Age structured SIR model for the spread of infectious diseases through indirect contacts
AAge structured SIR model for the spread of infectious diseases throughindirect contacts
Manoj Kumar, Syed AbbasSchool of Basic Sciences,Indian Institute of Technology Mandi,Kamand (H.P.) - 175005, IndiaEmail : [email protected]
Abstract : In this article, we discuss an age-structured SIR model in which disease not only spreadthrough direct person to person contacts for e.g. infection due to surface contamination but it can alsospread through indirect contacts. It is evident that age also plays a crucial role in SARS virus infectionincluding COVID-19 infection. We formulate our model as an abstract semilinear Cauchy problem in anappropriate Banach space to show the existence of solution and also show the existence of steady states.It is assumed in this work that the population is in a demographic stationary state and show that there isno disease-free equilibrium point as long as there is a transmission of infection due to the indirect contactsin the environment.
Key Words :SIR Model, Age structured population model, Riesz-Fr´echet-Kolmogorov theorem, Semi-groups of operators.
AMS Subject Classification : 00A71; 34G20; 47D03
Infectious diseases are one of threat to humanity. Due to increase in world population and mobility,pathogen transmission is easy and it is difficult to control the spread of disease. Viral transmission dependsboth on the interaction with host population and with the environment.Mathematical models can project how infectious diseases progress. The model can suggest the possibleoutcome of an epidemic which will help agencies to take well though measures. In 1927, Kermack andMcKendrick [3] introduced a model (called SIR model) by considering a given population having threecompartments. The compartments are divided into individuals in susceptible S ; infected I ;, and removed R class. It is very important to study infectious diseases and their possible nature of spread.Most of the cases it is assumed that the spread of infectious diseases is through person to person directcontact. But some infectious diseases can also spread through indirect contacts like contact with contami-nated surface having virus on it i.e. if a person touches their eyes, mouth or nose after touching fomites oranimals to human transmission. Through many studies it is observed that coronaviruses (including SARSCov2) may persist on objects or surfaces for some hours to many days. The persistence depends on differentfactors (e.g. surface type, humidity or temperature of the environment). Fomites consist of both permeable1 a r X i v : . [ q - b i o . P E ] J u l nd non permeable objects or surfaces that can be contaminated with pathogenic micro-organisms andserves as a vehicle in transmission. SARS-CoV-2, the coronavirus (CoV) causing COVID-19 is creatingthe most severe health issues for individuals above the age of 60 — with particularly fatal results for thoseindividuals having age above 80. In the United states, 31-59% of individuals ages 75 to 84 diagnosed withthe virus having svere symptoms due to which hospitalization is necesaary, in comparing with 14-21%of confirmed patients ages between 20 to 44. This data is based on US Centers for Disease Control andPrevention (CDC) report. So, it is natural to consider age structure while modeling the infectious diseasetransmission. The risk of transmission of infectious disease varies in different environments, for exampleat school, at home, at work place or in the community. [4] studied projected age-specific contact rates forcountries in different stages in development and with different demographic structures to those studied inPOLYMOD (a European Commission project), which provide validated approximations to social contactpatterns when directly measured data is not available. The data plotted in Fig. 1, Fig. 2, Fig. 3 and Fig. 4show the relation between age of individual and age of contact i.e. number of contacts made by individualsat all locations, at home, school and work respectively. Yellowish color on the diagonal of Fig. 1 and Fig.2 shows that same age individuals have more chances of direct contacts, so transmission coefficient will belarge for same age individuals. 