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Research article

An age- and sex-structured SIR model: Theory and an explicit-implicit numerical solution algorithm

  • Received: 26 May 2020 Accepted: 03 August 2020 Published: 31 August 2020
  • Since age and sex play an important role in transmission of diseases, we propose a SIR (susceptible-infectious-recovered) model for short-term predictions where the population is divided into subgroups based on both factors without taking into account vital dynamics. After stating our model and its underlining assumptions, we analyze its qualitative behavior thoroughly. We prove global existence and uniqueness, non-negativity, boundedness and certain monotonicity properties of the solution. Furthermore, we develop an explicit-implicit numerical solution algorithm and show that all properties of the continuous solution transfer to its time-discrete version. Finally, we provide one numerical example to illustrate our theoretical findings.

    Citation: Benjamin Wacker, Jan Schlüter. An age- and sex-structured SIR model: Theory and an explicit-implicit numerical solution algorithm[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5752-5801. doi: 10.3934/mbe.2020309

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  • Since age and sex play an important role in transmission of diseases, we propose a SIR (susceptible-infectious-recovered) model for short-term predictions where the population is divided into subgroups based on both factors without taking into account vital dynamics. After stating our model and its underlining assumptions, we analyze its qualitative behavior thoroughly. We prove global existence and uniqueness, non-negativity, boundedness and certain monotonicity properties of the solution. Furthermore, we develop an explicit-implicit numerical solution algorithm and show that all properties of the continuous solution transfer to its time-discrete version. Finally, we provide one numerical example to illustrate our theoretical findings.


    Mathematical models of disease spreading date back to the beginning of the twentieth century when Kermack and McKendrick published their famous epidemiological SIR model [1]. Since its invention, many researchers have relied heavily on these basic assumptions and have established more advances models [2,3,4]–only to name a few publications and references therein. Additionally, networks in epidemiology have been recently considered to describe dynamics of disease spreading and spreading patterns [5,6,7,8].

    Special attention has been currently attracted by structured models which take age or spatial structure into account [9,10]. However, transmission rates depend on age structure as well as sex structure in general. For that reason, we develop a simple age- and sex-structured SIR model for short-time prediction because we want to keep modeling as interpretable as possible [12]. Therefore, we structure our population by both sexes and same size age groups.

    Due to current epidemics like COVID-19 [13], we decided to stay with a SIR-typed model because data are suited for this type of models. If we take a closer look at data from Robert-Koch Institute in Germany, the assumption of same size age groups will be acceptable for current data. Theoretically, we have to consider continuous age-structure as presented in [10]. After this short motivational introduction, we can state our contributions in this article.

    Our contributions can be summarized as follows.

    1) We develop a time-continuous age- and sex-structured SIR model for short-term predictions with time-dependent transmission rates between susceptible and infectious people and time-dependent recovery rates.

    2) At first, we show certain properties such as non-negativity and boundedness of solutions.

    3) Additionally, we provide a thorough proof of global existence of solutions in time to our proposed system. We need non-negative and boundedness to conclude global existence and global uniqueness of the solution in time from inductive arguments based on Banach's fixed point theorem. This underlines usefulness of fixed point theorems for arguments regarding existence and uniqueness of solutions in different mathematical areas [11].

    4) Furthermore, we prove monotonicity properties of the global unique solution and investigate analytically that it convergences to a disease-free equilibrium.

    5) Afterwards, we introduce a time-discrete problem formulation which heavily relies on an explicit-implicit formulation of the right-hand-side function. As a consequence, our numerical solution scheme becomes unconditionally stable with respect to chosen time increments. We further show that all properties of the time-continuous formulation transfer to the time-discrete case.

    6) We finally summarize our numerical solution scheme in pseudo-code and one numerical example stresses our theoretical findings.

    Our article is structured as follows. After our motivational introduction of Section 1, we formulate the time-continuous age- and sex-structured SIR model in Section 2. Additionally, we analyze global existence and global uniqueness, non-negativity, boundedness, monotonicity and long-time behavior of the solution of this model. After that, we propose an explicit-implicit numerical solution scheme in Section 3. Here, we show that all properties of our time-continuous model transfer to our time-discrete problem formulation. We present one numerical example to illustrate our theoretical findings in Section 4 and finally, we conclude our article with some remarks on possible future research directions in Section 5.

    The aim of this section is the description and analysis of an age- and sex-structured SIR model. For that purpose, we briefly state our model and its assumptions. At first, we prove global existence based on a modified version of Gr{ö}nwall's Lemma. Afterwards, we provide proofs for non-negativity, boundedness, global uniqueness, monotonicity and long-time behavior of our model's solution.

    To especially state global existence and global uniqueness of the solution of our age- and sex-structured SIR model, we need to introduce some theoretical background material regarding nonlinear ordinary differential equations. Let us first recall Lipschitz continuity of a function on Euclidean spaces.

    Definition 2.1 ([14,Subsection 3.2]). Let d1,d2N. If SRd1, a defined function F:SRd2 is called Lipschitz continuous on S if there exists a non-negative constant L0 such that

    F(x)F(y)Rd2LxyRd1 (2.1)

    holds for all x,yS. Here, denotes a suitable norm on the corresponding Euclidean space.

    Let URd1 be open, let F:URd2. We shall call F locally Lipschitz continuous if for every point x0U there exists a neighborhood V of x0 such that the restriction of F to V is Lipschitz continuous on V.

    We consider an initial-value problem

    {z(t)=G(t,z(t)),z(0)=z0 (2.2)

    where z(t)=(x1(t),,xn(t)) denotes our solution vector. Our vectorial function is represented by G(t,z(t))=(g1(t,z(t)),,gn(t,z(t))) and z0Rn are our given initial conditions. To conclude global existence, we can apply the following theorem that is a direct consequence of Gr{ö}nwall's lemma.

    Theorem 2.2 ([14,Theorem 4.2.1]). If G:[0,)×RnRn is locally Lipschitz continuous and if there exist non-negative real constants B and K such that

    G(t,z(t))RnKz(t)Rn+B (2.3)

    holds for all z(t)Rn, then the solution of the initial value problem (2.2) exists for all time tR and moreover, it holds

    z(t)Rnz0Rnexp(K|t|)+BK(exp(K|t|)1) (2.4)

    for all tR.

    Finally, we need Banach's fixed point theorem to derive global uniqueness.

    Theorem 2.3 ([15,Theorem V.18]). Let (X,ϱ) be a complete metric space with the metric mapping ϱ:X×X[0,). Let T:XX be a strict contraction, i.e. there exists a constant K[0,1) such that ϱ(Tx,Ty)Kϱ(x,y) holds for all x,yX. Then the map T has a unique fixed point.

    At first, we define the supremum norm of a continuous function f:[0,)R. It is given by

    f:=supt[0,)|f(t)|.

    An equivalent definition can be given for continuous functions on intervals [a,b]. Let us now state the model's assumptions [10,16,17]:

    1) The population size N is fixed over time t, i.e. N(t)=N for all t[0,);

    2) We divide the population into three homogeneous subetaoups, namely susceptible people (S), infectious people (I) and recovered people (R). We can clearly assign every individual to exactly one subetaoup. Hence, we obtain

    N=S(t)+I(t)+R(t) (2.5)

    for all t[0,);

    3) We further distinguish our subetaoups. Let NaN be the number of age groups and let f and m be the subscripts for female and male persons respectively. Let k{1,,Na} be arbitrary. We denote the k-th female susceptible subetaoup by Sf,k and the k-th male susceptible subetaoup by Sm,k. Consequently, it is clear how we denote the infectious and recovered subetaoups;

    4) Additionally, no births and deaths occur;

    5) The time-varying transmission rates βSm,j,Is,k:[0,)(0,) are Lipschitz continuous and continuously differentiable for fixed j{1,,Na}, arbitrary k{1,,Na} and arbitrary s{f,m}. In addition to that, there exists a positive constant Mβ>0 such that βSm,j,Is,kMβ for all t0, arbitrary s{f,m} and arbitrary j,k{1,,Na};

    6) The time-varying recovery rates γIs,k:[0,)(0,) are Lipschitz continuous and continuously differentiable for arbitrary s{f,m} and arbitrary k{1,,Na}. Additionally, there are positive constants Mγ>0 and mγ>0 such that γIs,kMγ and γIs,k(t)mγ for all t0, arbitrary s{f,m} and arbitrary k{1,,Na}.

    For abbreviation, we write g(t):=dg(t)dt for the first derivative of a differentiable function g at time t. Our equations of the time-continuous age- and sex-structured SIR model read

    {Sf,j(t)=Nak=1{βSf,j,If,k(t)Sf,j(t)If,k(t)N+βSf,j,Im,k(t)Sf,j(t)Im,k(t)N},Sm,j(t)=Nak=1{βSm,j,If,k(t)Sm,j(t)If,k(t)N+βSm,j,Im,k(t)Sm,j(t)Im,k(t)N},If,j(t)=Nak=1{βSf,j,If,k(t)Sf,j(t)If,k(t)N+βSf,j,Im,k(t)Sf,j(t)Im,k(t)N}γIf,j(t)If,j(t),Im,j(t)=Nak=1{βSm,j,If,k(t)Sm,j(t)If,k(t)N+βSm,j,Im,k(t)Sm,j(t)Im,k(t)N}γIm,j(t)Im,j(t),Rf,j(t)=γIf,j(t)If,j(t),Rm,j(t)=γIm,j(t)Im,j(t) (2.6)

    with susceptible initial conditions Ss,j(0)=S1,s,j>0, infectious initial conditions Is,j(0)=I1,s,j0 and recovered initial conditions Rs,j(0)=R1,s,j0 for arbitrary s{f,m} and arbitrary j{1,,Na}. At least one initial condition of the infectious subetaoups should be positive. Obviously, it holds

    N(t)=Naj=1{Sf,j(t)+Sm,j(t)+If,j(t)+Im,j(t)+Rf,j(t)+Rm,j(t)}=0

    such that population size is preserved for all t0.

