IN THIS NOTE, WE PROVE THE EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR THE BOUNDARY VALUE PROBLEM MODELLING THE STATIONARY CASE OF THE BIOCONVECTIVE FLOW PROBLEM. THE BIOCONVECTIVE MODEL IS A BOUNDARY VALUE PROBLEM FOR A SYSTEM OF FOUR EQUATIONS: THE NONLINEAR STOKES EQUATION, THE INCOMPRESSIBILITY EQUATION, AND TWO TRANSPORT EQUATIONS. THE UNKNOWNS OF THE MODEL ARE THE VELOCITY OF THE FLUID, THE PRESSURE OF THE FLUID, THE LOCAL CONCENTRATION OF MICROORGANISMS, AND THE OXYGEN CONCENTRATION. WE DERIVE SOME APPROPRIATE A PRIORI ESTIMATES FOR THE WEAK SOLUTION, WHICH IMPLIES THE EXISTENCE, BY APPLICATION OF GOSSEZ THEOREM, AND THE UNIQUENESS BY STANDARD METHODOLOGY OF COMPARISON OF TWO ARBITRARY SOLUTIONS.

IN THIS WORK WE DEVELOP A STUDY OF POSITIVE PERIODIC SOLUTIONS FOR A MATHEMATICAL MODEL OF THE DYNAMICS OF COMPUTER VIRUS PROPAGATION. WE PROPOSE A GENERALIZED COMPARTMENT MODEL OF SEIR-KS TYPE, SINCE WE CONSIDER THAT THE POPULATION IS PARTITIONED IN FIVE CLASSES: SUSCEPTIBLE (S); EXPOSED (E); INFECTED (I); RECOVERED (R); AND KILL SIGNALS (K), AND ASSUME THAT THE RATES OF VIRUS PROPAGATION ARE TIME DEPENDENT FUNCTIONS. THEN, WE INTRODUCE A SUFFICIENT CONDITION FOR THE EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF THE GENERALIZED SEIR-KS MODEL. THE PROOF OF THE MAIN RESULTS ARE BASED ON A PRIORI ESTIMATES OF THE SEIR-KS SYSTEM SOLUTIONS AND THE APPLICATION OF COINCIDENCE DEGREE THEORY. MOREOVER, WE PRESENT AN EXAMPLE OF A GENERALIZED SYSTEM SATISFYING THE SUFFICIENT CONDITION.

IN THIS PAPER, WE STUDY THE PROBLEM OF COEFFICIENTS IDENTIFICATION IN POPULATION GROWTH MODELS. WE CONSIDER THAT THE DYNAMICS OF THE POPULATION IS DESCRIBED BY A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF SUSCEPTIBLE-INFECTIVE-RECOVERED (SIR) TYPE, AND WE ASSUME THAT WE HAVE A DISCRETE OBSERVATION OF INFECTIVE POPULATION. WE CONSTRUCT A CONTINUOUS OBSERVATION BY APPLYING TIME SERIES AND AN APPROPRIATE FITTING TO THE DISCRETE OBSERVATION DATA. THE IDENTIFICATION PROBLEM CONSISTS IN THE DETERMINATION OF DIFFERENT PARAMETERS IN THE GOVERNING EQUATIONS SUCH THAT THE INFECTIVE POPULATION OBTAINED AS SOLUTION OF THE SIR SYSTEM IS AS CLOSE AS TO THE OBSERVATION. WE INTRODUCE A REFORMULATION OF THE CALIBRATION PROBLEM AS AN OPTIMIZATION PROBLEM WHERE THE OBJECTIVE FUNCTION AND THE RESTRICTION ARE GIVEN BY THE COMPARISON IN THE L2-NORM OF THEORETICAL SOLUTION OF THE MATHEMATICAL MODEL AND THE OBSERVATION, AND THE SIR SYSTEM GOVERNING THE PHENOMENON, RESPECTIVELY. WE SOLVE NUMERICALLY THE OPTIMIZATION PROBLEM BY APPLYING THE GRADIENT METHOD WHERE THE GRADIENT OF THE COST FUNCTION IS OBTAINED BY INTRODUCING AN ADJOINT STATE. IN ADDITION, WE CONSIDER A NUMERICAL EXAMPLE TO ILLUSTRATE THE APPLICATION OF THE PROPOSED CALIBRATION METHOD.

IN THIS PAPER, WE INTRODUCE THE FUNCTIONAL FRAMEWORK AND THE NECESSARY CONDITIONS FOR THE WELL-POSEDNESS OF AN INVERSE PROBLEM ARISING FROM THE MATHEMATICAL MODELING OF DISEASE TRANSMISSION. THE DIRECT PROBLEM IS GIVEN BY AN INITIAL BOUNDARY VALUE PROBLEM FOR A REACTION-DIFFUSION SYSTEM. THE INVERSE PROBLEM CONSISTS IN THE DETERMINATION OF THE DISEASE AND RECOVERY TRANSMISSION RATES FROM OBSERVED MEASUREMENT OF THE DIRECT PROBLEM SOLUTION AT THE FINAL TIME. THE UNKNOWNS OF THE INVERSE PROBLEM ARE THE COEFFICIENTS OF THE REACTION TERM. WE FORMULATE THE INVERSE PROBLEM AS AN OPTIMIZATION PROBLEM FOR AN APPROPRIATE COST FUNCTIONAL. THEN, THE EXISTENCE OF SOLUTIONS OF THE INVERSE PROBLEM IS DEDUCED BY PROVING THE EXISTENCE OF A MINIMIZER FOR THE COST FUNCTIONAL. MOREOVER, WE ESTABLISH THE UNIQUENESS UP AN ADDITIVE CONSTANT OF THE IDENTIFICATION PROBLEM. THE UNIQUENESS IS A CONSEQUENCE OF THE FIRST ORDER NECESSARY OPTIMALITY CONDITION AND A STABILITY OF THE INVERSE PROBLEM UNKNOWNS WITH RESPECT TO THE OBSERVATIONS.