cuscuta, observable by eye in Cuscuta chinensis. We found stage-specific phytohormone expression and a likelihood of stem-wide phytohormone expression gradients. We then brought it all together by building models for the roles of phytohormones in Cuscuta growth dynamics and by performing ad hoc calculations which have suggested that the Cytokinin to Auxin ratio increases continuously in growing Cuscuta seedlings, to perhaps attain a maximum value beyond-which slow-down in growth begins.

Viral infection in cell culture and tissue is modeled with delay reaction-diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time-dependent viral load and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS-CoV-2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H^4 spaces using the fixed point technique where the elliptic equation contains fourth order differential operators with and without Fredholm property, generalizing the results of the preceding work [22].