    We examine non-negativity and boundedness of (2.6).

    Lemma 2.4. We obtain

    {0Ss,j(t)N,0Is,j(t)N,0Rs,j(t)N (2.7)

    for arbitrary s{f,m}, for all j{1,,Na} and for all t0 with respect to (2.6).

    Proof. We divide our proof into four parts. Let s{f,m} and j{1,,Na} be arbitrary in the following.

    1) We consider

    Ss,j(t)=Nak=1{βSs,j,If,k(t)Ss,j(t)If,k(t)N+βSs,j,Im,k(t)Ss,j(t)Im,k(t)N}=Ss,j(t)Nak=1{βSs,j,If,k(t)If,k(t)N+βSs,j,Im,k(t)Im,k(t)N}

    since Ss,j(t) is contained in both summands and does not depend on the summation index k. Hence, we can put this term outside our considered sum. Division by Ss,j(t) now yields

    Ss,j(t)Ss,j(t)=Nak=1{βSs,j,If,k(t)If,k(t)N+βSs,j,Im,k(t)Im,k(t)N}

    and since we are able to write Ss,j(t)=dSs,j(t)dt, we can rewrite this equation by

    dSs,j(t)Ss,j(t)=Nak=1{βSs,j,If,k(t)If,k(t)N+βSs,j,Im,k(t)Im,k(t)N}dt

    through separation of variables. By integration on the respective time interval [0,t], we observe that

    ln(Ss,j(t)S1,s,j)=t0Nak=1{βSs,j,If,k(τ)If,k(τ)N+βSs,j,Im,k(τ)Im,k(τ)N}dτ

    holds. We finally obtain

    Ss,j(t)=S1,s,jexp(t0Nak=1{βSs,j,If,k(τ)If,k(τ)N+βSs,j,Im,k(τ)Im,k(τ)N}dτ).

    Hence, it holds Ss,j(t)>0 for all t0 by our approach of separation of variables. This procedure is feasible because our initial conditions for susceptible people are positive.

    2) We examine

    Is,j(t)=Nak=1{βSs,j,If,k(t)Ss,j(t)If,k(t)N+βSs,j,Im,k(t)Ss,j(t)Im,k(t)N}γIs,j(t)Is,j(t),

    under the initial condition Is,j(0)=I1,s,j0 for arbitrary s{f,m} and arbitrary j{1,,Na}. Let us additionally assume that Is,k(0)=I1,s,k0 for arbitrary s{f,m} and arbitrary k{1,,Na} with kj. At least one initial condition I1,˜s,˜j should be positive. This implies

    Is,j(0)=Nak=1{βSs,j,If,k(0)Ss,j(T)If,k(0)N+βSs,j,Im,k(0)Ss,j(0)Im,k(0)N}γIs,j(0)Is,j(0)=0=Nak=1{βSs,j,If,k(0)Ss,j(0)If,k(0)N+βSs,j,Im,k(0)Ss,j(0)Im,k(0)N}βSs,j,I˜s,˜j(0)Ss,j(0)I˜s,˜j(0)N>0

    for all derivatives of initial conditions for infectious subetaoups where the initial conditions are zero at time t=0 since all Ss,j(0)>0 by assumption and all Is,k(0)0 with at least one positive function I˜s,˜j(0)>0 by assumption. Hence, there exists a time T1>0 such that Is,j(T1)>0 for all s{f,m} and all j{1,,Na}. Additionally, it holds Is,j(t)0 for all t[0,T1] for all s{f,m} and all j{1,,Na}.

    Now, we interpret T1>0 as our new starting point for our argument. We have to distinguish two cases.

    Case 1: Let T2>T1 and let Is1,j1(T2)=0 be one function of an infectious subetaoup which is non-negative for all t[0,T2]. This is feasible due to continuity of these functions. Let there be at least one function of infectious subetaoups which is positive at t=T2. As proven in the previous inequality, this implies Is1,j1(T2)>0. However, this yields the existence of a positive constant δ>0 such that Is1,j1(t)<0 for all t(T2δ,T2) by continuity. This contradicts our assumption. Hence, all functions of infectious subetaoups stay non-negative - even positive - in this case. By induction, this even holds on future time subintervals.

    Case 2: Let T2>T1 and let Is,j(T2)=0 for all s{f,m} and all j{1,,Na}. This implies the status of disease-free equilibrium for all future time points.

    Hence, (2.6) preserves non-negativity with respect to all infectious subetaoups.

    3) By our second property and integration of

    Rs,j(t)=γIs,j(t)Is,j(t)

    on the time interval [0,t], we obtain

    Rs,j(t)=R1,s,j+t0γIs,j(τ)Is,j(τ)dτ.

    It yields full non-negativity preservation of our non-linear ordinary differential equation system (2.6).

    4) Our upper bound is a direct consequence of

    N(t)=Naj=1{Sf,j(t)+Sm,j(t)+If,j(t)+Im,j(t)+Rf,j(t)+Rm,j(t)}=0

    for all t0 and our proof is complete.

    We now prove a global existence theorem of (2.6) based on Theorem 2.2.

    Theorem 2.5. The non-linear first order ordinary differential equation system (2.6) has at least one global solution, i.e. these possible solutions exist for all t0.

    Proof. We define the six vectors

    Sf(t)=(Sf,1(t),,Sf,Na(t))TRNa,Sm(t)=(Sm,1(t),,Sm,Na(t))TRNa,If(t)=(If,1(t),,If,Na(t))TRNa,Im(t)=(Im,1(t),,Im,Na(t))TRNa,Rf(t)=(Rf,1(t),,Rf,Na(t))TRNa,Rm(t)=(Rm,1(t),,Rm,Na(t))TRNa

    which build our solution vector

    z(t)=(Sf(t)Sm(t)If(t)Im(t)Rf(t)Rm(t))R6Na.

    Now, we define G:[0,)×R6NaRNa by (2.6) in a straightforward manner. By applying maximum norms, triangle inequalities, non-negativity and boundedness by Lemma 2.4, we obtain

    Sf,j(t)2Namax{Mβ,Mγ}z(t),Sm,j(t)2Namax{Mβ,Mγ}z(t),If,j(t)(2Na+1)max{Mβ,Mγ}z(t),Im,j(t)(2Na+1)max{Mβ,Mγ}z(t),Rf,j(t)max{Mβ,Mγ}z(t),Rm,j(t)max{Mβ,Mγ}z(t)

    for all j{1,,Na} and this yields

    G(t,z(t))(2Na+1)max{Mβ,Mγ}z(t).

    Hence, Theorem 2.2 implies global existence of the system's possible solutions in time.

    Now, we are able to prove global uniqueness of our time-continuous problem formulation (2.6).

    Theorem 2.6. The non-linear first order ordinary differential equation system (2.6) has exactly one global unique solution in time.

    Proof. 1) At first, we need one inequality for our proof. Let x1,x2,y1,y2R be arbitrary. By the triangle inequality, we obtain

    |x1y1x2y2|=|x1y1x2y1+x2y1x2y2||x1y1x2y1|+|x2y1x2y2|=|y1||x1x2|+|x2||y1y2|.

    2) Let

    z(t)=(Sf(t)Sm(t)If(t)Im(t)Rf(t)Rm(t))R6Naand˜z(t)=(~Sf(t)~Sm(t)~If(t)~Im(t)~Rf(t)~Rm(t))R6Na

    be two solutions of our initial value problem (2.6) with same time-varying coefficients and same initial value conditions. Let us consider

    ~Ss,j(τ)Ss,j(τ)=~Ss,j(0)Ss,j(0)=0τ0Nak=1{βSs,j,If,k(t)N(~Ss,j(t)~If,k(t)Ss,j(t)If,k(t))}dt+τ0Nak=1{βSs,j,Im,k(t)N(~Ss,j(t)~Im,k(t)Ss,j(t)Im,k(t))}dt

    for arbitrary s{f,m} and arbitrary j{1,,Na}. Application of the triangle inequality and assumptions on our time-varying coefficients yields

    |~Ss,j(τ)Ss,j(τ)|MβNτ0Nak=1|~Ss,j(t)~If,k(t)Ss,j(t)~If,k(t)+Ss,j(t)~If,k(t)Ss,j(t)If,k(t)|dt+MβNτ0Nak=1|~Ss,j(t)~Im,k(t)Ss,j(t)~Im,k(t)+Ss,j(t)~Im,k(t)Ss,j(t)Im,k(t)|dt.

    Since all functions are bounded above by the population size N, we obtain

    |~Ss,j(τ)Ss,j(τ)|Mβτ0Nak=1{2|~Ss,j(t)Ss,j(t)|+|~If,k(t)If,k(t)|+|~Im,k(t)Im,k(t)|}dt4Mβτ0Nak=1˜z(t)z(t)dt4MβNaτ˜z(t)z(t)

    by application of our inequality from the first step of this proof.

    3) Let us now consider

    ~Is,j(t)Is,j(t)={~Ss,j(t)γIs,j(t)~Is,j(t)}{Ss,j(t)γIs,j(t)Is,j(t)}=(Ss,j(t)~Ss,j(t))+γIs,j(t)(Is,j(t)~Is,j(t().

    By integration on the time interval [0,τ], we obtain

    ~Is,j(τ)Is,j(τ)=Ss,j(τ)~Ss,j(τ)+τ0γIs,j(t)(Is,j(t)~Is,j(t))dt.

    Application of the triangle inequality and the second part of this proof yields

    |~Is,j(τ)Is,j(τ)||Ss,j(τ)~Ss,j(τ)|+|τ0γIs,j(t)(Is,j(t)~Is,j(t))dt|4MβNaτ˜z(t)z(t)+Mγτ˜z(t)z(t)(4Na+1)max{Mβ,Mγ}τ˜z(t)z(t).