2rom the above heat maps, it is clear that it is natural to add age structure in ordinary differentialequation(ODE) based SIR models. So, after adding age structure, the ODE based SIR models becomepartial differential equation (PDE) models that are more complex to analyze. There is extensive litera-ture available on age-structured SIR models (for more details see [1, 2, 5–11]). In [12] an epidemiologicalmodel which study the impact of decline in population on the dynamics of infectious diseases especiallychildhood diseases is considered and also an example of measles in Italy is considered and [13] studied theSARS outbreak in Taiwan, using the data of daily reported cases from May 5 to June 4, 2003 to studythe spread of virus. H. Inaba [1] discussed threshold and stability results for an age structured SIR model,Andrea [2] generalized the work of [1] and also considered immigration of infective in all epidemiologicalcompartments. We considered an age structured SIR model in which individuals can also get infected dueto contaminated surfaces. We also assume that the net reproduction rate of the host population is unitywhich also makes our model different from the model considered in [2].Our work is divided into four sections. In section 2, we formulate our age structured SIR model. Insection 3, we discuss the existence of solution to our model. In section 4, we discuss steady state solutionsand show that there is no disease free steady state solution as long as there is transmission due to indirectcontacts in the environment. Let U ( a, t ) be the density of individuals of age a at time t . µ ( a ) and β ( a ) be age dependent mortality andfertility rates respectively. Let a m be the maximum age which an individual can attain i.e. the maximumlife span of an individual. Then the evolution of U ( a, t ) can be modeled by the following McKendrick-VonFoerster PDE with initial and boundary conditions: ∂U ( a,t ) ∂t + ∂U ( a,t ) ∂a = − µ ( a ) U ( a, t ) ( a, t ) ∈ (0 , a m ) × (0 , ∞ ) U (0 , t ) = (cid:82) a m β ( a ) U ( a, t ) da t ∈ (0 , ∞ ) U ( a,
0) = U ( a ) a ∈ (0 , a m ) , (2.1)where U (0 , t ) denotes the number of newborns per unit time at time t . We suppose that the mortality rate µ ∈ L loc ([0 , a m )) with the condition (cid:82) a m µ ( a ) da = + ∞ and the fertility rate β ∈ L ∞ (0 , a m ) . e − (cid:82) a µ ( s ) ds indicates the proportion of individuals who are still living at age a and (cid:82) a m β ( a ) e − (cid:82) a µ ( s ) ds da representsthe net reproduction rate. Let us assume that the net reproduction rate is 1. So, steady state solution isgiven by U ( a, t ) = U ( a ) = β e − (cid:82) a µ ( τ ) dτ , where β is given by β = (cid:82) a m U ( a ) da (cid:82) a m e − (cid:82) a µ ( τ ) dτ da . Let S ( a, t ) , I ( a, t ) and R ( a, t ) be the densities of susceptible, infective and recovered individuals of age a at time t . r ( a, b ) is the age dependent transmission coefficient which describes the contact processbetween susceptible and infective individuals i.e. r ( a, b ) S ( a, t ) I ( b, t ) dadb is the number of individuals whoare susceptibles with age lies between a and a + da and contract the disease after contact with an infectiveindividual aged between b and b + db . We assume the form of force of infection is given in the following3unctional form λ ( a, t ) = (cid:90) a m r ( a, η ) I ( η, t ) dη. Then the disease spread according to the following system of partial differential equations ∂S ( a,t ) ∂t + ∂S ( a,t ) ∂a = − λ ( a, t ) S ( a, t ) − c ( a ) S ( a, t ) − µ ( a ) S ( a, t ) ∂I ( a,t ) ∂t + ∂I ( a,t ) ∂a = λ ( a, t ) S ( a, t ) + c ( a ) S ( a, t ) − b ( a ) I ( a, t ) − µ ( a ) I ( a, t ) ∂R ( a,t ) ∂t + ∂R ( a,t ) ∂a = b ( a ) I ( a, t ) − µ ( a ) R ( a, t ) S (0 , t ) = (cid:82) a m β ( a )( S ( a, t ) + I ( a, t ) + R ( a, t )) da, I (0 , t ) = 0 , R (0 , t ) = 0 S ( a,
