SVLIAR age-of-infection and -immunity structured epidemic model of COVID-19 dynamics

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I. INTRODUCTION
This study is focused on a qualitative analysis of transmission dynamics of the COVID-19 induced disease in sub-classes of susceptible, vaccinated, latent, infected, asymptomatic and recovered individuals of population.The mostly used methods of theoretical analysis of transmission dynamics of infectious diseases in epidemic models are based on the age-structured models of population dynamics [1][2][3][4][5][6][7][8] which relate the age-dependent demographic parameters of susceptible, infected, and recovered sub-classes of population with characteristics of infection-induced disease transmission.Be-cause such models use pretty complex and accurate mathematical methods for simulation, they help us understand better the features of mechanisms, risks, dynamics and mitigation of pandemic diseases.
However, the characteristic time scale of many infectious diseases (including COVID-19) is about several months that is significantly less than the characteristic time scale of demographic processes of populationseveral dozen years.That is why such class of epidemic age-structured models neglects the age dependent modeling of demo-graphic processes and considers the unstructured equation of susceptible subclass dynamics and at the same time the age-of-infection structured equation of infected subclass dynamics [9][10][11].The new independent variable -age-of-infection can play a role of parameter of accurate adjustment of dynamical processes in susceptible and infected sub-classes.
Vitalii V. Akimenko, SVLIAR age-of-infection and -immunity structured epidemic model of COVID-19 dynamics In works [12,13] authors introduce successfully the age-of-infection variable in age-structured equations to describe disease progression through multiple infectious stages (as in the case of HIV, hepatitis B and hepatitis C) [12] and for partition of sub-classes of individuals infected with acute HBV and chronic HBV carriers in hepatitis B transmission model [13].
Another important aspect of studying the infectious disease transmission is the modelling of vaccination of population.Vaccination of susceptible subclass of population plays a crucial role for disease mitigation and decreasing of disease activity in practice, including COVID-19 disease.Such problems are studied theoretically in works [14,15] for epidemic model of SVLIAR-type based on unstructured equations of susceptible, vaccinated, latent, infected, asymptomatic and recovered individuals of population.Authors analyze the effect of vaccination in an SVLIAR model with demography by adding a compartment for vaccinated individuals and considering disease-induced death, imperfect and waning vaccination protection as well as waning infections-acquired immunity.
The new age-of-infection and -immunity structured SVLIAR-type model studied in this paper is based on the structured equations of sub-classes dynamics with introduced 5 in-dependent variables for each subclass (except subclass of susceptible individuals): a 1 -age of vaccine immunity of vaccinated individuals, a 2age of virus infection in organism during incubation period of latent individuals, a 3 -age (time period) of treatment of infectious disease of infected individuals, a 4 -age of asymptomatic infectious disease of asymptomatic individuals, a 5 -age (time period) of immunity of organism after recovering of recovered individuals.Each variable a i runs the internal subprocess that is specific for each particular subclass and allows us to adjust and synchronize all sub-processes with each other.Individuals can move from one subclass to another when these age variables take the maximum values, that is the processes in sub-classes are adjusted and synchronized by age variables.Such detailed age-structured epidemic model with adjustment of sub-processes provides the more accurate simulation of transmission dynamics of infectious disease and can be a basis for further theoretical analysis and simulation of different aspects of COVID-19 epidemic in more complex models.
Local asymptotic stability/instability of disease-free and endemic equilibria of age-of-infection andimmunity structured system is defined in the paper by new derived criteria which relate the maximum ages of each variable (maximum values of a i ) with demographic characteristics of population (birth and death rates), disease-induced death rate, death rate induced by the complications after COVID-19 disease, fraction of fully vaccinated susceptibles per unit of time, vaccination efficacy, rate of disease transmission and the other characteristics of the model.
Age-of-infection and -immunity structured SVLIAR epidemic model is considered in Section 2. Existence of disease-free equilibrium and its local asymptotic stability are studied in terms of the basic reproduction number in Section 3. Existence of endemic equilibrium and its local asymptotic stability [7,[16][17][18] are studied in terms of the basic reproduction number.Conditions of existence of a unique endemic equilibrium are derived in Sections 4, 5.This means, among other things, that additional compartment for vaccinated individuals has no effects on increasing of number of endemic equilibria but effects on its stability.Several concluding remarks are given in Section 6.The existence theorem, explicit recurrent formula for the solution of the agestructured SVLIAR model and numerical method with simulations (like in works [1,2]) are beyond the aim and scope of this paper due to the complexity of the model and will be the subject of our further study.

