Processing math: 100%
Research article

Evaluating persistence of shape information using a matching protocol

  • Many laboratories have studied persistence of shape information, the goal being to better understand how the visual system mediates recognition of objects. Most have asked for recognition of known shapes, e.g., letters of the alphabet, or recall from an array. Recognition of known shapes requires access to long-term memory, so it is not possible to know whether the experiment is assessing short-term encoding and working memory mechanisms, or has encountered limitations on retrieval from memory stores. Here we have used an inventory of unknown shapes, wherein a string of discrete dots forms the boundary of each shape. Each was displayed as a target only once to a given respondent, with recognition being tested using a matching task. Analysis based on signal detection theory was used to provide an unbiased estimate of the probability of correct decisions about whether comparison shapes matched target shapes. Four experiments were conducted, which found the following: a) Shapes were identified with a high probability of being correct with dot densities ranging from 20% to 4%. Performance dropped only about 10% across this density range. b) Shape identification levels remained very high with up to 500 milliseconds of target and comparison shape separation. c) With one-at-a-time display of target dots, varying the total time for a given display, the proportion of correct decisions dropped only about 10% even with a total display time of 500 milliseconds. d) With display of two complementary target subsets, also varying the total time of each display, there was a dramatic decline of proportion correct that reached chance levels by 500 milliseconds. The greater rate of decline for the two-pulse condition may be due to a mechanism that registers when the number of dots is sufficient to create a shape summary. Once a summary is produced, the temporal window that allows shape information to be added may be more limited.

    Citation: Ernest Greene, Michael J. Hautus. Evaluating persistence of shape information using a matching protocol[J]. AIMS Neuroscience, 2018, 5(1): 81-96. doi: 10.3934/Neuroscience.2018.1.81

    Related Papers:

    [1] Sumati Kumari Panda, Abdon Atangana, Juan J. Nieto . Correction: New insights on novel coronavirus 2019-nCoV/SARS-CoV-2 modelling in the aspect of fractional derivatives and fixed points. Mathematical Biosciences and Engineering, 2022, 19(2): 1588-1590. doi: 10.3934/mbe.2022073
    [2] Tahir Khan, Roman Ullah, Gul Zaman, Jehad Alzabut . A mathematical model for the dynamics of SARS-CoV-2 virus using the Caputo-Fabrizio operator. Mathematical Biosciences and Engineering, 2021, 18(5): 6095-6116. doi: 10.3934/mbe.2021305
    [3] A. D. Al Agha, A. M. Elaiw . Global dynamics of SARS-CoV-2/malaria model with antibody immune response. Mathematical Biosciences and Engineering, 2022, 19(8): 8380-8410. doi: 10.3934/mbe.2022390
    [4] Somayeh Fouladi, Mohammad Kohandel, Brydon Eastman . A comparison and calibration of integer and fractional-order models of COVID-19 with stratified public response. Mathematical Biosciences and Engineering, 2022, 19(12): 12792-12813. doi: 10.3934/mbe.2022597
    [5] Rahat Zarin, Usa Wannasingha Humphries, Amir Khan, Aeshah A. Raezah . Computational modeling of fractional COVID-19 model by Haar wavelet collocation Methods with real data. Mathematical Biosciences and Engineering, 2023, 20(6): 11281-11312. doi: 10.3934/mbe.2023500
    [6] A. M. Elaiw, Raghad S. Alsulami, A. D. Hobiny . Global dynamics of IAV/SARS-CoV-2 coinfection model with eclipse phase and antibody immunity. Mathematical Biosciences and Engineering, 2023, 20(2): 3873-3917. doi: 10.3934/mbe.2023182
    [7] Biplab Dhar, Praveen Kumar Gupta, Mohammad Sajid . Solution of a dynamical memory effect COVID-19 infection system with leaky vaccination efficacy by non-singular kernel fractional derivatives. Mathematical Biosciences and Engineering, 2022, 19(5): 4341-4367. doi: 10.3934/mbe.2022201
    [8] Adnan Sami, Amir Ali, Ramsha Shafqat, Nuttapol Pakkaranang, Mati ur Rahmamn . Analysis of food chain mathematical model under fractal fractional Caputo derivative. Mathematical Biosciences and Engineering, 2023, 20(2): 2094-2109. doi: 10.3934/mbe.2023097
    [9] M. Botros, E. A. A. Ziada, I. L. EL-Kalla . Semi-analytic solutions of nonlinear multidimensional fractional differential equations. Mathematical Biosciences and Engineering, 2022, 19(12): 13306-13320. doi: 10.3934/mbe.2022623
    [10] Noura Laksaci, Ahmed Boudaoui, Seham Mahyoub Al-Mekhlafi, Abdon Atangana . Mathematical analysis and numerical simulation for fractal-fractional cancer model. Mathematical Biosciences and Engineering, 2023, 20(10): 18083-18103. doi: 10.3934/mbe.2023803
  • Many laboratories have studied persistence of shape information, the goal being to better understand how the visual system mediates recognition of objects. Most have asked for recognition of known shapes, e.g., letters of the alphabet, or recall from an array. Recognition of known shapes requires access to long-term memory, so it is not possible to know whether the experiment is assessing short-term encoding and working memory mechanisms, or has encountered limitations on retrieval from memory stores. Here we have used an inventory of unknown shapes, wherein a string of discrete dots forms the boundary of each shape. Each was displayed as a target only once to a given respondent, with recognition being tested using a matching task. Analysis based on signal detection theory was used to provide an unbiased estimate of the probability of correct decisions about whether comparison shapes matched target shapes. Four experiments were conducted, which found the following: a) Shapes were identified with a high probability of being correct with dot densities ranging from 20% to 4%. Performance dropped only about 10% across this density range. b) Shape identification levels remained very high with up to 500 milliseconds of target and comparison shape separation. c) With one-at-a-time display of target dots, varying the total time for a given display, the proportion of correct decisions dropped only about 10% even with a total display time of 500 milliseconds. d) With display of two complementary target subsets, also varying the total time of each display, there was a dramatic decline of proportion correct that reached chance levels by 500 milliseconds. The greater rate of decline for the two-pulse condition may be due to a mechanism that registers when the number of dots is sufficient to create a shape summary. Once a summary is produced, the temporal window that allows shape information to be added may be more limited.