0) = S ( a ) , I ( a,
0) = I ( a ) and R ( a,
0) = R ( a ) . (2.2) b ( a ) is the recovery rate of individuals and c ( a ) is the proportion of individuals which are infected dueto surface contamination. This factor c ( a ) depends on the proportion of frontline workers as they aresusceptible to viral infection from indirect contacts even during lockdown situation (if lockdown is imposed).Here we are assuming that spread of disease already started and fomites are present in the environmenteven if transmission coefficient r ( a, b ) is zero. Assume that b, c ∈ L ∞ (0 , a m ) and r ∈ L ∞ ((0 , a m ) × (0 , a m ))and also assume that all are non negative. [14] studied how respiratory and viral disease spread in thepresence of fomites. It is observed that enveloped respiratory viruses remain viable for less time and thenonenveloped enteric viruses remain viable for longer time. They calculated the inactivations coefficientsof various respiratory viruses. Fig. 5 shows the respiratory virus inactivation rates ( K j ). In Fig. 5, wehave used the short forms flu and cov for influenza and coronavirus respectively.We impose the following conditions on our model:i)Although there may be incubation period for some diseases but here we are assuming that there isno incubation period and the individuals become infected instantaneously after contact with infectedindividuals or fomites. 4i) We assume that the age zero individuals can not be infected.iii) Transmission coefficient r ( a, b ) only summarizes the contact process between susceptible and infectedindividuals.iv) Population is in stationary demographic state.v) The susceptible individuals who got infected due to contact with infected individuals have not infecteddue to the contact with fomites and vice versa.Let S ( a, t ) , I ( a, t ) and R ( a, t ) be defined in the following way S ( a, t ) = S ( a, t ) U ( a, t ) , I ( a, t ) = I ( a, t ) U ( a, t ) and R ( a, t ) = R ( a, t ) U ( a, t )and the force of infection is given by λ ( a, t ) = (cid:90) a m r ( a, η ) U ( η ) I ( η, t ) dη. Then our new system becomes ∂S ( a,t ) ∂t + ∂S ( a,t ) ∂a = − λ ( a, t ) S ( a, t ) − c ( a ) S ( a, t ) ∂I ( a,t ) ∂t + ∂I ( a,t ) ∂a = λ ( a, t ) S ( a, t ) + c ( a ) S ( a, t ) − b ( a ) I ( a, t ) ∂R ( a,t ) ∂t + ∂R ( a,t ) ∂a = b ( a ) I ( a, t ) S (0 , t ) = 1 , I (0 , t ) = 0 , R (0 , t ) = 0 S ( a,
0) = S ( a ) , I ( a,
0) = I ( a ) and R ( a,
0) = R = ( a ) S ( a, t ) + I ( a, t ) + R ( a, t ) = 1 . (2.3)So, new transformations reduced our system into a simpler form i.e. boundary conditions now becomeconstant and there is no term involving natural mortality rate. If we observe system (2.3) carefully, then it is clear that once susceptible and infected individuals areknown, recovered individuals can be obtained easily so, it is enough to show the existence of solution tothe below SI system instead of full SIR system ∂S ( a,t ) ∂t + ∂S ( a,t ) ∂a = − λ ( a, t ) S ( a, t ) − c ( a ) S ( a, t ) ∂I ( a,t ) ∂t + ∂I ( a,t ) ∂a = λ ( a, t ) S ( a, t ) + c ( a ) S ( a, t ) − b ( a ) I ( a, t ) S (0 , t ) = 1 , I (0 , t ) = 0 . (3.1)We will analyze the system (3.1) only, because force of infection does not explicitly depend on recoveredindividuals. Let ˜ S ( a, t ) = S ( a, t ) − , ˜ I ( a, t ) = I ( a, t ) , then the system (3.1) reduces to ∂ ˜ S ( a,t ) ∂t + ∂ ˜ S ( a,t ) ∂a = − λ ( a, t )(1 + ˜ S ( a, t )) − c ( a )(1 + ˜ S ( a, t )) ∂ ˜ I ( a,t ) ∂t + ∂ ˜ I ( a,t ) ∂a = λ ( a, t )(1 + ˜ S ( a, t )) + c ( a )(1 + ˜ S ( a, t )) − b ( a ) ˜ I ( a, t )˜ S (0 , t ) = 0 , ˜ I (0 , t ) = 0 .