    4) Furthermore, it holds

    ~Rs,j(τ)Rs,j(τ)=τ0γIs,j(t)(~Is,j(t)Is,j(t))dt.

    We obtain

    |~Rs,j(τ)Rs,j(τ)|Mγτ˜z(t)z(t).

    5) Combining the previous steps, we conclude

    ˜z(t)z(t)4(Na+1)max{Mβ,Mγ}τ˜z(t)z(t)

    on the time interval [0,τ]. Choose τ:=18(Na+1)max{Mβ,Mγ}. This implies

    ˜z(t)z(t)4(Na+1)max{Mβ,Mγ}8(Na+1)max{Mβ,Mγ}˜z(t)z(t)=12˜z(t)z(t)

    and hence, the solution is unique on the time interval [0,τ] by Banach's fixed point theorem. Inductively, all previous steps hold on following time intervals [kτ,(k+1)] with arbitrary kN and initial conditions at time point t=kτ. Therefore, we conclude that the solution is unique for all t0 which proves our assertion.

    We conclude our analysis of our time-continuous problem formulation (2.6) by an investigation of monotonicity and long-time behavior.

    Theorem 2.7. We obtain the following properties for arbitrary s{f,m} and for all j{1,,Na}:

    1) Ss,j is monotonically decreasing and there exists a number Ss,j0 such that limtSs,j(t)=Ss,j holds. Additionally, we obtain Ss,j>0;

    2) Rs,j is monotonically increasing and there exists a number Rs,j0 such that limtRs,j(t)=Rs,j;

    3) Is,j is Lebesgue-integrable on [0,) and we get limtIs,j(t)=0;

    4) Our system (2.6) always converges to a disease-free equilibrium

    for all solution functions of (2.6).

    Proof. We divide our proof in four parts. Let s{f,m} and j{1,,Na} be arbitrary.

    1) Since 0Ss,j(t)N and 0Is,j(t)N hold for all t0 by Lemma 2.4, we obtain Ss,j(t)0 for all t0. By separation of variables, we know that

    Ss,j(t)=S1,s,jexp(t0Nak=1{βSs,j,If,k(τ)If,k(τ)N+βSs,j,Im,k(τ)Im,k(τ)N}dτ)

    is valid and this implies

    Ss,j(t)S1,s,jexp(2MβNat)>0.

    Since Ss,j is monotonically decreasing, bounded below by zero and

    Ss,j(t)S1,s,jexp(2MβNat)>0,

    there exists a positive real number Ss,j such that we obtain the limit limtSs,j(t)=Ss,j.

    2) By considering Rs,j(t)=γIs,j(t)Is,j(t)0 from Lemma 2.4, we conclude that Rs,j is monotonically increasing. Since Rs,j is further bounded above by N according to Lemma 2.4, there exists a positive real number Rs,j such that limtRs,j(t)=Rs,j.

    3) We have Rs,j(t)=γIs,j(t)Is,j(t) according to our non-linear differential equation system (2.6). Integration on [0,) yields

    Rs,jR1,s,j=0γIs,j(τ)Is,j(τ)dτmγ0Is,j(τ)dτ.

    This yields

    0|Is,j(τ)|dτ=0Is,j(τ)dτRs,jR1,s,jmγNmγ

    and hence, Is,j is Lebesgue-integrable on [0,). This shows limtIs,j(t)=0.

    4) Remember the notation introduced at the beginning of the proof of Theorem 2.5. By our three aforementioned properties, we obtain the limiting vector

    z=limtz(t)=limt(Sf(t)Sm(t)If(t)Im(t)Rf(t)Rm(t))=(limtSf(t)limtSm(t)limtIf(t)limtIm(t)limtRf(t)limtRm(t))=(SfSm0RNa0RNaIfIm)R6Na

    and this vector represents the disease-free equilibrium. Hence, our non-linear differential equation system converges to the disease-free equilibrium. This finishes our proof.

    Here, we develop an explicit-implicit time-discrete variant of our time-continuous age- and sex-structured SIR model. We organize this section similar to the previous one. Our constructive goal in this section is to present a numerical solution scheme that captures as many properties of its continuous analogue as possible.

    Let us assume that our time interval [0,T] can be divided by a strictly increasing sequence {tp}Mp=1 for MN with t1=0 and tM=T. To distinguish continuous and time-discrete solutions, all time-discrete functions are denoted by Snums,j(tp) for example. We additionally assume that time-continuous and time-discrete time-varying transmission rates and recovery rates coincide for all times.

    Here, we state our explicit-implicit time-discrete problem formulation

    {Snumf,j(tp+1)Snumf,j(tp)tp+1tp=Nak=1{βSnumf,j,Inumf,k(tp+1)Snumf,j(tp+1)Inumf,k(tp)N+βSnumf,j,Inumm,k(tp+1)Snumf,j(tp+1)Inumm,k(tp)N},Snumm,j(tp+1)Snumm,j(tp)tp+1tp=Nak=1{βSnumm,j,Inumf,k(tp+1)Snumm,j(tp+1)Inumf,k(tp)N+βSnumm,j,Inumm,k(tp+1)Snumm,j(tp+1)Inumm,k(tp)N},Inumf,j(tp+1)Inumf,j(tp)tp+1tp=Nak=1{βSnumf,j,Inumf,k(tp+1)Snumf,j(tp+1)Inumf,k(tp)N+βSnumf,j,Inumm,k(tp+1)Snumf,j(tp+1)Inumm,k(tp)N}γInumf,j(tp+1)Inumf,j(tp+1),Inumm,j(tp+1)Inumm,j(tp)tp+1tp=Nak=1{βSnumm,j,Inumf,k(tp+1)Snumm,j(tp+1)Inumf,k(tp)N+βSnumm,j,Inumm,k(tp+1)Snumm,j(tp+1)Inumm,k(tp)N}γInumm,j(tp+1)Inumm,j(tp+1),Rnumf,j(tp+1)Rnumf,j(tp)tp+1tp=γInumf,j(tp+1)Inumf,j(tp+1),Rnumm,j(tp+1)Rnumm,j(tp)tp+1tp=γInumm,j(tp+1)Inumm,j(tp+1) (3.1)

    of the time-continuous SIR model (2.6) for all p{1,,M1} and for all subscripts of age groups j{1,,Na}. Our initial conditions read

    Snums,j(t1)>0andInums,j(t1)0andRnums,j(t1)0

    for arbitrary s{f,m} and all j{1,,Na} with at least one initial condition of infectious subetaoups to be positive. For abbreviation, we write in short Δp+1=(tp+1tp) for all p{1,,M1} in the following. This explicit-implicit time-discrete problem formulation obviously fulfills

    N=Naj=1{Snumf,j(tp+1)+Snumm,j(tp+1)+Inumf,j(tp+1)+Inumm,j(tp+1)+Rnumf,j(tp+1)+Rnumm,j(tp+1)}=Naj=1{Snumf,j(tp)+Snumm,j(tp)+Inumf,j(tp)+Inumm,j(tp)+Rnumf,j(tp)+Rnumm,j(tp)} (3.2)

    for all p{1,,M1}.

    Let us proceed with unique solvability of our numerical scheme (3.1).

    1) We observe from

    Snums,j(tp+1)Snums,j(tp)Δp+1=Nak=1{βSnums,j,Inumf,k(tp+1)Snums,j(tp+1)Inumf,k(tp)N+βSnums,j,Inumm,k(tp+1)Snums,j(tp+1)Inumm,k(tp)N}

    that

    Snums,j(tp+1)=Snums,j(tp)1+Δp+1NSsum,nums,j(tp+1) (3.3)

    holds for arbitrary s{f,m}, for all j{1,,Na} and for all p{1,,M1}. Here, the sum in the denominator is given by

    Ssum, nums,j(tp+1)=Nak=1{βSnums,j,Inumf,k(tp+1)Inumf,k(tp)+βSnums,j,Inumm,k(tp+1)Inumm,k(tp)}.

    2) We see from

    Inums,j(tp+1)Inums,j(tp)Δp+1=Nak=1{βSnums,j,Inumf,k(tp+1)Snums,j(tp+1)Inumf,k(tp)N+βSnums,j,Inumm,k(tp+1)Snums,j(tp+1)Inumm,k(tp)N}γInums,j(tp+1)Inums,j(tp+1)

    that

    Inums,j(tp+1)=Inums,j(tp)1+Δp+1γInums,j(tp+1)+Δp+1Nak=1{βSnums,j,Inumf,k(tp+1)Snums,j(tp+1)Inumf,k(tp)N+βSnums,j,Inumm,k(tp+1)Snums,j(tp+1)Inumm,k(tp)N}1+Δp+1γInums,j(tp+1) (3.4)

    holds for arbitrary s{f,m}, for all j{1,,Na} and for all p{1,,M1}.

    3) We conclude from

    Rnums,j(tp+1)Rnums,j(tp)Δp+1=γInums,j(tp+1)Inums,j(tp+1)

    that

    Rnums,j(tp+1)=Rnums,j(tp)+γInums,j(tp+1)Δp+1Inums,j(tp+1) (3.5)

    holds for arbitrary s{f,m}, for all j{1,,Na} and for all p{1,,M1}.

    4) Hence, all our computations demonstrate that our numerical solution scheme (3.1) is uniquely solvable. We even infer that, in contrast to typical explicit Euler-time stepping schemes, it is unconditionally stable and we avoid non-linearities as in implicit Euler-time stepping schemes. We summarize our computations and our observations in the following theorem.

    Theorem 3.1. Our numerical solution scheme (3.1) is uniquely solvable for all time steps. Additionally, it is also unconditionally stable.

    Proof. Follow the above computations in Subsection 3.2.