II. THE MODEL
Age-of-infection and -immunity structured SVLIAR epidemic model considers transmission dynamics of the COVID-19 virus disease in population which consists of the following sub-classes (Figure 1): -susceptible (non-infected), -vaccinated (people are immune after 1st and 2nd vaccinations), -latent (infected individuals without disease symptoms when virus develops within its incubation period), -infected (ill individuals with explicit symptoms), -asymptomatic (individuals with symptoms free form of infectious disease), -recovered (people are immune after infectious disease).
The quantity of susceptible individuals is described by S(t).The age-specific density of vaccinated subclasses of individuals is V (a 1 , t), a 1 ∈ [0, a The age-specific density of latent individuals is L(a 2 , t), a 2 ∈ [0, a d ], t ≥ 0, where a d is a maximum incubation period of virus infection in organ-ism.The number of all latent individuals is The age-specific density of infected individuals, which have the symptoms of disease, is I(a 3 , t), a 3 ∈ [0, a The age-specific density of asymptomatic individuals, which are infected, sick and do not have the symptoms of disease, is A(a 4 , t), a 4 ∈ [0, a d ], t ≥ 0, where a (4) d is a maximum period of asymptomatic infectious disease.The number of asymptomatic individuals is The age-specific density of recovered individuals is R(a 5 , t), a 5 ∈ [0, a d ], t ≥ 0, where a (5) d is a period when disease-induced immunity of individuals starts to wain after recovering (maximum age of disease-induced immunity).The number of recovered individuals is defined as We will assume further that a d , that is the sum of infection incubation period and period of disease-induced immunity waning in recovered individuals is bigger than period of vaccination-induced immunity waning in vaccinated individuals.The series of buster vaccinations and cases with new COVIDmutations for which the current vaccine is not efficient are not considered in this study.We arrive to the autonomous SVLIAR age-structured epidemic model where µ is a natural death rate, b is a birth rate, γ is a disease-induced death rate, v is a death rate induced by the complications after disease, q is a fraction of fully vaccinated susceptibles per unit of time, σ is a vaccination efficacy.Vaccinated individuals become susceptibles when they lost immunity after vaccination at age of vaccination a d .Recovered individuals become susceptibles when they lost immunity after full treatment at the maximum age of after-disease immunity a (5) d .The force of infection is defined as: Biomath 13 (2024), 2404266, https://doi.org/10.55630/j.biomath.2024.04.266 3/15 Vitalii V. Akimenko, SVLIAR age-of-infection and -immunity structured epidemic model of COVID-19 dynamics where β > 0 is a rate of transmission, η > 0 is a modification of transmission for asymptomatic [14].
Eqs. ( 1)-( 6) are completed by the non-negative initial values: Density of newly vaccinated individuals V (0, t) is the sum of the number of vaccinated arrivals and susceptibles per unit of time: Density of new latent individuals L(0, t) is defined through the sum of the number of infected susceptibles and infected vaccinated individuals with low immunity due to the weak efficacy of vaccine: Density of just infected individuals I(0, t) is a (1 − ρ)fraction of a density of latent individuals which have the symptoms of disease after the incubation period of infection: Density of new asymptomatic individuals A(0, t) is a ρ-fraction of density of latent individuals which do not have the symptoms of disease after the incubation period of infection: Density of new recovered individuals R(0, t) is the sum of densities of fully treated infected individuals and recovered asymptomatic individuals: When system (1)-( 13) degenerates to the system of nonlinear ODE, it becomes: Summarizing Eqs. ( 14)-( 19) yields the balance equation for the system (1)-( 13): Thus, the change in total population size over the given time period is due to the difference between newborn individuals and those who died during COVID-19 illness, died from complication after suffering COVID-19 illness and died of natural causes (or other, non-COVID-19 disease) over the given time period.

III. TRIVIAL AND DISEASE-FREE EQUILIBRIA
It is easy to verify that trivial equilibrium of the system (1)-( 13) always exists.The disease-free equilibrium (DFE): satisfies the system: which has a solution: d ) from the second equation into the first one yields: Thus, we arrive at Statement 1.
there exists the disease-free equilibrium of the system (1)-( 13): It is easy to verify that DFE (26) is a particular stationary solution of system (1)-( 13) with initial values d ) , that is susceptible subclass is extinguishing and system has only trivial equilibrium.While condition R 0 > 1 holds if birth rate is relatively large b > µ + q 1 − exp(−µa d ) , that is susceptible subclass is a growing population.
Proof of Theorem 1 is given in Appendix B.
V. LOCAL ASYMPTOTIC STABILITY OF ENDEMIC
Proof of Theorem 2 is given in Appendix D.

VI. DISCUSSION AND CONCLUSIONS
This article is focused on the qualitative analysis of age-of-infection and -immunity structured SVLIARtype model with 5 independent age variables for incubation period of COVID-19, period of vaccine immunity, period of COVID-19 disease treatment, period of asymptomatic COVID-19 disease and period of immunity after recovering.Either, there is only the diseasefree equilibrium when the basic reproduction number equals to one R 0 = 1, or unique positive endemic equilibrium exists when the basic reproduction number is bigger than one R 0 > 1 [1,6,8,18].
The criterion of local asymptotic stability of diseasefree equilibrium which controls the transition of system to the endemic or trivial equilibrium when diseasefree equilibrium is unstable is derived.We proved that system has at most one endemic equilibrium when R 0 > 1.
Criterion of local asymptotic stability of such equilibrium is pretty complex, contains all coefficients of the model and relates the demographic characteristics of population (birth and death rates) with maximums of all age variables: a     This result is a direct extension of similar results for unstructured epidemic SVLIAR model [14,15].Our results show that efficacy of vaccination and fraction of vaccinated susceptibles play a crucial role in stabilization of COVID-19 disease transmission among population.Thus, similarly to unstructured epidemic SVLIAR models, vaccination can cause asymptotic stability of endemic equilibrium.

( 1 )
d (period of time when vaccinationinduced immunity starts to wain (maximum age of vaccination-induced immunity)), a

( 3 )
d (maximum period of infectious disease (or disease treatment)), a