    Differential equations, especially those possessing non-negative solutions, play an important role in many areas of sciences such as biology [1,2,3,4,5], chemistry [6,7,8], epidemiology [9,10,11] or population dynamics [12,13,14] to name a few of these fields. Since we focus on examining a time-discrete, mathematical model of epidemiology in our work, we limit ourselves to a concise review of literature in this area.

    In this article, we want to consider the time-continuous non-autonomous susceptible-infected-recovered (SIR) model

    $ {S(t)=α(t)S(t)I(t)N,I(t)=α(t)S(t)I(t)Nβ(t)I(t),R(t)=β(t)I(t),S(0)=S0,I(0)=I0,R(0)=R0
    $
    (1.1)

    in epidemiology thoroughly investigated by Wacker and Schlüter in 2020 [15]. We provide further assumptions regarding this time-continuous model in Section 2. In 1927, one of the most famous papers in mathematical epidemiology, written by Kermack and McKendrick, was published [16]. Previously, Ross and Hudson proposed important foundations for epidemiological models [17,18]. Additional well-written books and review articles on mathematical epidemiology are authored by Hethcote [19] and Brauer and Castillo-Chávez [20]. Since the establishment of the classical SIR model, many modifications have been suggested and, due to the on-going COVID-19 pandemic, many articles have been published with respect to this topic and we can only summarize a brief selection of these works.

    To this day, there are works which provide closed solution formulas for simple models in epidemiology [21,22,23]. However, more articles consider numerical approximations due to appearing nonlinearities. Especially during the COVID-19 pandemic, different approaches have been published [15,24]. In 2022, Chen and co-authors suggested a susceptible-exposed-infected-unreported-recovered (SEIUR) model for the spread of COVID-19 with piecewise constant dynamical parameters [25] as one possible extension of classical SIR models. Taking age- and sex-structure of a population into account, Wacker and Schlüter proposed an extension of classical SIR models by multiple groups which can be considered as, for example, age groups or locally related groups [26]. Using fractional differential operators, Xu and co-authors analyzed the spread of COVID-19 in 2020 with respect to the United States by means of a fractional-order susceptible-exposed-infected-recovered (SEIR) model [27]. Regarding parameter identification problems in epidemiological models, Marinov and Marinova designed adaptive SIR models to solve these inverse problems [28,29]. In context of inverse problems, Comunian, Gaburro and Giudici concluded that high-quality data are needed to calibrate constant parameters of epidemiological models [30]. Other interesting lines of research concern the consideration of epidemics on networks [31,32,33,34,35] or application of deep-learning approaches to, for example, forecast time-series data of epidemics by these methods [36,37,38].

    One important aspect in epidemiological models is preservation of positivity. It plays a huge role in different scientific problems such as wave equations [39], simulations of chemical reactions [7], population dynamics [12,13,14] or other biological systems [3]. However, many classical time discretization schemes suffer from conserving non-negativity for arbitraty time step sizes [40]. Hence, designing non-negativity preserving numerical solution schemes for such problems can be regarded as an art as stated by Mickens [41]. For these reasons, Wacker and Schlüter examined a numerical solution scheme

    $ {Sh,j+1Sh,jhj+1=αj+1Sh,j+1Ih,j+1N,Ih,j+1Ih,jhj+1=αj+1Sh,j+1Ih,j+1Nβj+1Ih,j+1,Rh,j+1Rh,jhj+1=βj+1Ih,j+1,Sh,1=S0,Ih,1=I0,Rh,1=R0
    $
    (1.2)

    for (1.1) based on implicit Eulerian time discretization in [15] whose algorithm is presented in Algorithm 1.

    Algorithm 1 Algorithmic summary for implicit Eulerian finite-difference-method (1.2) for (1.1)
    1: Inputs
    2: Time vector $ \left(t_{1}, \ldots, t_{M} \right)^{T} \in \mathbb{R}^{M} $
    3: Time-varying transmission rates $ \left(\alpha_{1}, \ldots, \alpha_{M} \right)^{T} \in \mathbb{R}^{M} $
    4: Time-varying recovery rates $ \left(\beta_{1}, \ldots, \beta_{M} \right)^{T} \in \mathbb{R}^{M} $
    5: Initialize $ \textbf{S} : = \textbf{0}, \textbf{I} : = \textbf{0}, \textbf{R} : = \textbf{0} \in \mathbb{R}^{M} $
    6: Initial conditions $ \textbf{S}_{1} : = S_{1} $, $ \textbf{I}_{1} : = I_{1} $ and $ \textbf{R}_{1} : = R_{1} $
    7: Total population size $ N : = \textbf{S}_{1} + \textbf{I}_{1} + \textbf{R}_{1} $
    8: Implicit Eulerian Finite-Difference-Method
    9: for $ j \in \left\{ 1, \ldots, M - 1 \right\} $ do
    10:  Define $ h_{j + 1} : = t_{j + 1} - t_{j} $ as current time step size
    11:  Define $ A_{j + 1} : = \left(1 + \beta_{j + 1} \cdot h_{j + 1} \right) \cdot \left(\alpha_{j + 1} \cdot h_{j + 1} \right) $
    12:  Define $ B_{j + 1} : = \dfrac{\left(1 + \beta_{j + 1} \cdot h_{j + 1} \right) \cdot N - \alpha_{j + 1} \cdot h_{j + 1} \cdot \left(S_{j} + I_{j} \right)}{2} $
    13:  Compute $ I_{j + 1} : = - \dfrac{B_{j + 1}}{A_{j + 1}} + \sqrt{\dfrac{B_{j + 1}^{2}}{A_{j + 1}^{2}} + \dfrac{N \cdot I_{j}}{A_{j + 1}}} $
    14:  Compute $ S_{j + 1} : = \dfrac{S_{j}}{1 + \alpha_{j + 1} \cdot h_{j + 1} \cdot \dfrac{I_{j + 1}}{N}} $
    15:  Compute $ R_{j + 1} : = R_{j} + \beta_{j + 1} \cdot h_{j + 1} \cdot I_{j + 1} $
    16: end for
    17: Outputs
    18: Calculated vectors $ \textbf{S}, \textbf{I}, \textbf{R} \in \mathbb{R}^{M} $