λ ( a, t ) = (cid:82) a m r ( a, η ) U ( η ) ˜ I ( η, t ) dη. (3.2)5et X = L (0 , a m ; C ) equipped with the L norm and linear operator A be defined as( A ξ )( a ) = ( − dda ξ ( a ) − c ( a ) ξ ( a ) , − dda ξ ( a ) − b ( a ) ξ ( a ) − c ( a ) ξ ( a ))where ξ = ( ξ ( a ) , ξ ( a )) ∈ D ( A ) D ( A ) = { ξ = ( ξ , ξ ) ∈ X | ξ , ξ ∈ AC [0 , a m ] , ξ (0) = (0 , } AC [0 , a m ] is the set of absolutely continuous functions.Suppose that r ( a, b ) ∈ L ∞ ((0 , a m ) × (0 , a m )) and( F ξ )( a ) = ( − ( P ξ )( a )(1 + ξ ( a )) − c ( a ) , ( P ξ )( a )(1 + ξ ( a )) + c ( a )) , ξ ∈ X, where bounded linear operator P is defined by( P ψ )( a ) = (cid:90) a m r ( a, η ) U ( η ) ψ ( η ) dη, ψ ∈ L (0 , a m ) . Now, system (3.2) can be written as an abstract semilinear Cauchy problem in Banach space
Xddt Z ( t ) = A ( t ) Z ( t ) + F ( Z ( t )) , Z (0) = Z ∈ Z where Z ( t ) = ( ˜ S ( · , t ) , ˜ I ( · , t )) ∈ Z, Z ( a ) = ( ˜ S ( a ) , ˜ I ( a ))In the same manner as proved in [1], we can prove that A generates a C semigroup S ( t ) , t ≥ F iscontinuously Fr´echet differentiable on X .So, for each Z ∈ X , there exists a maximal interval of existence [0 , t ) and a unique solution t −→ Z ( t ; Z )which is continuous from [0 , t ) to X such that Z ( t, Z ) = S ( t ) Z + (cid:90) t S ( t − σ ) F ( Z ( σ ; Z )) dσ ∀ t ∈ [0 , t ] . Moreover, if Z ∈ D ( A ) , then Z ( t ; Z ) ∈ D ( A ) for 0 ≤ t < t and t −→ Z ( t ; Z ) is continuously differen-tiable and satisfies (3.2) on [0 , t ) . dS ( a ) da = − λ ( a ) S ( a ) − c ( a ) S ( a ) dI ( a,t ) da = λ ( a ) S ( a ) + c ( a ) S ( a ) − b ( a ) I ( a ) S (0) = 1 , I (0) = 0 (4.1)with λ ( a ) = (cid:82) a m r ( a, η ) U ( η ) I ( η ) dη .Steady state solution can be obtained as S ( a ) = exp (cid:18) − (cid:90) a ( λ ( σ ) + c ( σ ) dσ (cid:19) I ( a ) = (cid:90) a exp (cid:18) − (cid:90) aσ b ( η ) dη (cid:19) ( λ ( σ ) + c ( σ )) exp (cid:18) − (cid:90) σ ( λ ( η ) + c ( η )) dη (cid:19) dσ. λ ( a ) = (cid:90) a m r ( a, ζ ) U ( ζ ) I ( ζ ) dζ (4.2)= (cid:90) a m r ( a, ζ ) U ( ζ ) (cid:90) ζ exp (cid:18) − (cid:90) ζσ b ( η ) dη (cid:19) ( λ ( σ ) + c ( σ )) exp (cid:18) − (cid:90) σ ( λ ( η ) + c ( η )) dη (cid:19) dσdζ. = (cid:90) a m φ ( a, σ )( λ ( σ ) + c ( σ )) exp (cid:18) − (cid:90) σ ( λ ( η ) + c ( η )) dη (cid:19) dσ (4.3)where φ ( a, σ ) = (cid:90) a m σ r ( a, ζ ) U ( ζ ) exp (cid:18) − (cid:90) ζσ b ( η ) dη (cid:19) dζ (4.4)Using (4.2), we can get the following estimate | λ ( a ) |≤ U (cid:107) r (cid:107) ∞ (cid:107) I (cid:107) where (cid:107)(cid:107) ∞ and (cid:107)(cid:107) are the L ∞ and L norms respectively and U is the total population.Therefore, λ ∈ L ∞ (0 , a m ) . It is clear that there is no disease free equilibrium as long as there is transmission due to fomites in theenvironment. That means if there are fomites present in the environment contaminated with pathogenicmicro-organisms, disease still can spread without direct contact between susceptible and infected individ-uals.On Banach space E = L (0 , a m ), with positive cone E + = { ψ ∈ E | ψ ≥ a.e. } , let us defineΦ( ψ )( a ) = (cid:90) a m φ ( a, σ )( ψ ( σ ) + c ( σ )) exp (cid:18) − (cid:90) σ ( ψ ( η ) + c ( η )) dη (cid:19) dσ (4.5)Suppose that we have the following assumptions(A1) r ( · , · ) satisfies lim h −→ (cid:82) a m (cid:107) r ( a + h, s ) − r ( a, s ) (cid:107) da = 0 uniformly for s ∈ R with r ( · , · ) extended bydefining r ( a, s ) = 0 for a.e. a, s ∈ ( −∞ , ∪ ( a m , ∞ ) . (A2) There exist m > , < α < a m such that r ( a, b ) ≥ m for a.e. ( a, b ) ∈ (0 , a m ) × ( a m − α, a m ).(A3) There exist a , a satisfying 0 ≤ a < a ≤ a m such that c ( a ) > a ∈ ( a , a ).Observe that Φ(0)( a ) = (cid:90) a m φ ( a, σ ) exp (cid:18) − (cid:90) σ c ( η ) dη (cid:19) c ( σ ) dσ. (4.6)Since force of infection is non negative, we have λ ( a ) ≥ Φ(0)( a ) a.e. a ∈ (0 , a m )and because of assumption (A2), we have Φ(0)( a ) >