    Let us first remark that our initial conditions are non-negative. By induction, it follows that

    Snums,j(tp)0,Inums,j(tp)0andRnums,j(tp)0

    hold from (3.3) - (3.5) for all s{f,m}, all j{1,,Na} and all p{1,,M}. Boundedness is a consequence of (3.2). Thus, we can state the following lemma.

    Lemma 3.2. We obtain

    0Snums,j(tp)N,0Inums,j(tp)Nand0Rnums,j(tp)N

    for arbitrary s{f,m}, for all j{1,,Na} and for all p{1,,M}.

    We continue this section with our theorem on monotonicity and long-time behavior of the solution of our explicit-implicit numerical scheme (3.1).

    Theorem 3.3. We have the following properties:

    1) The sequence {Snums,j(tp)}Mp=1 is monotonically decreasing and there exists a non-negative real number S,num such that limpSnums,j(tp)=S,num;

    2) The sequence {Rnums,j(tp)}Mp=1 is monotonically increasing and there exists a non-negative real number R,num such that limpRnums,j(tp)=R,num;

    3) The sequence {Inums,j(tp)}Mp=1 fulfills limpInums,j(tp)=I,num=0

    for arbitrary s{f,m} and for all j{1,,Na}.

    Proof. 1) By Lemma 3.2, we know that the sequence {Snums,j(tp)}Mp=1 is bounded. Again by Lemma 3.2 and (3.3) - (3.5), we get

    Snums,j(tp+1)=Snums,j(tp)1+Δp+1NSsum,nums,j(tp+1)Snums,j(tp)

    for arbitrary s{f,m}, for all j{1,,Na} and for all p{1,,M1}. Hence, the sequence {Snums,j(tp)}Mp=1 is monotonically decreasing and it thus converges. This implies the existence of a non-negative real number S,num such that limpSnums,j(tp)=S,num holds.

    2) By Lemma 3.2, we know that the sequence {Rnums,j(tp)}Mp=1 is bounded. Again by Lemma 3.2 and (3.3) - (3.5), we conclude

    Rnums,j(tp+1)=Rnums,j(tp)+γInums,j(tp+1)Δp+1Inums,j(tp+1)Rnums,j(tp)

    for arbitrary s{f,m}, for all j{1,,Na} and for all p{1,,M1}. Hence, the sequence {Rnums,j(tp)}Mp=1 is monotonically increasing and it thus converges. This yields the existence of a non-negative real number R,num such that limpRnums,j(tp)=R,num holds.

    3) Let us assume the contrary. This implies the existence of a positive real number I,num such that limpInums,j(tp)=I,num holds. By (3.4), we then know that all values of the sequence are positive from a certain sequence index. Hence, there exists a positive real number ˜Inum, min such that Inums,j(tp)˜Inum, min. Considering

    Rnums,j(tp+1)Rnums,j(tp)=γInums,j(tp+1)Δp+1Inums,j(tp+1)

    from (3.5), we obtain

    Rnums,j(tp+1)Rnums,j(tp)γInums,j(tp+1)Δp+1˜Inum, minmγΔp+1˜Inum, min

    and summation by parts yields

    R,numRnums,j(tL)limpmγtp+1˜Inum, minmγtL˜Inum, minp

    from the mentioned time index L as our summation beginning. However, this contradicts our second property. Hence, limpInums,j(tp)=I,num=0 holds.

    Here, we want to discuss convergence of our proposed numerical scheme (3.1).

    Theorem 3.4. In addition to the assumptions of Subsection 2.2, all solution functions Ss,j,Is,j,Rs,j:[0,)[0,N] are assumed to be continuously differentiable twice with globally bounded first and second derivatives. Additionally, all first derivatives of time-varying transmission rates and time-varying recovery rates are assumed to be globally bounded as well. Let Δp1 for all pN. If maxpNΔp0 holds, the discrete solution of the numerical scheme (3.1) converges linearly towards the global unique continuous solution on a considered time interval [0,T].

    Proof. Since this proof become relatively technical, we briefly describe our strategy. At first, local errors between continuous and time-discrete solutions are considered. Afterwards, we need to take into account that errors propagate in time. Finally, we investigate cumulation of these errors which finalizes our proof. We adapt ideas from [18] and [19]. In general, we follow [19,Satz 74.1] and modify ideas for explicit Eulerian time-stepping schemes because our scheme is a mixture of explicit-implicit parts.

    1) For investigation of local errors, we assume that

    (tp,Snums,j(tp))=(tp,Ss,j(tp)),(tp,Inums,j(tp))=(tp,Is,j(tp))and(tp,Rnums,j(tp))=(tp,Rs,j(tp))

    hold for arbitrary s{f,m} and arbitrary j{1,,Na} and we consider the time interval [tp,tp+1]. Here, we thus only consider one time step and denote solutions by ~Snums,j(tp+1), ~Inums,j(tp+1) and ~Rnums,j(tp+1) respectively.

    1.1) It first holds

    ~Snums,j(tp+1)=Ss,j(tp)Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}

    and solving this equation for ~Snums,j(tp+1) yields

    ~Snums,j(tp+1)=Ss,j(tp)1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}=Ss,j(tp)Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}.

    We consider |Ss,j(tp+1)~Snums,j(tp+1)|. It holds

    |Ss,j(tp+1)~Snums,j(tp+1)|=|Ss,j(tp+1){Ss,j(tp)Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}}|.

    Zero addition and application of the triangle inequality implies

    |Ss,j(tp+1)~Snums,j(tp+1)|=|Ss,j(tp+1)Ss,j(tp)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}+Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}||Ss,j(tp+1)Ss,j(tp)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}|+|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}+Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}|.

    We define the two terms

    Ia=|Ss,j(tp+1)Ss,j(tp)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}|

    and

    Ib=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}+Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}|.

    For Ia, we obtain

    Ia=|Ss,j(tp+1)Ss,j(tp)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}|=|tp+1tpSs,j(τ)dτΔp+1Ss,j(tp)|=|tp+1tpSs,j(τ)dτtp+1tpSs,j(tp)dτ|=|tp+1tp{Ss,j(τ)Ss,j(tp)}dτ|.

    Application of the mean value theorem of calculus yields the existence of ξa(tp,tp+1) such that

    Ss,j(ξa)=Ss,j(τ)Ss,j(tp)τtp

    holds. This implies

    Ia=|tp+1tp{Ss,j(τ)Ss,j(tp)}dτ|=|tp+1tp(τtp)Ss,j(τ)Ss,j(tp)τtpdτ|=|tp+1tp(τtp)Ss,j(ξa)dτ|maxt[tp,tp+1]|Ss,j(t)||tp+1tp(τtp)dτ|Δ2p+12Ss,j.

    For Ib, we obtain

    Ib=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}+Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}|=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}Δ2p+1{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}}{Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}+Δp+1Ss,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}|.

    Application of the triangle inequality and rearranging yields

    Ib|Δp+1{Nak=1q{f,m}{(βSs,j,Iq,k(tp+1)βSs,j,Iq,k(tp))Ss,j(tp)Itp(tp)N}}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}|+|Δ2p+1{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}}{Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}}1+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}|.

    Since

    11+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}

    is valid, we obtain

    Ib|Δp+1{Nak=1q{f,m}{(βSs,j,Iq,k(tp+1)βSs,j,Iq,k(tp))Ss,j(tp)Itp(tp)N}}|+Δ2p+1|{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}}{Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Iq,k(tp)N}}|.

    By the mean value theorem of calculus, there exists ξb(tp,tp+1) such that

    βSs,j,Iq,k(ξb)=βSs,j,Iq,k(tp+1)βSs,j,Iq,k(tp)tp+1tp

    holds. This implies

    Ib|Δ2p+1{Nak=1q{f,m}{(βSs,j,Iq,k(tp+1)βSs,j,Iq,k(tp))tp+1tpSs,j(tp)Itp(tp)N}}|+Δ2p+1{Nak=1q{f,m}MβN}{Nak=1q{f,m}Mβ}|Δ2p+1{Nak=1q{f,m}{βSs,j,Iq,k(ξb)Ss,j(tp)Itp(tp)N}}|+Δ2p+1{2MβNaN}{2MβNa}Δ2p+12NaNβ+Δ2p+14M2βN2aN=Δ2p+1{2NaNβ+4M2βN2aN}.

    Here, β denotes the vector of all derivatives of time-varying transmission rates. We conclude

    |Ss,j(tp+1)~Snums,j(tp+1)|Ia+IbΔ2p+12Ss,j+Δ2p+1{2NaNβ+4M2βN2aN}Δ2p+1{Ss,j+2NaNβ+4M2βN2aN}:=Cs,loc

    and summarizing our results, this implies

    |Ss,j(tp+1)~Snums,j(tp+1)|Cs,locΔ2p+1. (3.6)

    1.2) From

    ~Inums,j(tp+1)=Is,j(tp)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}Δp+1γIs,j(tp+1)~Inums,j(tp+1),

    we obtain

    ~Inums,j(tp+1)=Is,j(tp)1+Δp+1γIs,j(tp+1)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)=Is,j(tp)Δp+1γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1).

    We consider |Is,j(tp+1)~Inums,j(tp+1)|. It holds

    |Is,j(tp+1)~Inums,j(tp+1)|=|Is,j(tp+1)Is,j(tp)+Δp+1γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|=|Is,j(tp+1)Is,j(tp)Δp+1{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}γIs,j(tp)Is,j(tp)}+Δp+1{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}γIs,j(tp)Is,j(tp)}+Δp+1γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    Rearranging of these terms and application of the triangle inequality yields

    |Is,j(tp+1)~Inums,j(tp+1)|=|{Is,j(tp+1)Is,j(tp)Δp+1{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}γIs,j(tp)Is,j(tp)}}+{Δp+1{γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)}}+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)||Is,j(tp+1)Is,j(tp)Δp+1{Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}γIs,j(tp)Is,j(tp)}|+|Δp+1{γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)}|+|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|=|tp+1tpIs,j(τ)dτΔp+1Is,j(tp)|+|Δp+1{γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)}|+|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    We define the following three terms

    Ic:=|tp+1tpIs,j(τ)dτΔp+1Is,j(tp)|,Id:=|Δp+1{γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)}|

    and

    Ie:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    Ic can be rewritten as

    Ic=|tp+1tpIs,j(τ)dτtp+1tpIs,j(tp)dτ|=|tp+1tp(Is,j(τ)Is,j(tp))dτ|.