    Such non-standard finite-difference-methods for time discretization are also important in other application areas such as magnetohydrodynamics [42]. Here, we use $ f \left(t_{j} \right) : = f_{j} $ as an abbreviation for time-continuous functions at time points $ t_{j} $ and $ f_{h} \left(t_{j} \right) : = f_{h, j} $ as an abbreviation for time-discrete functions at time time points $ t_{j} $ on possibly non-equidistant meshes $ t_{1} < t_{2} < \ldots < t_{M - 1} < t_{M} $. Although Algorithm 1 preserves all desired properties of the time-continuous model (1.1) as shown in [15], there are still some disadvantages:

    $ 1) $ From its formulation (1.1), there is to solve a quadratic equation

    $ Aj+1I2j+1+Bj+1Ij+1NIj=0
    $

    where the a-priori knowledge of non-negative solutions from the time-continuous model is used to rule out one of both possible solutions for Algorithm 1 to obtain unique solvability;

    $ 2) $ In each time step, we must recalculate the auxiliary quantities $ A_{j + 1} $ and $ B_{j + 1} $;

    $ 3) $ By both aforementioned arguments, we conclude that Algorithm 1 does not intrinsically calculate biologically correct solutions and, from an algorithm point of view, recalculation of these auxiliary quantities is undesirable.

    We want to propose a different discretization by non-standard finite-difference-methods based on non-local approximations. To explain the idea of non-standard finite-difference-methods based on non-local approximations, we consider a time-continuous dynamical system

    $ y(t)=f(t,y(t)).
    $

    Standard time-discretizations normally approximate the right-hand-side function by function values of $ \textbf{y} \left(t \right) $ at one previous or the current time point. Non-standard finite-difference-methods based on non-local approximations, on the contrary, approximate $ \textbf{f} \left(t, \textbf{y} \left(t \right) \right) $ by function values from more then solely one previous or the current time point. For further details, we refer to the book [41] by Mickens. For the aforementioned three reasons, we suggest a non-standard finite-difference-method (NSFDM) given by

    $ {Sh,j+1Sh,jhj+1=αj+1Sh,j+1Ih,jN,Ih,j+1Ih,jhj+1=αj+1Sh,j+1Ih,jNβj+1Ih,j+1,Rh,j+1Rh,jhj+1=βj+1Ih,j+1,Sh,1=S0,Ih,1=I0,Rh,1=R0
    $
    (1.3)

    in Section 3 as our first main contribution, after summarizing analytical properties of the time-continuous problem formulation (1.1) in Section 2. In Section 3, we additionally prove unconditional preservation of non-negativity and linear convergence of (1.3) towards the time-continuous solution of (1.1). Additionally, (1.3) transfers to a uniquely solvable algorithm which preserves non-negativity intrinsically. Based on (1.3), we design a numerical algorithm to solve the inverse problem of finding the time-varying parameter functions in Section 4 as our next main contribution. Finally, we strengthen our theoretical findings in Section 5 by providing some numerical examples, especially one numerical example where linear convergence is stressed, in contrast to [15], where linear convergence was not shown numerically.

    Following [15], we take the following assumptions into consideration with regard to our time-continuous model (1.1):

    $ (1). $ The total population size $ N $ is fixed for all time $ t \geq 0 $, i.e., it holds

    $ N(t)=N=S0+I0+R0
    $
    (2.1)

    for all $ t \geq 0 $ and we assume $ I_{0} > 0 $ to really get an evolution in time of the infected subgroup of the population;

    $ (2). $ We divide the total population into three homogeneous subgroups, namely susceptible people (S), infected people (I) and recovered people (R). The equation

    $ N(t)=S(t)+I(t)+R(t)
    $
    (2.2)

    is valid for all $ t \geq 0 $;

    $ (3). $ The time-varying transmission rate $ \alpha \colon \left[0, \infty \right) \longrightarrow \left[0, \infty \right) $ is at least continuously differentiable once and there exist positive constants $ \alpha_{ \text{min}} > 0 $ and $ \alpha_{ \text{max}} > 0 $ such that

    $ αminα(t)αmax
    $
    (2.3)

    holds for all $ t \geq 0 $;

    $ (4). $ The time-varying recovery rate $ \beta \colon \left[0, \infty \right) \longrightarrow \left[0, \infty \right) $ is at least continuously differentiable once and there exist positive constants $ \beta_{ \text{min}} > 0 $ and $ \beta_{ \text{max}} > 0 $ such that

    $ βminβ(t)βmax
    $
    (2.4)

    holds for all $ t \geq 0 $.

    We summarize the following analytical properties of the time-continuous problem formulation (1.1) from article [15]:

    Theorem 1. Let all assumptions from (2.1)(2.4) for our time-continuous model (1.1) be fulfilled. The following statements hold true:

    (1). Possible solutions of (1.1) remain non-negative and bounded above, i.e.,

    $ {0S(t)N;0I(t)N;0R(t)N
    $
    (2.5)

    hold for all $ t \geq 0 $;

    (2). The time-continuous problem formulation (1.1) possesses exactly one globally unique solution for all $ t \geq 0 $;

    (3). First, $ S $ is monotonically decreasing and there exists $ S^{\star} > 0 $ such that $ \lim\limits_{t \to \infty} S \left(t \right) = S^{\star} $. Secondly, $ R $ is monotonically increasing and there exists $ R^{\star} > 0 $ such that $ \lim\limits_{t \to \infty} R \left(t \right) = R^{\star} $. Finally, it holds $ \lim\limits_{t \to \infty} I \left(t \right) = 0 $.

    Proof. $ (1). $ This statement's proof can be found in [15, Theorem 4].

    $ (2). $ Globally unique existence in time is proven in [15, Theorem 6].

    $ (3). $ Proof of the solution components' monotonicity and its long-time behavior is given in [15, Theorem 8].

    We want to note that $ S^{\star} $ and $ R^{\star} $ depend on the initial conditions.