0. Now, we will prove an important theorem whichwill help us to show the existence of fixed point to (4.5).7 heorem 4.1.
Let D = { ψ ∈ L (0 , a m ) | (cid:107) ψ (cid:107) ≤ M, M is a positive constant } and suppose that the as-sumptions (A1)-(A3) hold, then(a) D is bounded, closed, convex and also Φ( D ) ⊆ D .(b) Φ is completely continuous .Hence, Schauder’s principle gives existence of fixed point of (4.5).Proof. Boundedness of set D is clear and also for any ψ , ψ ∈ D , ≤ p ≤ pψ + (1 − p ) ψ ∈ D . Closedness also follows from the definition of D . Now, we will show that Φ( D ) ⊆ D .Φ( ψ )( a ) ≤ (cid:107) φ (cid:107) ∞ (cid:90) a m ( ψ ( σ ) + c ( σ )) exp (cid:18) − (cid:90) σ ( ψ ( η ) + c ( η )) dη (cid:19) dσ ≤ M (cid:107) φ (cid:107) ∞ (cid:107) c (cid:107) ∞ (cid:90) a m exp (cid:18) − (cid:90) a m ψ ( η ) dη (cid:19) dσ + M (cid:107) φ (cid:107) ∞ (cid:90) a m ψ ( σ ) exp (cid:18) − (cid:90) a m ψ ( η ) dη (cid:19) dσ = M (cid:107) φ (cid:107) ∞ (cid:107) c (cid:107) ∞ (cid:90) a m exp (cid:18) − (cid:90) a m ψ ( η ) dη (cid:19) dσ + M (cid:107) φ (cid:107) ∞ (cid:20) − exp (cid:18) − (cid:90) a m ψ ( s ) ds (cid:19)(cid:21) where M is an upper bound on exp (cid:0) − (cid:82) σ c ( η ) dη (cid:1) . Now , using the fact that | ψ | ≤ M , we can easily provethat | Φ( ψ )( a ) | ≤ M for some generic constant M . Now, (4.7)(Φ( ϕ ))( a ) − (Φ( ϕ ))( a ) = (cid:90) a m (cid:104) ϕ e − (cid:82) σ ϕ ( η ) dη − ϕ e − (cid:82) σ ϕ ( η ) dη (cid:105) φ ( a, σ ) e − (cid:82) σ c ( η ) dη dσ + (cid:90) a m (cid:104) e − (cid:82) σ ϕ ( η ) dη − e − (cid:82) σ ϕ ( η ) dη (cid:105) φ ( a, σ ) e − (cid:82) σ c ( η ) dη dσ. Let us firstly estimate the first integral as follows (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) a m (cid:104) ϕ e − (cid:82) σ ϕ ( η ) dη − ϕ e − (cid:82) σ ϕ ( η ) dη (cid:105) φ ( a, σ ) e − (cid:82) σ c ( η ) dη dσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) φ (cid:107) ∞ M (cid:104) e − (cid:82) am ϕ ( η ) dη − − e − (cid:82) am ϕ ( η ) dη + 1 (cid:105) = M (cid:107) φ (cid:107) ∞ (cid:16) e −(cid:107) ϕ (cid:107) − e −(cid:107) ϕ (cid:107) (cid:17) ≤ M (cid:107) φ (cid:107) ∞ (cid:107) ϕ − ϕ (cid:107) ≤ M (cid:107) ϕ − ϕ (cid:107) M is generic constant. Similarly, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) a m (cid:104) e − (cid:82) σ ϕ ( η ) dη − e − (cid:82) σ ϕ ( η ) dη (cid:105) φ ( a, σ ) e − (cid:82) σ c ( η ) dη dσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ M (cid:107) φ (cid:107) ∞ (cid:90) a m (cid:104) e − (cid:82) am ϕ ( η ) dη e (cid:82) amσ ϕ ( η ) dη − e − (cid:82) am ϕ ( η ) dη e (cid:82) amσ ϕ ( η ) dη (cid:105) dσ = M (cid:107) φ (cid:107) ∞ (cid:90) a m (cid:104) e −(cid:107) ϕ (cid:107) e (cid:82) amσ ϕ ( η ) dη − e −(cid:107) ϕ (cid:107) e (cid:82) amσ ϕ ( η ) dη (cid:105) dσ ≤ M (cid:16) e (cid:107) ϕ (cid:107) − e (cid:107) ϕ (cid:107) (cid:17) ≤ M (cid:107) ϕ − ϕ (cid:107) which proves the continuity of Φ.Now we will prove that Φ is compact operator, so let us define T , T : L (0 , a m ) −→ L (0 , a m ) by T ( ψ )( a ) = (cid:82) a m ψ ( σ ) k ( a, σ ) dσ (4.8) T ( ψ )( a ) = (cid:82) a m ψ ( σ ) k ( a, σ ) dσ (4.9) k ( a, σ ) = φ ( a, σ ) exp (cid:0) − (cid:82) σ c ( η ) dη (cid:1) , k ( a, σ ) = φ ( a, σ ) c ( σ ) exp (cid:0) − (cid:82) σ c ( η ) dη (cid:1) . (4.10)The operators T , T are linear, continuous and positive. By applying Riesz-Fr´echet-Kolmogorov theo-rem on compactness in L , we can conclude that T , T are compact operators. Now, let us define nonlinearoperators F , F : L (0 , a m ) −→ L (0 , a m ) by F ( ψ )( σ ) = ψ ( σ ) exp (cid:0) − (cid:82) σ ψ ( τ ) dτ (cid:1) (4.11) F ( ψ )( σ ) = exp (cid:0) − (cid:82) σ ψ ( τ ) dτ (cid:1) . (4.12)Here, F , F are continuous and hence T ◦ F , T ◦ F are compact operators in L (0 , a m ).Therefore, Φ = T ◦ F + T ◦ F is compact operator. Hence, Schauder’s principle gives existence of fixedpoint of (4.5).Let T = Φ (cid:48) (0) denote the Fr´echet derivative of Φ at 0 i.e. T ( ψ )( a ) = (cid:90) a m φ ( a, σ ) ψ ( σ ) exp (cid:18) − (cid:90) σ c ( η ) dη (cid:19) dσ for a ∈ (0 , a m ) , ψ ∈ L (0 , a m ) . Clearly, T is a positive linear, continuous and also compact operator. Let us define T ( ψ )( a ) = (cid:82) a m φ ( a, σ ) ψ ( σ ) dσ (4.13) T n ( ψ )( a ) = (cid:82) a m φ ( a, σ ) ψ ( σ ) exp (cid:0) − (cid:82) σ c n ( η ) dη (cid:1) dσ (4.14)where c n is the sequence of the proportion of individuals infected due to indirect contacts. The spectralradius ( ρ ( T )) of the operator T plays an important role in deciding the nature of equilibrium solutions i.e.whether disease free equilibrium solution exists or not. In our case if there is a proportion of individualswho are infected due to fomites, disease free equilibrium point will not exists. Our aim is to prove thefollowing theorem: Theorem 4.2.
Let T be as defined in (4.13) and Φ n be analogous to Φ in which c is replaced by c n . a) If spectral radius ρ ( T ) ≤ , then the sequence { ψ n } of fixed points of Φ n converges to zero.(b) If spectral radius of T is larger than , then ∃ γ > such that (cid:107) ψ n (cid:107)≥ γ ∀ n ∈ N . Our aim is also to prove that lim n →∞ ρ ( T n ) = ρ ( T )which gives dependence of force of infection on c n . Before proving the above theorem, we will prove somelemmas and also state some theorems. Definition 4.3.