    By the mean value theorem of calculus, there exists ξc(tp,tp+1) such that

    Is,j(ξc)=Is,j(τ)Is,j(tp)τtp

    holds. This implies

    Ic=|tp+1tp(Is,j(τ)Is,j(tp))dτ|=|tp+1tp(τtp)Is,j(τ)Is,j(tp)τtpdτ|=|tp+1tp(τtp)Is,j(ξc)dτ|Δ2p+12Is,j.

    For Id, we obtain

    Id:=|Δp+1{γIs,j(tp+1)Is,j(tp)1+Δp+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)}|=|Δp+1Is,j(tp){γIs,j(tp+1)γIs,j(tp)}1+Δp+1γIs,j(tp+1)Δ2p+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)|.

    Application of the triangle inequality implies

    Id|Δp+1Is,j(tp){γIs,j(tp+1)γIs,j(tp)}1+Δp+1γIs,j(tp+1)|+|Δ2p+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)||Δp+1Is,j(tp){γIs,j(tp+1)γIs,j(tp)}|+|Δ2p+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)|.

    By the mean value theorem of calculus, there is ξd(tp,tp+1) such that

    γIs,j(ξd)=γIs,j(tp+1)γIs,j(tp)tp+1tp

    holds. Hence, we conclude

    Id|Δ2p+1Is,j(tp)γIs,j(tp+1)γIs,j(tp)tp+1tp|+|Δ2p+1γIs,j(tp+1)γIs,j(tp)Is,j(tp)||Δ2p+1Is,j(tp)γIs,j(ξd)|+Δ2p+1M2γNΔ2p+1Nγ+Δ2p+1M2γN.

    Here, γ denotes the vector containing all derivatives of time-varying recovery rates. We consider

    Ie:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    By zero addition, we obtain

    Ie:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp+1)Iq,k(tp)N}+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(tp+1)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)(Ss,j(tp)Ss,j(tp+1))Iq,k(tp)N}+Δp+1Nak=1q{f,m}{(βSs,j,Iq,k(tp)βSs,j,Iq,k(tp+1))Ss,j(tp+1)Iq,k(tp)N}+Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    Application of the triangle inequality yields

    Ie|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)(Ss,j(tp)Ss,j(tp+1))Iq,k(tp)N}|+|Δp+1Nak=1q{f,m}{(βSs,j,Iq,k(tp)βSs,j,Iq,k(tp+1))Ss,j(tp+1)Iq,k(tp)N}|+|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    We define the following three terms

    Ie,1:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)(Ss,j(tp)Ss,j(tp+1))Iq,k(tp)N}|,Ie,2:=|Δp+1Nak=1q{f,m}{(βSs,j,Iq,k(tp)βSs,j,Iq,k(tp+1))Ss,j(tp+1)Iq,k(tp)N}|

    and

    Ie,3:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    Considering Ie,1, there exists ξe,1(tp,tp+1) such that

    Ss,j(ξe,1)=Ss,j(tp+1)Ss,j(tp)tp+1tp

    holds due to the mean value theorem of calculus. Hence, we obtain

    Ie,1:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp)(Ss,j(tp)Ss,j(tp+1))Iq,k(tp)N}|=|Δ2p+1Nak=1q{f,m}{βSs,j,Iq,k(tp)(Ss,j(tp)Ss,j(tp+1))tptp+1Iq,k(tp)N}|=|Δ2p+1Nak=1q{f,m}{βSs,j,Iq,k(tp)Ss,j(ξe,1)Iq,k(tp)N}|Δ2p+12NaβSs,j.

    By the mean value theorem of calculus, there exists ξe,2(tp,tp+1) such that

    βSs,j,Iq,k(ξe,2)=βSs,j,Iq,k(tp+1)βSs,j,Iq,k(tp)tp+1tp

    is valid. Application of the triangle inequality yields

    Ie,2:=|Δp+1Nak=1q{f,m}{(βSs,j,Iq,k(tp)βSs,j,Iq,k(tp+1))Ss,j(tp+1)Iq,k(tp)N}|=|Δ2p+1Nak=1q{f,m}{(βSs,j,Iq,k(tp+1)βSs,j,Iq,k(tp))tp+1tpSs,j(tp+1)Iq,k(tp)N}|=|Δ2p+1Nak=1q{f,m}{βSs,j,Iq,k(ξe,2)Ss,j(tp+1)Iq,k(tp)N}|Δ2p+12NaNβ.

    Now, we consider

    Ie,3:=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|.

    By application of the triangle inequality, we obtain

    Ie,3=|Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)+Δ2p+1γIs,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)Δp+1Nak=1q{f,m}{βSs,j,Iq,k(tp+1)~Snums,j(tp+1)Iq,k(tp)N}1+Δp+1γIs,j(tp+1)|Δ2p+1|γIs,j(tp)Nak=1q{f,m}{βSs,j,Iq,k(tp+1)Ss,j(tp+1)Iq,k(tp)N}|+Δp+1|Nak=1q{f,m}{βSs,j,Iq,k(tp+1)(Ss,j(tp+1)~Snums,j(tp+1))Iq,k(tp)N}|Δ2p+12NaNMβMγ+Δp+12NaMβ|Ss,j(tp+1)~Snums,j(tp+1)|.

    By inequality (3.6) from Step 1.1), we know that

    |Ss,j(tp+1)~Snums,j(tp+1)|Cs,locΔ2p+1

    holds. This implies

    Ie,3Δ2p+12NaNMβMγ+Δp+12NaMβ|Ss,j(tp+1)~Snums,j(tp+1)|Δ2p+12NaNMβMγ+Δp+12NaMβCs,locΔ2p+1=Δ2p+12NaNMβMγ+Δ3p+12NaMβCs,loc=Δ2p+1{2NaNMβMγ+2NaMβCs,locΔp+1}Δ2p+1{2NaNMβMγ+2NaMβCs,loc}.

    Combining our results, we obtain

    |Is,j(tp+1)~Inums,j(tp+1)|Ic+Id+IeIc+Id+Ie,1+Ie,2+Ie,3Δ2p+12Is,j+Δ2p+1Nγ+Δ2p+1M2γN+Δ2p+12NaβSs,j+Δ2p+12NaNβ+Δ2p+1{2NaNMβMγ+2NaMβCs,loc}=Δ2p+1{Is,j2+Nγ+M2γN+Δ2p+12NaβSs,j+2NaNβ+2NaNMβMγ+2NaMβCs,loc}

    We define

    CI,loc:={Is,j2+Nγ+M2γN+Δ2p+12NaβSs,j+2NaNβ+2NaNMβMγ+2NaMβCs,loc}.

    We conclude

    |Is,j(tp+1)~Inums,j(tp+1)|Δ2p+1CI,loc. (3.7)

    1.3) It holds

    ~Rnums,j(tp+1)=Rs,j(tp)+Δp+1γIs,j(tp+1)~Inums,j(tp+1).

    We consider |Rs,j(tp+1)~Rnums,j(tp+1)| and obtain

    |Rs,j(tp+1)~Rnums,j(tp+1)|=|Rs,j(tp+1)Rs,j(tp)Δp+1γIs,j(tp+1)~Inums,j(tp+1)|.

    Application of zero addition and the triangle inequality yields

    |Rs,j(tp+1)~Rnums,j(tp+1)|=|tp+1tpRs,j(τ)dτΔp+1γIs,j(tp)Is,j(tp)+Δp+1γIs,j(tp)Is,j(tp)Δp+1γIs,j(tp)Is,j(tp+1)+Δp+1γIs,j(tp)Is,j(tp+1)Δp+1γIs,j(tp+1)Is,j(tp+1)+Δp+1γIs,j(tp+1)Is,j(tp+1)Δp+1γIs,j(tp+1)~Inums,j(tp+1)||tp+1tpRs,j(τ)dτΔp+1γIs,j(tp)Is,j(tp)|+|Δp+1γIs,j(tp)Is,j(tp)Δp+1γIs,j(tp)Is,j(tp+1)|+|Δp+1γIs,j(tp)Is,j(tp+1)Δp+1γIs,j(tp+1)Is,j(tp+1)|+|Δp+1γIs,j(tp+1)Is,j(tp+1)Δp+1γIs,j(tp+1)~Inums,j(tp+1)|=|tp+1tp(Rs,j(τ)Rs,j(tp))dτ|:=If,1+|Δp+1γIs,j(tp)(Is,j(tp)Is,j(tp+1))|:=If,2+|Δp+1Is,j(tp+1)(γIs,j(tp)γIs,j(tp+1))|:=If,3+|Δp+1γIs,j(tp+1)(Is,j(tp+1)~Inums,j(tp+1))|:=If,4.