    Let $ 0 = t_{1} < t_{2} < \ldots < t_{M} = T $ be an arbitrary and possibly non-equidistant partition of the simulation interval $ \left[0, T \right] $ with final simulation time $ T $. As abbreviations, we write $ f \left(t_{j} \right) : = f_{j} $ for time-continuous functions and $ f_{h} \left(t_{j} \right) : = f_{h, j} $ for time-discrete functions at time points $ t_{j} $ for all $ j \in \left\{ 1, 2, \ldots, M \right\} $. The time-discrete problem formulation of our time-continuous model (1.1) is given by (1.3) for all $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $. Here, $ h_{j + 1} : = t_{j + 1} - t_{j} $ denote non-equidistant time step sizes for all $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $. Additionally, the supremum norm on $ \left[0, T \right] $ for time-continuous functions is denoted by

    $ f(t):=supt[0,T]|f(t)|
    $

    while the same supremum norm on $ \left[t_{p}, t_{p + 1} \right] $ reads

    $ f(t),p+1:=supt[tp,tp+1]|f(t)|.
    $

    As our first result, our time-discrete problem formulation (1.3) conserves total population size $ N $ for all time points $ t_{j} \in \left[0, T \right] $ with $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    Theorem 2. Possible solutions of (1.3) fulfill

    $ N=Sh,j+Ih,j+Rh,j
    $
    (3.1)

    for all $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    Proof. This statement can be proven by induction. For $ j = 1 $, it is clear by our initial conditions that

    $ N=Sh,1+Ih,1+Rh,1
    $

    holds. Let us assume that

    $ N=Sh,j+Ih,j+Rh,j
    $

    is valid for an arbitrary $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $. Adding all three equations of (1.3), we obtain

    $ (Sh,j+1Sh,j)+(Ih,j+1Ih,j)+(Rh,j+1Rh,j)hj+1=0
    $

    and this implies

    $ Sh,j+1+Ih,j+1+Rh,j+1=Sh,j+Ih,j+Rh,j
    $

    which proves our assertion (3.1).

    As our next result, we show that approximate solutions of our time-continuous model (1.1) from our discretization model (1.3) remain non-negative and bounded.

    Theorem 3. Possible solutions of (1.3) remain non-negative and bounded for all $ t_{j} \in \left[0, T \right] $ with arbitrary $ j \in \left\{ 1, 2, \ldots, M \right\} $, i.e., these solultions fulfill

    $ {0Sh,jN;0Ih,jN;0Rh,jN
    $
    (3.2)

    for all $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    Proof. $ (1). $ By the first three equations of (1.3), we obtain

    $ Sh,j+1=Sh,j1+hj+1αj+1Ih,jN
    $
    (3.3)

    for all $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $,

    $ Ih,j+1=Ih,j1+hj+1αj+1Sh,j+1Ih,jN1+hj+1βj+1
    $
    (3.4)

    for all $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $ and

    $ Rh,j+1=Rh,j+hj+1βj+1Ih,j+1
    $
    (3.5)

    for all $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $.

    $ (2). $ By our assumption of non-negative initial values $ S_{h, 1}, I_{h, 1}, R_{h, 1} $ and by the three equations (3.3)–(3.5), we conclude that

    $ Sh,j0;Ih,j0;Rh,j0
    $

    hold for all $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    $ (3). $ By non-negativity and conservation of total population size (3.1) from Theorem 2, it follows that

    $ Sh,jN;Ih,jN;Rh,jN
    $

    hold for all $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    $ (4). $ Hence, all possible solutions of (1.3) remain non-negative and bounded.

    It directly follows that our numerical solution scheme (1.3) is uniquely solvable for all $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    Theorem 4. The discretization scheme (1.3), based on the explicit solutions (3.3)(3.5), is well-defined and uniquely solvable for all $ j \in \left\{ 1, 2, \ldots, M \right\} $.

    Proof. (3.3)–(3.5) prove that our numerical solution scheme (1.3) is well-defined. Unique solvability is a direct consequence of these equations as well. Hence, our assertion is proven.

    Remark 1. We briefly want to note why we call solutions (3.3)–(3.5) explicit. It obviously holds

    $ Sh,j+1=Sh,j1+hj+1αj+1Ih,jN=NSh,jN+hj+1αj+1Ih,j
    $
    (3.6)

    for all $ j \in \left\{ 1, 2, \ldots, M \right\} $ explicitly. Plugging (3.6) into (3.4) yields

    $ Ih,j+1=Ih,j(1+hj+1αj+1Sh,j+1Ih,jN)1+hj+1βj+1=Ih,j(1+hj+1αj+1Sh,jIh,jN+hj+1αj+1Ih,j)1+hj+1βj+1
    $
    (3.7)

    for all $ j \in \left\{ 1, 2, \ldots, M \right\} $ as an explicit formula. Finally, by using (3.7), we get

    $ Rh,j+1=Rh,j+hj+1βj+1Ih,j+1=Rh,j+hj+1βj+1Ih,j(1+hj+1αj+1Sh,jIh,jN+hj+1αj+1Ih,j)1+hj+1βj+1
    $
    (3.8)

    for all $ j \in \left\{ 1, 2, \ldots, M \right\} $. For these reasons, we talk about an explicit numerical solution scheme.

    We show that monotonicity properties and long-time behavior of the time-continuous problem formulation (1.1) translates to our time discretization (1.3).

    Theorem 5. 1). The sequence $ \left\{ S_{h, j} \right\}_{j = 1}^{M} $ is monotonically decreasing and there exists a non-negative constant $ S_{h}^{\star} $ such that $ \lim\limits_{j \to \infty} S_{h, j} = S_{h}^{\star} $ holds.

    2). The sequence $ \left\{ R_{h, j} \right\}_{j = 1}^{M} $ is monotonically increasing and there exists a non-negative constant $ R_{h}^{\star} $ such that $ \lim\limits_{j \to \infty} R_{h, j} = R_{h}^{\star} $ holds.

    3). It holds $ \lim\limits_{j \to \infty} I_{h, j} = 0 $ if also $ T \to \infty $ and $ h_{j + 1} \not = 0 $ are assumed.

    Proof. $ (1). $ Since $ \left\{ S_{h, j} \right\}_{j = 1}^{M} $ is non-negative and bounded above by the total population size $ N $, the inequality

    $ Sh,j+1=Sh,j1+hj+1αj+1Ih,jNSh,j
    $

    from (3.6) implies that this sequence is also monotonically decreasing which proves our first claim.

    $ (2). $ Since $ \left\{ R_{h, j} \right\}_{j = 1}^{M} $ is non-negative and bounded above by the total population size $ N $, the inequality

    $ Rh,j+1=Rh,j+hj+1βj+1Ij+1Rh,j
    $

    from (3.5) yields that this sequence is monotonically increasing which shows our second assertion.

    $ (3). $ We conclude

    $ 0Ij+1=Rh,j+1Rh,jhj+1βj+1Rh,j+1Rh,jhj+1βmin
    $

    from (3.5) and this shows that our third claim is also true.