Let E + ⊂ E be a cone in Banach space E , then the cone E + is called total if the followingset { f − g : f, g ∈ E + } is dense in Banach space E . Theorem 4.4. (Krein-Rutman (1948)) Let E be a real Banach space and E + be total order cone in E .Let A : E −→ E be positive linear and compact operator w.r.t. E + and also ρ ( A ) > . Then ρ ( A ) is aneigen value of A and A ∗ with eigen vectors in E + and E ∗ + respectively. In SIR model without fomites transmission coefficient c , Inaba [1] proved the following results: Theorem 4.5. ( [1] Proposition 4.6) Let T be the Fr´echet derivative of Φ at (a) If spectral radius ρ ( T ) ≤ , then there is a disease free fixed point ψ = 0 to the operator Φ .(b) If spectral radius ρ ( T ) > , then there exist atleast one non zero fixed point of Φ . Theorem 4.6. ( [15] Theorem V6.6) Let E = L p ( µ ) , p ∈ [1 , ∞ ] and ( Z, S , µ ) be a σ − finite measurespace. Suppose A ∈ L ( E ) is defined by A g ( t ) = (cid:90) K ( s, t ) g ( s ) dµ ( s ) , g ∈ L p ( µ ) , non negative K is S × S measurable kernel which satisfy the following assumptions(a) Some power of A is compact.(b) C ∈ S and µ ( C ) > , µ ( Z \ C ) > ⇒ (cid:90) Z \ C (cid:90) C K ( s, t ) dµ ( s ) dµ ( t ) > . Then ρ ( A ) > is an eigen value of A with a unique normalized eigen function g satisfying g ( C ) > µ -a.e.; moreover if K ( s, t ) > µ ⊗ µ -a.e, then every other eigen value λ of A has the bound | λ | < ρ ( A ) . Let { c n } be a sequence in L ∞ + (0 , a m ) such that c n ( a ) −→ n −→ ∞ a.e. a ∈ (0 , a m ) i.e. proportionof individuals who are susceptible to fomite infection are becoming less and Φ is defined as in (4 .
5) with c = 0. Propostion 4.7.
There exist a converging subsequence { ψ n k } of { ψ n } such that if ψ = lim k →∞ ψ n k , then ψ is the fixed point of Φ . roof. Because Φ is compact and 0 ≤ (cid:107) ψ n (cid:107)≤ M, ∃ a converging subsequence { Φ ( ψ n k ) } and let ψ = lim k →∞ Φ ( ψ n k ) . Because Φ n k ( ψ n k ) − Φ ( ψ n k ) = (cid:90) a m φ ( a, σ ) ψ ( σ ) exp (cid:18) − (cid:90) σ c n k ( η ) dη (cid:19) dσ − (cid:90) a m φ ( a, σ ) ψ ( σ ) dσ = (cid:90) a m φ ( a, σ ) ψ ( σ ) (cid:20) exp (cid:18) − (cid:90) σ c n k ( η ) dη (cid:19) − (cid:21) dσ ∵ lim n →∞ c n ( a ) = 0 a.e. a ∈ (0 , a m )we have lim k →∞ [Φ n k ( ψ n k ) − Φ ( ψ n k )] = 0 . ∴ lim k →∞ ψ n k = lim k →∞ Φ n k ( ψ n k )= lim k →∞ [Φ ( ψ n k ) + Φ n k ( ψ n k ) − Φ ( ψ n k )] = ψ. Because Φ is continuous, we have lim k →∞ Φ ( ψ n k ) = Φ ( ψ ) = ψ which proves that Φ ( ψ ) = ψ. Lemma 4.8.