    By the mean value theorem of calculus, there are ξf,1,ξf,2,ξf,3,ξf,4(tp,tp+1) such that

    Rs,j(ξf,1)=Rs,j(τ)Rs,j(tp)τtp,Is,j(ξf,2)=Is,j(tp+1)Is,j(tp)tp+1tp,γIs,j(ξf,3)=γIs,j(tp+1)γIs,j(tp)tp+1tp

    hold. This implies

    If,1:=|tp+1tp(Rs,j(τ)Rs,j(tp))dτ|=|tp+1tp(τtp)Rs,j(τ)Rs,j(tp)τtpdτ|=|tp+1tp(τtp)Rs,j(ξf,1)dτ|Δ2p+12Rs,j,If,2:=|Δp+1γIs,j(tp)(Is,j(tp)Is,j(tp+1))|=|Δ2p+1γIs,jIs,j(tp+1)Is,j(tp)tp+1tp|=|Δ2p+1γIs,jIs,j(ξf,2)|Δ2p+1βIs,j

    and

    If,3:=|Δp+1Is,j(tp+1)(γIs,j(tp)γIs,j(tp+1))|=|Δ2p+1Is,j(tp+1)γIs,j(tp+1)γIs,j(tp)tp+1tp|=|Δ2p+1Is,j(tp+1)γIs,j(ξf,3)|Δ2p+1NγIs,j.

    By inequality (3.7) from Step 1.2), we know that

    |Is,j(tp+1)~Inums,j(tp+1)|Δ2p+1CI,loc

    is valid. We infer that

    \begin{eqnarray*} I_{f, 4} & = & \left| \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \left( I_{s, j} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right) \right| \\ & \leq & \Delta_{p + 1} \cdot \lVert \gamma_{I_{s, j}} \rVert_{\infty} \cdot \Delta_{p + 1}^{2} \cdot C_{I, \text{loc}} \leq \Delta_{p + 1}^{3} \cdot C_{I, \text{loc}} \cdot M_{\gamma} \end{eqnarray*}

    holds. Summarizing our results, we obtain

    \begin{eqnarray*} \left| R_{s, j} \left( t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| & \leq & I_{f, 1} + I_{f, 2} + I_{f, 3} + I_{f, 4} \\ & \leq & \dfrac{\Delta_{p + 1}^{2}}{2} \cdot \lVert R_{s, j}^{\prime \prime} \rVert_{\infty} + \Delta_{p + 1}^{2} \cdot \lVert { \beta } \rVert_{\infty} \cdot \lVert I_{s, j}^{\prime} \rVert_{\infty} \\ & & + \Delta_{p + 1}^{2} \cdot N \cdot \lVert \gamma_{I_{s, j}}^{\prime} \rVert_{\infty} + \Delta_{p + 1}^{3} \cdot C_{I, \text{loc}} \cdot M_{\gamma} \\ & = & \Delta_{p + 1}^{2} \cdot \left\{ \dfrac{\lVert R_{s, j}^{\prime \prime} \rVert_{\infty}}{2} + \lVert { \beta } \rVert_{\infty} \cdot \lVert I_{s, j}^{\prime} \rVert_{\infty} + N \cdot \lVert \gamma_{I_{s, j}}^{\prime} \rVert_{\infty} + \Delta_{p + 1} \cdot C_{I, \text{loc}} \cdot M_{\gamma} \right\} \\ & \leq & \Delta_{p + 1}^{2} \cdot \underbrace{\left\{ \dfrac{\lVert R_{s, j}^{\prime \prime} \rVert_{\infty}}{2} + \lVert { \beta } \rVert_{\infty} \cdot \lVert I_{s, j}^{\prime} \rVert_{\infty} + N \cdot \lVert \gamma_{I_{s, j}}^{\prime} \rVert_{\infty} + C_{I, \text{loc}} \cdot M_{\gamma} \right\}}_{: = C_{R, \text{loc}}} \\ & = & \Delta_{p + 1}^{2} \cdot C_{R, \text{loc}}. \end{eqnarray*}

    and

    \begin{equation} \left| R_{s, j} \left( t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \leq \Delta_{p + 1}^{2} \cdot C_{R, \text{loc}} \end{equation} (3.8)

    in a short manner.

    1.4) Conclusively, we obtain

    \begin{equation} \begin{split} & \, \, \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| I_{s, j} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| R_{s, j} \left( t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \right\} \\ & \leq \Delta_{p + 1}^{2} \cdot \underbrace{\max \left\{ C_{S, \text{loc}}, C_{I, \text{loc}}, C_{R, \text{loc}} \right\}}_{: = C_{ \text{loc}}} = \Delta_{p + 1}^{2} \cdot C_{ \text{loc}} \end{split} \end{equation} (3.9)

    from the inequalities (3.6), (3.7) and (3.8).

    2) In reality, the points \left(t_{p}, S_{s, j}^{ \text{num}} \left(t_{p} \right) \right) , \left(t_{p}, I_{s, j}^{ \text{num}} \left(t_{p} \right) \right) and \left(t_{p}, R_{s, j}^{ \text{num}} \left(t_{p} \right) \right) do not lie on the continuous solution graph. For that reason, we must investigate how procedural errors S_{s, j}^{ \text{num}} \left(t_{p} \right) - S_{s, j} \left(t_{p} \right) , I_{s, j}^{ \text{num}} \left(t_{p} \right) - I_{s, j} \left(t_{p} \right) and R_{s, j}^{ \text{num}} \left(t_{p} \right) - R _{s, j} \left(t_{p} \right) propagate to the \left(p + 1 \right) -th time step. These estimates are going to be carried out in the following steps 2) and 3) of this proof.

    2.1) At first, we must consider \left| S_{s, j}^{ \text{num}} \left(t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left(t_{p + 1} \right)} \right| . Remember that \widetilde{S_{s, j}^{ \text{num}} \left(t_{p} \right)} = S_{s, j} \left(t_{p} \right) . Note that

    S_{s, j}^{ \text{num} } \left( t_{p + 1} \right) = S_{s, j}^{ \text{num} } \left( t_{p} \right) - \dfrac{\Delta_{p + 1} \cdot S_{s, j}^{ \text{num} } \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \cdot \dfrac{I_{q, k}^{ \text{num} } \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num} } \left( t_{p} \right)}{N} \right\}}

    and

    \widetilde{S_{s, j}^{ \text{num} } \left( t_{p + 1} \right)} = S_{s, j} \left( t_{p} \right) - \dfrac{\Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\}}

    are valid. Hence, we obtain

    \begin{eqnarray*} & & \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & = & \left| \dfrac{S_{s, j}^{ \text{num}} \left( t_{p} \right)}{1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}} - \dfrac{S_{s, j} \left( t_{p} \right)}{1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\}} \right| \\ & = & \left| \dfrac{\left\{ S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right\} + \Delta_{p + 1} \cdot S_{s, j}^{ \text{num}} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\}}{\left\{ 1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\} \right\} \cdot \left\{ 1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right\}} \right. \\ & & \left. - \dfrac{\Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{\left\{ 1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\} \right\} \cdot \left\{ 1 + \Delta_{p + 1} \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right\}} \right|. \end{eqnarray*}

    Application of the triangle inequality and zero addition yields

    \begin{eqnarray*} & & \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \left| \Delta_{p + 1} \cdot S_{s, j}^{ \text{num}} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right. \\ & & \left. - \Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\} \right| \\ & = & \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \left| \Delta_{p + 1} \cdot S_{s, j}^{ \text{num}} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right. \\ & & \left. - \Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right. \\ & & \left. + \Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right. \\ & & \left. - \Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\} \right| \\ & \leq & \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \\ & & + \left| \Delta_{p + 1} \cdot \left( S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right)}{N} \right\} \right| \\ & & + \left| \Delta_{p + 1} \cdot S_{s, j} \left( t_{p} \right) \cdot \sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{I_{q, k} \left( t_{p} \right) - I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\} \right| \\ & \leq & \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k} \left( t_{p} \right) - I_{q, k}^{ \text{num}} \left( t_{p} \right) \right| \right\}. \end{eqnarray*}

    Summarizing this result, we obtain

    \begin{equation} \begin{split} & \, \, \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \\ & \, \, + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k} \left( t_{p} \right) - I_{q, k}^{ \text{num}} \left( t_{p} \right) \right| \right\}. \end{split} \end{equation} (3.10)

    2.2) Now, we consider \left| I_{s, j}^{ \text{num}} \left(t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left(t_{p + 1} \right)} \right| . We first observe that

    \begin{eqnarray*} I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) & = & I_{s, j}^{ \text{num}} \left( t_{p} \right) - \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot I_{s, j}^{ \text{num}} \left( t_{p} \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \\ & & + \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \end{eqnarray*}

    and

    \begin{eqnarray*} \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} & = & \widetilde{I_{s, j} \left( t_{p} \right)} - \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \widetilde{I_{s, j}^{ \text{num}} \left( t_{p} \right)}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \\ & & + \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot \widetilde{I_{q, k}^{ \text{num}} \left( t_{p} \right)}}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \\ & = & I_{s, j} \left( t_{p} \right) - \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot I_{s, j} \left( t_{p} \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \\ & & + \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot I_{q, k} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \end{eqnarray*}

    are valid from step 1.2). Application of the triangle inequality and zero addition yields

    \begin{eqnarray*} & & \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & = & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot I_{s, j}^{ \text{num}} \left( t_{p} \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. + \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. - I_{s, j} \left( t_{p} \right) + \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot I_{s, j} \left( t_{p} \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. - \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot I_{q, k} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \left| \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \left( I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & & + \left| \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. - \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot I_{q, k} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & = & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \left| \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \left( I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & & + \left| \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. - \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. + \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right. \\ & & \left. - \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot I_{q, k} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \left| \dfrac{\Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \left( I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right)}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & & + \left| \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\left( S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right) \cdot I_{q, k}^{ \text{num}} \left( t_{p} \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & & + \left| \Delta_{p + 1} \cdot \dfrac{\sum\limits_{k = 1}^{N_{a}} \sum\limits_{q \in \left\{ f, m \right\}} \left\{ \beta_{S_{s, j}, I_{q, k}} \left( t_{p + 1} \right) \cdot \dfrac{\widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \cdot \left( I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right)}{N} \right\}}{1 + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\}. \end{eqnarray*}

