Competition Between Self-birth and Catalyzed Death in Aggregation Growth with Catalysis Injection

We propose two irreversible aggregation growth models of aggregates of two distinct species (A and B) to study the interactions between virus aggregates and medicine efficacy aggregates in the virus-medicine cooperative evolution system. The A-species aggregates evolve driven by self monomer birth andB-species aggregate-catalyzed monomer death in model I and by self birth, catalyzed death, and self monomer exchange reactions in model II, while the catalyst B-species aggregates are assumed to be injected into the system sustainedly or at a periodic time-dependent rate. The kinetic behaviors of the A-species aggregates are investigated by the rate equation approach based on the mean-field theory with the self birth rate kernel I_A(K)=Ik, catalyzed death rate kernel J_AB(k)=Jk and self exchange rate kernel K_A(k,l)=Kkl. The kinetic behaviors of the A-species aggregates are mainly dominated by the competition between the two effects of the self birth (with the effective rate I) and the catalyzed death (with the effective rate JB₀), while the effects of the self exchanges of the A-species aggregates which appear in an effectiverate KA₀ play important roles in the cases of I>JB₀ and I=JB₀. The evolution behaviors of the total mass M^A(t)₁ and the total aggregate number M^A(t)₀ are obtained, and the aggregate size distribution a_k(t) of species A is found toapproach a generalized scaling form in the case of I ≧ JB₀ and a special modified scaling form in the case of I0. The periodical evolution of the B-monomers concentration plays an exponential form of the periodic modulation.