    Remark 2. Both Theorems 1 and 5 imply that our time-continuous problem formulation (1.1) and our time-discrete problem formulation (1.3) converge towards the disease-free equilibrium state in the long run.

    We start this subsection by listing all our assumptions for our main result regarding linear convergence of the time-discrete solution towards the time-continuous one:

    ($ A_{1}1 $) We consider the time interval $ \left[0, T \right] $ where

    $ t1=0<t2<<tM1<tM:=T
    $

    holds;

    ($ A_{1}2 $) The initial conditions of the time-continuous and of the time-discrete problems coincide;

    ($ A_{1}3 $) The time-continuous solution functions $ S, I, R \, \colon \, \left[0, T \right] \longrightarrow \mathbb{R} $ should be twice continuously differentiable;

    ($ A_{1}4 $) The time-varying transmission rate $ \alpha \, \colon \, \left[0, T \right] \longrightarrow \left[0, \infty \right) $ and the time-varying recovery rate $ \beta \, \colon \, \left[0, T \right] \longrightarrow \left[0, \infty \right) $ should be once continuously differentiable;

    ($ A_{1}5 $) The time-varying transmission rate and the time-varying recovery rate should be bounded, i.e., there are non-negative constants $ \alpha_{ \text{min}}, \alpha_{ \text{max}}, \beta_{ \text{min}}, \beta_{ \text{max}} $ such that

    $ 0αminα(t)αmax
    $

    and

    $ 0βminβ(t)βmax
    $

    hold for all $ t \in \left[0, T \right] $;

    ($ A_{1}6 $) Set $ h : = \max\limits_{p \in \mathbb{N}} h_{p} $.

    Theorem 6. Let all assumptions ($ A_{1}1 $)–($ A_{1}6 $) be fulfilled. Then the time-discrete solution convergences linearly towards the time-continuous solution if $ h \to 0 $.

    Proof. We shortly explain our proof's strategy because it is relatively technical. If we first assume that our time-discrete solution coincides with the time-continuous solution at a certain time point $ t_{p} \in \left[0, T \right] $ for an arbitrary $ p \in \left\{ 1, 2, \ldots, M - 1 \right\} $, then we investigate the local error propagation to the next time point $ t_{p + 1} $. Secondly, we consider error propagation in time and finally, we examine accumulation of these errors. We denote time-discrete solutions, for example, by $ S_{h, p} $ and time-continuous solutions by $ S \left(t_{p} \right) $ at the same time point $ t_{p} $.

    1) For investigation of local errors, we assume that

    $ (tp,Sh,p)=(tp,S(tp));(tp,Ih,p)=(tp,I(tp))and(tp,Rh,p)=(tp,R(tp))
    $

    hold for an arbitrary $ p \in \left\{ 1, 2, \ldots, M - 1 \right\} $ on the time-interval $ \left[t_{p}, t_{p + 1} \right] $. Since we only take one time step into account, we denote corresponding time-discrete solutions by $ \widetilde{S_{h, p + 1}} $, $ \widetilde{I_{h, p + 1}} $ and $ \widetilde{R_{h, p + 1}} $.

    1.1) We have by (3.6) from Remark 1 that

    $ ~Sh,p+1=S(tp)(1+αp+1hp+1I(tp)N)=S(tp)S(tp)αp+1hp+1I(tp)N(1+αp+1hp+1I(tp)N)
    $
    (3.9)

    holds. This implies

    $ |S(tp+1)~Sh,p+1|(3.9)=|S(tp+1)S(tp)+S(tp)αp+1hp+1I(tp)N(1+αp+1hp+1I(tp)N)||tp+1tpS(τ)dτhp+1S(tp)|=:IS,1+|hp+1S(tp)+S(tp)αp+1hp+1I(tp)N(1+αp+1hp+1I(tp)N)|=:IIS,1.
    $
    (3.10)

    By the mean value Theorem, there exists $ \xi_{S, 1} \in \left(t_{p}, t_{p + 1} \right) $ such that we obtain

    $ |S(ξS,1)|=|S(τ)S(tp)τtp|S(t).
    $
    (3.11)

    This yields

    $ IS,1=|tp+1tp{S(τ)S(tp)}dτ|=|tp+1tp(S(τ)S(tp))(τtp)(τtp)dτ|(3.11)12h2p+1S(t).
    $
    (3.12)

    For $ II_{S, 1} $, we notice that

    $ IIS,1=|hp+1S(tp)+S(tp)αp+1hp+1I(tp)N(1+αp+1hp+1I(tp)N)|(1.1)=|αphp+1S(tp)I(tp)N+S(tp)αp+1hp+1I(tp)N(1+αp+1hp+1I(tp)N)|=|hp+1S(tp)N||αpI(tp)+αp+1I(tp)(1+αp+1hp+1I(tp)N)|hp+1|αpI(tp)(1+αp+1hp+1I(tp)N)+αp+1I(tp)(1+αp+1hp+1I(tp)N)|hp+1|(αp+1αp)I(tp)(1+αp+1hp+1I(tp)N)|+hp+1|αpαp+1hp+1I(tp)I(tp)N(1+αp+1hp+1I(tp)N)|hp+1N|αp+1αp|=:IIIS,1+h2p+1α2maxN
    $

    and hence,

    $ IIS,Ihp+1N|αp+1αp|=:IIIS,1+h2p+1α2maxN
    $
    (3.13)

    holds due to boundedness of time-continuous solutions. Using again the mean value theorem for $ III_{S, 1} $, we see that there is $ \xi_{\alpha, 1} \in \left(t_{p}, t_{p + 1} \right) $ such that we obtain

    $ |α(ξα,1)|=|αp+1αptp+1tp|hp+1hp+1α(t).
    $
    (3.14)

    Hence, we conclude

    $ IIIS,1=|(αp+1αp)|(3.14)hp+1α(t)
    $
    (3.15)

    for $ III_{S, 1} $. Plugging (3.15) into (3.13) yields

    $ IIS,1hp+1NIIIS,1+h2p+1α2maxN(3.15)h2p+1Nα(t)+h2p+1α2maxN=h2p+1(α(t)N+α2maxN).
    $
    (3.16)

    Thus, we finally obtain

    $ |S(tp+1)~Sh,p+1|(3.10)=IS,1+IIS,112h2p+1S(t)+h2p+1(α(t)N+α2maxN)=(12S(t)+α(t)N+α2maxN)=:Cloc,Sh2p+1.
    $
    (3.17)