    Using (3.10), we obtain

    \begin{eqnarray*} & & \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k} \left( t_{p} \right) - I_{q, k}^{ \text{num}} \left( t_{p} \right) \right| \right\} \right\} \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \\ & = & \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \\ & & + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \\ & & + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \end{eqnarray*}

    and the result reads

    \begin{equation} \begin{split} & \, \, \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \\ & \, \, + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \\ & \, \, + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \\ & \, \, + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\}. \end{split} \end{equation} (3.11)

    2.3) We consider \left| R_{s, j}^{ \text{num}} \left(t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left(t_{p + 1} \right)} \right| . From step 1.3), we know that

    R_{s, j}^{ \text{num} } \left( t_{p + 1} \right) = R_{s, j}^{ \text{num} } \left( t_{p} \right) + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot I_{s, j}^{ \text{num} } \left( t_{p + 1} \right)

    and

    \widetilde{R_{s, j}^{ \text{num} } \left( t_{p + 1} \right)} = R_{s, j} \left( t_{p} \right) + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \widetilde{I_{s, j}^{ \text{num} } \left( t_{p + 1} \right)}

    hold. By application of the triangle inequality, this implies

    \begin{eqnarray*} & & \left| R_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & = & \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) + \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - R_{s, j} \left( t_{p} \right) - \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| + \left| \Delta_{p + 1} \cdot \gamma_{I_{s, j}} \left( t_{p + 1} \right) \cdot \left( I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right) \right| \\ & \leq & \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|. \end{eqnarray*}

    Using inequality (3.11), we obtain

    \begin{eqnarray*} & & \left| R_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq & \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left\{ \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right. \\ & & \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right\}. \end{eqnarray*}

    We conclude that

    \begin{equation} \begin{split} & \, \, \left| R_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \\ & \leq \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left\{ \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \right. \\ & \, \, \left. + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & \, \, \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & \, \, \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right. \\ & \, \, \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right\}. \end{split} \end{equation} (3.12)

    holds.

    2.4) Now, we want to combine our results. Since s \in \left\{ f, m \right\} and j \in \left\{ 1, \ldots, N_{a} \right\} are arbitrary indices, we infer by inequalities (3.10), (3.11) and (3.12) that

    \begin{eqnarray*} & & \, \, \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| R_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \right\} \\ & \leq & \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k} \left( t_{p} \right) - I_{q, k}^{ \text{num}} \left( t_{p} \right) \right| \right\}, \right. \\ & & \left. \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \\ & & \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right. \\ & & \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\}, \right. \\ & & \left. \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left\{ \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| \right. \right. \\ & & \left. \left. + \Delta_{p + 1} \cdot M_{\gamma} \cdot \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right| + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \right. \\ & & \left. \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right| \right. \right. \\ & & \left. \left. + \Delta_{p + 1}^{2} \cdot 4 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right. \right. \\ & & \left. \left. + \Delta_{p + 1} \cdot 2 \cdot N_{a} \cdot M_{\beta} \cdot \max\limits_{\substack{k \in \left\{ 1, \ldots, N_{a} \right\} \\ q \in \left\{ f, m \right\}}} \left\{ \left| I_{q, k}^{ \text{num}} \left( t_{p} \right) - I_{q, k} \left( t_{p} \right) \right| \right\} \right\} \right\} \\ & \leq & \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right|, \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right|, \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| \right\} \\ & & \times \left\{ 1 + \Delta_{p + 1} \cdot \left\{ 2 \cdot M_{\gamma} + 4 \cdot N_{a} \cdot M_{\beta} + 8 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \cdot \Delta_{p + 1} \right\} \right\} \\ & \leq & \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right|, \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right|, \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| \right\} \\ & & \times \left\{ 1 + \Delta_{p + 1} \cdot \left\{ 2 \cdot M_{\gamma} + 4 \cdot N_{a} \cdot M_{\beta} + 8 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \right\} \right\} \end{eqnarray*}

    holds because \Delta_{p + 1} \leq 1 by assumption. This yields

    \begin{equation} \begin{split} & \, \, \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| R_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \right\} \\ & \leq \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p} \right) - S_{s, j} \left( t_{p} \right) \right|, \left| I_{s, j}^{ \text{num}} \left( t_{p} \right) - I_{s, j} \left( t_{p} \right) \right|, \left| R_{s, j}^{ \text{num}} \left( t_{p} \right) - R_{s, j} \left( t_{p} \right) \right| \right\} \\ & \, \, \times \left\{ 1 + \Delta_{p + 1} \cdot \underbrace{\left\{ 2 \cdot M_{\gamma} + 4 \cdot N_{a} \cdot M_{\beta} + 8 \cdot N_{a}^{2} \cdot M_{\beta}^{2} \right\}}_{: = C_{ \text{prop}}} \right\}. \end{split} \end{equation} (3.13)

    3) Finally, we can finish our proof of convergence. For abbreviation, we write

    \begin{eqnarray*} & & \lVert \mathbf{z}^{ \text{num}} \left( t_{p + 1} \right) - \mathbf{z} \left( t_{p + 1} \right) \rVert_{ \text{conv}} \\ & : = & \max\limits_{\substack{j \in \left\{ 1, \ldots, N_{a} \right\} \\ s \in \left\{ f, m \right\}}} \left\{ \left| S_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{S_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| I_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{I_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right|, \left| R_{s, j}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{R_{s, j}^{ \text{num}} \left( t_{p + 1} \right)} \right| \right\} \end{eqnarray*}

    where \mathbf{z} \in \mathbb{R}^{6 \cdot N_{a}} is defined as in the proof of Theorem 2.5. Our proof is heavily based on the inequality

    1 + x \leq \text{exp} \left( x \right)

    for all x \geq 0 . Note that t_{1} = 0 < t_{2} < \ldots < t_{M - 1} < t_{M} = T .

    3.1) At first, we want to inductively prove that

    \begin{equation} \begin{split} \lVert \mathbf{z}^{ \text{num}} \left( t_{p + 1} \right) - \mathbf{z} \left( t_{p + 1} \right) \rVert_{ \text{conv}} & \leq \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{1} \right\} \right) \\ & \, \, + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p + 1} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{k} \right\} \right) \end{split} \end{equation} (3.14)

    holds for all p \in \left\{ 0, \ldots, M - 1 \right\} . Let p = 0 first. The inequality (3.14) is fulfilled. Let p = 1 to understand the concept. By application of the triangle inequality and inequalities (3.9) and (3.13), we see that

    \begin{eqnarray*} & & \lVert \mathbf{z}^{ \text{num}} \left( t_{2} \right) - \mathbf{z} \left( t_{2} \right) \rVert_{ \text{conv}} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{2} \right) - \widetilde{\mathbf{z}^{ \text{num}} \left( t_{2} \right)} \rVert_{ \text{conv}} + \lVert \widetilde{\mathbf{z}^{ \text{num}} \left( t_{2} \right)} - \mathbf{z} \left( t_{2} \right) \rVert_{ \text{conv}} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \left\{ 1 + C_{ \text{prop}} \cdot \Delta_{2} \right\} + C_{ \text{loc}} \cdot \Delta_{2}^{2} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \Delta_{2} \right) + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{2} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{2} - t_{k} \right\} \right) \\ & = & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \Delta_{2} \right) + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{2} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{2} - t_{k} \right\} \right) \\ & = & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{2} - t_{1} \right\} \right) + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{2} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{2} - t_{k} \right\} \right) \end{eqnarray*}

    is valid. We now assume that

    \begin{equation*} \begin{split} \lVert \mathbf{z}^{ \text{num}} \left( t_{p} \right) - \mathbf{z} \left( t_{p} \right) \rVert_{ \text{conv}} & \leq \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p} - t_{1} \right\} \right) \\ & \, \, + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p} - t_{k} \right\} \right) \end{split} \end{equation*}

    holds. We now want to show that (3.14) follows. We see that

    \begin{eqnarray*} & & \lVert \mathbf{z}^{ \text{num}} \left( t_{p + 1} \right) - \mathbf{z} \left( t_{p + 1} \right) \rVert_{ \text{conv}} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{p + 1} \right) - \widetilde{\mathbf{z}^{ \text{num}} \left( t_{p + 1} \right)} \rVert_{ \text{conv}} + \lVert \widetilde{\mathbf{z}^{ \text{num}} \left( t_{p + 1} \right)} - \mathbf{z} \left( t_{p + 1} \right) \rVert_{ \text{conv}} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{p} \right) - \mathbf{z} \left( t_{p} \right) \rVert_{ \text{conv}} \cdot \left\{ 1 + C_{ \text{prop}} \cdot \Delta_{p + 1} \right\} + C_{ \text{loc}} \cdot \Delta_{p + 1}^{2} \\ & \leq & \left\{ \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p} - t_{1} \right\} \right) + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p} - t_{k} \right\} \right) \right\} \\ & & \times \, \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{p} \right\} \right) + C_{ \text{loc}} \cdot \Delta_{p + 1}^{2} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{1} \right\} \right) \\ & & + \left( C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p} - t_{k} \right\} \right) \right) \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{p} \right\} \right) + C_{ \text{loc}} \cdot \Delta_{p + 1}^{2} \\ & = & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{1} \right\} \right) \\ & & + \left( C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{k} \right\} \right) \right) + C_{ \text{loc}} \cdot \Delta_{p + 1}^{2} \\ & = & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{1} \right\} \right) \\ & & + \left( C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p + 1} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{k} \right\} \right) \right) \end{eqnarray*}

    holds. This proves (3.14) by induction.