Suppose T be as defined in (4.13), then ρ ( T ) is an eigen value of both T and T ∗ withunique strictly positive normalized eigen vectors ψ and f respectively.Proof. We know that T ( ψ )( a ) = (cid:90) a m φ ( a, η ) ψ ( η ) dη and is a compact operator by Theorem 4.1. Comparing T with A , conditions of Theorem 4.6 are satisfiedand therefore ρ ( T ) > T with a unique normalized eigen vector ψ ∈ L (0 , a m ),satisfying ψ ( a ) > a.e. and every other eigen value λ of T satisfy | λ | < ρ ( T ). Also T and T ∗ both havesame non zero eigen values with same multiplicities. Since ρ ( T ) is the only eigen value of T with a uniquenormalized eigen vector ψ , ρ ( T ) is also an algebraically simple eigen value of T ∗ with unique normalizedeigen function f . Now our task is to prove that eigen function f is strictly positive. Suppose functionˆ f ∈ L ∞ + \ { } representing the functional f be defined as (cid:104) f, ψ (cid:105) = (cid:90) a m ˆ f ( η ) ψ ( η ) dη ∀ ψ ∈ L (0 , a m ) . Now , T ∗ ( ϕ )( a ) = (cid:90) a m ϕ ( η ) φ ( η, a ) dη ∀ ϕ ∈ L ∞ (0 , a m ) , g : [0 , a m ] −→ R which is continuous and g ( a ) > ∀ a ∈ [0 , a m ) and vanishes at a m (because of assumption (A2)) such that φ ( η, a ) ≥ g ( a ) a.e. η, a ∈ (0 , a m ) . Then , ˆ f ( a ) = 1 ρ ( T ) T ∗ ( ˆ f )( a ) ≥ ρ ( T ) g ( a ) (cid:90) a m ˆ f ( η ) dη > f is strictly positive as ˆ f ∈ L ∞ + (0 , a m ) \ { } . Lemma 4.9.
Let T and T n be as defined in (4.13) and (4.14) respectively. Then lim n →∞ ρ ( T n ) = ρ ( T ) and ρ ( T n ) ≥ ρ ( T ) ∀ n. Proof.
Clearly T n −→ T uniformly. Since T and T n are compact operators and ρ ( T ) and ρ ( T n ) aresimple eigen values of T and T n respectively, we have the conclusion of our lemma.Now, we are ready to prove our Theorem 4.2 Proof.
We know that any converging subsequence { ψ n k } of { ψ n } converges to ψ , the fixed point of Φ . ByTheorem 4.5, for ρ ( T ) ≤ , Φ has only one fixed point which is 0. So, every convergent subsequence of { ψ n } converges to zero, i.e. the sequence { ψ n } converges to zero. Now, we will prove the part (b) of thetheorem.Given ρ ( T ) >
1, by lemma 4.9 we have ρ ( T n ) > ∀ n. Let f n ∈ ( L (0 , a m )) ∗ \ { } be the strictly positive eigen vector of T ∗ n with eigen value ρ ( T n ). Then for all n , we have (cid:104) f n , ψ n (cid:105) = (cid:104) f n , Φ n ψ n (cid:105) = (cid:104) f n , ¯Φ n ψ n + u n (cid:105) where ¯Φ n = Φ n − u n and u n is defined by integral on R.H.S. of (4.6) with c replaced by c n . Observe that exp( −(cid:107) ψ (cid:107) ) T ψ ≤ ¯Φ ψ ≤ T ψ ∀ ψ ∈ L (0 , a m )Therefore, (cid:104) f n , ψ n (cid:105) ≥ (cid:104) f n , exp( −(cid:107) ψ n (cid:107) ) T n ψ n + u n (cid:105) > (cid:104) f n , exp( −(cid:107) ψ n (cid:107) ) T n ψ n (cid:105) = exp( −(cid:107) ψ n (cid:107) ) (cid:104) T ∗ n f n , ψ n (cid:105) = exp( −(cid:107) ψ n (cid:107) ) ρ ( T n ) (cid:104) f n , ψ n (cid:105) Therefore, exp( −(cid:107) ψ n (cid:107) ) ρ ( T n ) < ∀ ni.e. (cid:107) ψ n (cid:107) > log( ρ ( T n )) ≥ log( ρ ( T ))choose γ = log( ρ ( T )) > Discussion
The figures on data related to interaction show that the age plays a crucial role in SARS diseases andespecially in COVID-19 infection as well as in recovery. So, we have studied an age structured SIR modelin which susceptible individuals not only get infected due to direct contact with infected person, but canalso get infected due to contact with contaminated surfaces. We proved that there is no disease freeequilibrium as long as there is transmission due to indirect contacts in the environment. That means forinstance if there are fomites present in the environment contaminated with pathogenic micro-organisms,disease still can spread without direct contact between susceptible and infected individuals. So, removingfomites present on the surfaces is one of the effective measure to slow the infection. Hence sanitization ofsurfaces and proper care to frontline workers will help to fight with such diseases.
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