    1.2) By (3.7) from Remark 1, we get

    $ ~Ih,p+1=I(tp)(1+αp+1hp+1~Sh,p+1N)(1+βp+1hp+1).
    $
    (3.18)

    Hence, it follows

    $ |I(tp+1)~Ih,p+1|(3.18)=|I(tp+1)I(tp)(1+αp+1hp+1~Sh,p+1N)(1+βp+1hp+1)|=|{I(tp+1)I(tp)}+{I(tp)I(tp)(1+αp+1hp+1~Sh,p+1N)(1+βp+1hp+1)}|=|tp+1tpI(τ)dτ+I(tp)(βp+1hp+1αp+1hp+1~Sh,p+1N)(1+βp+1hp+1)||tp+1tpI(τ)dτhp+1I(tp+1)|=:II,1+|hp+1I(tp+1)+I(tp)(βp+1hp+1αp+1hp+1~Sh,p+1N)(1+βp+1hp+1)|=:III,1.
    $
    (3.19)

    By the mean value theorem, there exists $ \xi_{I, 1} \in \left(t_{p}, t_{p + 1} \right) $ such that

    $ I(ξI,1)=I(τ)I(tp+1)τtp+1
    $
    (3.20)

    is valid. This implies

    $ II,1=|tp+1tp(I(τ)I(tp+1))(τtp+1)(τtp+1)dτ|(3.20)=|tp+1tpI(ξI,1)(τtp+1)dτ|12I(t)h2p+1.
    $
    (3.21)

    For $ II_{I, 1} $, we observe

    $ III,1=|hp+1I(tp+1)+I(tp)(βp+1hp+1αp+1hp+1~Sh,p+1N)(1+βp+1hp+1)|=hp+1|αp+1I(tp+1)S(tp+1)Nβp+1I(tp+1)+I(tp)(βp+1hp+1αp+1hp+1~Sh,p+1N)(1+βp+1hp+1)|=hp+11+βp+1hp+1|αp+1I(tp+1)S(tp+1)N(1+βp+1hp+1)βp+1I(tp+1)(1+βp+1hp+1)+βp+1I(tp)αp+1I(tp)~Sh,p+1N|
    $
    (3.22)

    and can conclude

    $ III,1(3.22)hp+1|αp+1N(I(tp+1)S(tp+1)I(tp)~Sh,p+1)|=:IIII,1+hp+1|αp+1βp+1hp+1I(tp+1)S(tp+1)N|=:IVI,1+hp+1|βp+1(I(tp)I(tp+1))|=:VI,1+hp+1|β2p+1I(tp+1)hp+1|=:VII,1.
    $
    (3.23)

    By boundedness and the mean value theorem, we receive the following estimates

    $ VII,1hp+1Nβ2max,VI,1hp+1βmaxI(t),IVI,1hp+1Nαmaxβmax
    $
    (3.24)

    for $ VI_{I, 1} $, $ V_{I, 1} $ and $ IV_{I, 1} $. By boundedness, the triangle inequality and the mean value theorem, we can deduce

    $ IIII,1=|αp+1N||{I(tp+1)S(tp+1)I(tp)S(tp+1)}+{I(tp)S(tp+1)I(tp)~Sh,p+1}|αmaxN|I(tp+1)I(tp)||S(tp+1)|+αmaxN|S(tp+1)~Sh,p+1||I(tp)|(3.17)αmaxhp+1I(t)+αmaxh2p+1Cloc,S
    $
    (3.25)

    and consequently, this implies

    $ III,1hp+1{IIII,1+IVI,1+VI,1+VII,1}hp+1{αmaxhp+1I(t)+αmaxh2p+1Cloc,S+αmaxβmaxhp+1N+βmaxhp+1I(t)+β2maxhp+1N}.
    $
    (3.26)

    Hence, we conclude that

    $ |I(tp+1)~Ih,p+1|(3.19)II,1+III,1(3.26)12h2p+1I(t)+h2p+1{αmaxI(t)+αmaxhp+1Cloc,S+αmaxβmaxN+βmaxI(t)+β2maxN}
    $
    (3.27)

    holds. By defining

    $ Cloc,I:={12I(t)+αmaxI(t)+αmaxhp+1Cloc,S+αmaxβmaxN+βmaxI(t)+β2maxN},
    $
    (3.28)

    we finally deduce

    $ |I(tp+1)~Ih,p+1|(3.28)h2p+1Cloc,I
    $
    (3.29)

    from (3.27).

    1.3) By (3.8) from Remark 1, we get

    $ ~Rh,p+1=R(tp)+hp+1βp+1~Ih,p+1.
    $
    (3.30)

    Hence, it follows

    $ |R(tp+1)~Rh,p+1|(3.30)=|R(tp+1)R(tp)hp+1βp+1~Ih,p+1||tp+1tpR(τ)dτhp+1βp+1I(tp+1)|+hp+1βmax|I(tp+1)~Ih,p+1|.
    $
    (3.31)

    By using similar argument as in previous steps with the help of the mean value theorem and (3.29), we can deduce

    $ |R(tp+1)~Rh,p+1|12h2p+1R(t)+h3p+1βmaxCloc,I=h2p+1{12R(t)+βmaxCloc,Ihp+1}=:Cloc,R.
    $
    (3.32)

    1.4) By setting

    $ Cloc:=max{Cloc,S,Cloc,I,Cloc,R},
    $
    (3.33)

    we conclude

    $ Ap+1:=max{|S(tp+1)~Sh,p+1|,|I(tp+1)~Ih,p+1|,|R(tp+1)~Rh,p+1|}Cloch2p+1.
    $
    (3.34)

    2) Normally, the points $ \left(t_{p}, S_{p} \right) $, $ \left(t_{p}, I_{p} \right) $ and $ \left(t_{p}, R_{p} \right) $ do not exactly lie on the graph of the time-continuous solution. We then must investigate how procedural errors such as $ \left| S_{h, p} - S \left(t_{p} \right) \right| $, $ \left| I_{h, p} - I \left(t_{p} \right) \right| $ or $ \left| R_{h, p} - R \left(t_{p} \right) \right| $ propagate to the $ \left(p + 1 \right) $-th time step.