    3.2) Concluding our proof, we consider

    \begin{equation*} \begin{split} \lVert \mathbf{z}^{ \text{num}} \left( t_{p + 1} \right) - \mathbf{z} \left( t_{p + 1} \right) \rVert_{ \text{conv}} & \leq \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{1} \right\} \right) \\ & \, \, + C_{ \text{loc}} \cdot \sum\limits_{k = 2}^{p + 1} \Delta_{k}^{2} \cdot \text{exp} \left( C_{ \text{prop}} \cdot \left\{ t_{p + 1} - t_{k} \right\} \right) \end{split} \end{equation*}

    from (3.14). We define \Delta : = \max\limits_{r \in \left\{ 2, \ldots, M \right\}} \Delta_{r} . We infer that

    \begin{eqnarray*} & & \lVert \mathbf{z}^{ \text{num}} \left( t_{p + 1} \right) - \mathbf{z} \left( t_{p + 1} \right) \rVert_{ \text{conv}} \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot T \right) + C_{ \text{loc}} \cdot \Delta \cdot \sum\limits_{k = 2}^{p + 1} \Delta_{k} \cdot \text{exp} \left( C_{ \text{prop}} \cdot T \right) \\ & \leq & \lVert \mathbf{z}^{ \text{num}} \left( t_{1} \right) - \mathbf{z} \left( t_{1} \right) \rVert_{ \text{conv}} \cdot \text{exp} \left( C_{ \text{prop}} \cdot T \right) + C_{ \text{loc}} \cdot \Delta \cdot T \cdot \text{exp} \left( C_{ \text{prop}} \cdot T \right) \end{eqnarray*}

    holds. If the initial conditions of our continuous and our time-discrete problem formulation coincide and \Delta \to 0 , the time-discrete solution convergences linearly towards the continuous solution. This proves our assertion.

    We briefly summarize our numerical solution algorithm for the time-discrete explicit-implicit numerical scheme (3.1) in Table 1. This summary is intended to give a brief overview of aspects which need to be considered during implementation. Especially, we state all inputs which are important for our time-discrete numerical scheme.

    Table 1.  Algorithmic summary of our time-discrete explicit-implicit numerical solution scheme for the age- and sex-structured SIR model.
    Input: - Population size N
    - Increasing sequence of time points t_{1} = 0 < t_{2} < \ldots < t_{M - 1} < t_{M} = T
    - Initial condition of susceptible people S_{s, j} \left(t_{1} \right) for arbitrary s \in \left\{ f, m \right\}
    and all j \in \left\{ 1, \ldots, N_{a} \right\}
    - Initial condition of infected people I_{s, j} \left(t_{1} \right) for arbitrary s \in \left\{ f, m \right\}
    and all j \in \left\{ 1, \ldots, N_{a} \right\}
    - Initial condition of recovered people R_{s, j} \left(t_{1} \right) for arbitrary s \in \left\{ f, m \right\}
    and all j \in \left\{ 1, \ldots, N_{a} \right\}
    - Time-varying transmission rates \beta_{S_{s, j}, I_{q, k}} \colon \left[0, \infty \right) \longrightarrow \left[0, \infty \right)
    for arbitrary s, q \in \left\{ f, m \right\} and arbitrary j, k \in \left\{ 1, \ldots, N_{a} \right\}
    - Time-varying recovery rates \gamma_{I_{s, j}} \colon \left[0, \infty \right) \longrightarrow \left[0, \infty \right)
    for arbitrary s \in \left\{ f, m \right\} and arbitrary j \in \left\{ 1, \ldots, N_{a} \right\}
    Steps: For all p \in \left\{ 1, \ldots, M - 1 \right\} do the following:
    - Compute S_{s, j} \left(t_{p + 1} \right) for arbitrary s \in \left\{ f, m \right\} and all
    j \in \left\{ 1, \ldots, N_{a} \right\} by (3.3)
    - Compute I_{s, j} \left(t_{p + 1} \right) for arbitrary s \in \left\{ f, m \right\} and all
    j \in \left\{ 1, \ldots, N_{a} \right\} by (3.4)
    - Compute R_{s, j} \left(t_{p + 1} \right) for arbitrary s \in \left\{ f, m \right\} and all
    j \in \left\{ 1, \ldots, N_{a} \right\} by (3.5)
    Output: - Sequence of susceptible people \left\{ S_{s, j} \left(t_{p} \right) \right\}_{p = 1}^{M} for arbitrary s \in \left\{ f, m \right\}
    and all j \in \left\{ 1, \ldots, N_{a} \right\}
    - Sequence of infected people \left\{ I_{s, j} \left(t_{p} \right) \right\}_{p = 1}^{M} for arbitrary s \in \left\{ f, m \right\}
    and all j \in \left\{ 1, \ldots, N_{a} \right\}
    - Sequence of recovered people \left\{ R_{s, j} \left(t_{p} \right) \right\}_{p = 1}^{M} for arbitrary s \in \left\{ f, m \right\}
    and all j \in \left\{ 1, \ldots, N_{a} \right\}

     | Show Table
    DownLoad: CSV

    In this section, we illustrate our theoretical findings by one synthetic data example. At first, we sum up all important information to set calculations up. Finally, we show the results and discuss these findings with respect to our theoretical results.

    Let us provide our setting. In Table 2, we summarize the corresponding indices of population subetaoups. The total population is divided into six subetaoups. Now, we report the (time-varying) transmission rates \beta_{S_{s, j}, I_{q, k}} \colon \left[0, \infty \right) \longrightarrow \left[0, \infty \right) and (time-varying) recovery rates \gamma_{I_{s, j}} \colon \left[0, \infty \right) \longrightarrow \left[0, \infty \right) for arbitrary s, q \in \left\{ f, m \right\} and arbitrary j, k \in \left\{ 1, \ldots, N_{a} \right\} . These data can be found in Tables 3 and 4. This is an imaginary disease which spreads mainly in the adult population. All initial conditions of populations subetaoups are described in Table 5. The final time is set T = 180 with an equidistant time sequence

    t_{1} = 0 \lt t_{2} = 1 \lt \ldots \lt t_{180} = 179 \lt t_{181} = 180
    Table 2.  Indices of corresponding population subgroups.
    Young Adult Elder
    Female f, y f, a f, e
    Male m, y m, a m, e

     | Show Table
    DownLoad: CSV
    Table 3.  (Time-varying) transmission rates.
    \beta_{S, I} I_{f, y} I_{f, a} I_{f, e} I_{m, y} I_{m, a} I_{m, e}
    S_{f, y} 0.10 0.08 0.04 0.10 0.08 0.04
    S_{f, a} 0.08 0.20 0.02 0.08 0.20 0.02
    S_{f, e} 0.04 0.02 0.01 0.04 0.02 0.01
    S_{m, y} 0.10 0.08 0.04 0.10 0.08 0.04
    S_{m, a} 0.08 0.20 0.02 0.08 0.20 0.02
    S_{m, e} 0.04 0.02 0.01 0.04 0.02 0.01

     | Show Table
    DownLoad: CSV
    Table 4.  (Time-varying) recovery rates.
    I_{f, y} I_{f, a} I_{f, e} I_{m, y} I_{m, a} I_{m, e}
    \gamma_{I} 0.20 0.10 0.05 0.20 0.10 0.05

     | Show Table
    DownLoad: CSV
    Table 5.  Initial conditions for all population subgroups.
    f, y f, a f, e m, y m, a m, e
    S \left(0 \right) 10000 20000 19900 10000 20000 19900
    I \left(0 \right) 35 35 30 35 35 30
    R \left(0 \right) 0 0 0 0 0 0

     | Show Table
    DownLoad: CSV

    and this implies M = 181 . The total population size reads N = 100000 due to Table 5. Hence, all data are available for our numerical simulation.

    Here, we present the results of our setting described before. In Figure 1, the temporal development of all susceptible population subetaoups is depicted. It can be clearly seen that the resulting graphs are decreasing in time. In Figure 2, all graphs of the temporal development with regard to all infectious subetaoups are portrayed. Figure 3 illustrates the temporal development of all recovered population subetaoups. As expected, these curves are increasing in time. Finally, conservation of the total population size for our implicit-explicit numerical solution scheme is shown in Figure 4.

    Figure 1.  Results for all susceptible population subgroups.
    Figure 2.  Results for all infectious population subgroups.
    Figure 3.  Results for all recovered population subgroups.
    Figure 4.  Conservation of total population size.

    We introduced an age- and sex-structured SIR model for short-term predictions in Section 2. We established global existence, global uniqueness, non-negativity and boundedness of the solution. Additionally, we showed some monotonicity properties and proved convergence to a disease-free equilibrium in the continuous setting. Afterwards, we proposed an explicit-implicit numerical solution scheme in Section 3. We were able to demonstrate that all aforementioned properties transfer to this time-discrete setting of the age- and sex-structured SIR model for short-term predictions. We also concluded that this scheme is linearly convergent towards the continuous solution. For short-term predictions, effects of demography and transmission between age groups can be simplified or neglected in this case.

    To continue this work and extend it to long-term predictions that definitely play an important role, it might be fruitful to additionally take birth rates and death rates into account. The works [20,21] can serve as examples for extensions of our work. Incubation times also lead to delays from transfer between different compartments. Hence, introduction of delays in our system might be another possible future research direction. Examples can be seen in [22]. Furthermore, spatial inhomogeneities should also be considered because spreading of diseases differ in regions depending on social status for example [23,24], which leads to ODE-PDE coupled systems. Application of higher-order methods might be considerable as well [25,26].

    Finally, we stress the fact that the inverse problem in dynamics of biological systems needs further investigation [27,28,29,30].

    Both authors conceived and designed the research. Benjamin Wacker analyzed the time-continuous problem formulation. Benjamin Wacker analyzed the time-discrete problem formulation. Benjamin Wacker implemented the explicit-implicit numerical solution scheme. Both authors discussed the numerical example. Both authors drafted and edited this manuscript.

    Both authors declare that they have no conflict of interest.



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