    2.1) We observe that

    $ |Sh,p+1~Sh,p+1|=|(Sh,pSh,pαp+1hp+1Ih,pN1+αp+1hp+1Ih,pN)(S(tp)S(tp)αp+1hp+1I(tp)N1+αp+1hp+1I(tp)N)||Sh,pS(tp)|+αmaxhp+1NIS,2
    $
    (3.35)

    holds where $ I_{S, 2} $ is given and estimated by

    $ IS,2=|Sh,pIh,p(1+αp+1hp+1I(tp)N)S(tp)I(tp)(1+αp+1hp+1Ih,pN)(1+αp+1hp+1Ih,pN)(1+αp+1hp+1I(tp)N)||Sh,pIh,pS(tp)I(tp)|=:IIS,2+αmaxhp+1N|Sh,pIh,pI(tp)S(tp)I(tp)Ih,p|=:IIIS,2.
    $
    (3.36)

    We have

    $ IIS,2=|Sh,pIh,pS(tp)I(tp)||Sh,p(Ih,pI(tp))|+|I(tp)(Sh,pS(tp))|N|Ih,pI(tp)|+N|Sh,pS(tp)|
    $
    (3.37)

    and

    $ IIIS,2=|Sh,pIh,pI(tp)S(tp)I(tp)Ih,p|=|I(tp)(Sh,pIh,pS(tp)Ih,p)|N|Ih,p(Sh,pS(tp))|N2|Sh,pS(tp)|.
    $
    (3.38)

    Plugging (3.37) and (3.38) into (3.36) yields

    $ IS,2IIS,2+αmaxhp+1NIIIS,2N|Ih,pI(tp)|+N|Sh,pS(tp)|+αmaxhp+1N|Sh,pS(tp)|=N|Ih,pI(tp)|+N(1+αmaxhp+1)|Sh,pS(tp)|.
    $
    (3.39)

    Using (3.35), we obtain

    $ |Sh,p+1~Sh,p+1||Sh,pS(tp)|+αmaxhp+1NIS,2(1+αmaxhp+1+(αmaxhp+1)2)|Sh,pS(tp)|+αmaxhp+1|Ih,pI(tp)|.
    $
    (3.40)

    2.2) We see that

    $ |Ih,p+1~Ih,p+1|=|(Ih,pIh,p(βp+1hp+1αp+1hp+1Sh,p+1N)1+βp+1hp+1)(I(tp)I(tp)(βp+1hp+1αp+1hp+1~Sh,p+1N)1+βp+1hp+1)|
    $
    (3.41)

    holds and get

    $ |Ih,p+1~Ih,p+1||Ih,pI(tp)|+βmaxhp+1|Ih,pI(tp)|+αmaxhp+1N|Ih,pSh,p+1I(tp)~Sh,p+1||Ih,pI(tp)|+βmaxhp+1|Ih,pI(tp)|+αmaxhp+1N|Ih,pSh,p+1I(tp)Sh,p+1|+αmaxhp+1N|I(tp)Sh,p+1I(tp)~Sh,p+1|(1+hp+1(αmax+βmax))|Ih,pI(tp)|+αmaxhp+1|Sh,p+1~Sh,p+1|.
    $
    (3.42)

    Plugging (3.40) into (3.42) yields

    $ |Ih,p+1~Ih,p+1|(1+hp+1(αmax+βmax))|Ih,pI(tp)|+αmaxhp+1{(1+αmaxhp+1+(αmaxhp+1)2)|Sh,pS(tp)|+αmaxhp+1|Ih,pI(tp)|}=(1+hp+1(αmax+βmax)+α2maxh2p+1)|Ih,pI(tp)|+(αmaxhp+1+(αmaxhp+1)2+(αmaxhp+1)3)|Sh,pS(tp)|.
    $
    (3.43)

    2.3) We obtain

    $ |Rh,p+1~Rh,p+1||Rh,pR(tp)|+βmaxhp+1|Ih,p+1~Ih,p+1|(3.43)|Rh,pR(tp)|+(βmaxhp+1+βmaxh2p+1(αmax+βmax)+βmaxα2maxh3p+1)|Ih,pI(tp)|+(βmaxαmaxh2p+1+βmaxα2maxh3p+1+βmaxα3maxh4p+1)|Sh,pS(tp)|
    $
    (3.44)

    2.4) For our summary, we define

    $ Bp+1:=max{|Sh,p+1~Sh,p+1|,|Ih,p+1~Ih,p+1|,|Rh,p+1~Rh,p+1|}
    $
    (3.45)

    for all $ p \in \left\{ 0, 1, \ldots, M - 1 \right\} $ and

    $ Dp:=max{|Sh,pS(tp)|,|Ih,pI(tp)|,|Rh,pR(tp)|}
    $
    (3.46)

    for all $ p \in \left\{ 1, \ldots, M \right\} $. By (3.40), (3.43) and (3.44), we define the constants

    $ CS,prop:=1+hp+12αmax+h2p+1α2max=1+hp+1(2αmax+hp+1α2max)=:~CS,prop;CI,prop:=1+hp+1(2(αmax+βmax)+2α2maxhp+1+α3maxh2p+1)=:~CI,prop;CR,prop:=1+hp+1(βmax+2βmax(αmax+βmax)hp+1+2βmaxα2maxh2p+1+βmaxα3maxh3p+1)=:1+hp+1~CR,prop
    $

    and

    $ Cprop:=max{~CS,prop,~CI,prop,~CR,prop}.
    $
    (3.47)

    Hence, it holds

    $ Bp+1(1+hp+1Cprop)Dp.
    $
    (3.48)

    3) We obtain

    $ Dp+1Bp+1+Ap+1(1+hp+1Cprop)Dp+Cloch2p+1(1+hp+1Cprop)((1+hpCprop)Dp1+Cloch2p)+Cloch2p+1
    $
    (3.49)

    and inductively

    $ Dp+1exp(CpropT)D1+exp(CpropT)ClochT
    $
    (3.50)

    which proves linear convergence of the time-discrete solution towards the time-continuous solution.

    Our algorithm for our proposed non-standard finite-difference-method for (1.1) is portrayed in Algorithm 2 below.

    Algorithm 2 Algorithmic summary for our proposed non-standard finite-difference-method (3.2) for (1.1)
    1: Inputs
    2: Time vector $ \left(t_{1}, \ldots, t_{M} \right)^{T} \in \mathbb{R}^{M} $
    3: Time-varying transmission rates $ \left(\alpha_{1}, \ldots, \alpha_{M} \right)^{T} \in \mathbb{R}^{M} $
    4: Time-varying recovery rates $ \left(\beta_{1}, \ldots, \beta_{M} \right)^{T} \in \mathbb{R}^{M} $
    5: Initialize $ \textbf{S} : = \textbf{0}, \textbf{I} : = \textbf{0}, \textbf{R} : = \textbf{0} \in \mathbb{R}^{M} $
    6: Initial conditions $ \textbf{S}_{1} : = S_{1} $, $ \textbf{I}_{1} : = I_{1} $ and $ \textbf{R}_{1} : = R_{1} $
    7: Total population size $ N : = \textbf{S}_{1} + \textbf{I}_{1} + \textbf{R}_{1} $
    8: Non-Standard Finite-Difference-Method
    9: for $ j \in \left\{ 1, \ldots, M - 1 \right\} $ do
    10:  $ h_{j + 1} : = t_{j + 1} - t_{j} $
    11:  $ \textbf{S}_{j + 1} : = \dfrac{\textbf{S}_{j}}{1 + h_{j + 1} \cdot \alpha_{j + 1} \cdot \dfrac{\textbf{I}_{j}}{N}} $ from (3.6)
    12:  $ \textbf{I}_{j + 1} : = \dfrac{\textbf{I}_{j} + h_{j + 1} \cdot \alpha_{j + 1} \cdot \dfrac{\textbf{S}_{j + 1} \cdot \textbf{I}_{j}}{N}}{1 + h_{j + 1} \cdot \beta_{j + 1}} $ from (3.7)
    13:  $ \textbf{R}_{j + 1} : = \textbf{R}_{j} + h_{j + 1} \cdot \beta_{j + 1} \cdot \textbf{I}_{j + 1} $ from (3.8)
    14: end for
    15: Outputs
    16: Calculated vectors $ \textbf{S}, \textbf{I}, \textbf{R} \in \mathbb{R}^{M} $

    We consider the parameter identification problem (1.3) for all $ j \in \left\{ 1, 2, \ldots, M - 1 \right\} $ where we take the following assumptions into account:

    ($ A_{2}1 $) The sequences $ \left\{ S_{h, p} \right\}_{p = 1}^{M} $, $ \left\{ I_{h, p} \right\}_{p = 1}^{M} $ and $ \left\{ R_{h, p} \right\}_{p = 1}^{M} $ are given;

    ($ A_{2}2 $) It holds $ 0 \leq S_{h, p} \leq N $ for all $ p \in \left\{ 1, \ldots, M \right\} $ and the sequence $ \left\{ S_{h, p} \right\}_{p = 1}^{M} $ is monotonically decreasing;

    ($ A_{2}3 $) It holds $ 0 \leq I_{h, p} \leq N $ for all $ p \in \left\{ 1, \ldots, M \right\} $;

    ($ A_{2}4 $) It holds $ 0 \leq R_{h, p} \leq N $ for all $ p \in \left\{ 1, \ldots, M \right\} $ and the sequence $ \left\{ R_{h, p} \right\}_{p = 1}^{M} $ is monotonically increasing;

    ($ A_{2}5 $) We assume $ S_{h, p} \not = 0 $, $ I_{h, p} \not = 0 $ and $ R_{h, p} \not = 0 $ for all $ p \in \left\{ 1, \ldots, M \right\} $.

    Our idea now is to reduce (1.3) to

    $ {Sh,p+1Sh,php+1=αp+1Sh,p+1Ih,pN;Rh,p+1Rh,php+1=βp+1Ih,p+1
    $
    (4.1)

    for all $ p \in \left\{ 1, \ldots, M - 1 \right\} $. Hence, we obtain the sought time-varying transmission rate and the sought time-varying recovery rate by

    $ {αp+1=NSh,p+1Ih,p(Sh,pSh,p+1)hp+1;βp+1=1Ih,p+1(Rh,p+1Rh,p)hp+1
    $
    (4.2)

    for all $ p \in \left\{ 1, \ldots, M - 1 \right\} $. Finally, we have explicit formulas for the time-varying transmission and recovery rates at every time point which we can evaluate by the known data from the numbers of susceptible, infected and recovered people. We have the following theorem.

    Theorem 7. Let all assumptions ($ A_{2}1 $)–($ A_{2}5 $) be fulfilled. (4.1) for parameter identification is uniquely solvable for all $ p \in \left\{ 1, \ldots, M - 1 \right\} $ and it holds $ \alpha_{p + 1} \geq 0 $ and $ \beta_{p + 1} \geq 0 $ for all $ p \in \left\{ 1, \ldots, M - 1 \right\} $.

    Proof. 1) Unique solvability is a direct consequence of (4.2).

    2) $ \left\{ S_{h, p} \right\}_{p = 1}^{M} $ is monotonically decreasing. Hence, $ S_{h, p} - S_{h, p + 1} \geq 0 $ for all $ p \in \left\{ 1, \ldots, M - 1 \right\} $ and we have

    $ αp+1=NSh,p+1Ih,p(Sh,pSh,p+1)hp+10.
    $

    3) $ \left\{ R_{h, p} \right\}_{p = 1}^{M} $ is monotonically increasing. Hence, $ R_{h, p + 1} - R_{h, p} \geq 0 $ for all $ p \in \left\{ 1, \ldots, M - 1 \right\} $ and we have

    $ βp+1=1Ih,p+1(Rh,p+1Rh,p)hp+10.
    $

    This finishes our claim's proof.

    Our algorithmic summary for our proposed parameter identification problem (4.2) reads:

    Algorithm 3 Algorithmic summary for our proposed parameter identification algorithm for (4.2)
    1: Inputs
    2: Time vector $ \left(t_{1}, \ldots, t_{M} \right)^{T} \in \mathbb{R}^{M} $
    3: Sequence of susceptible population $ \left\{ S_{h, p} \right\}_{p = 1}^{M} $ saved in one vector $ \textbf{S} \in \mathbb{R}^{M} $
    4: Sequence of infected population $ \left\{ I_{h, p} \right\}_{p = 1}^{M} $ saved in one vector $ \textbf{I} \in \mathbb{R}^{M} $
    5: Sequence of recovered population $ \left\{ R_{h, p} \right\}_{p = 1}^{M} $ saved in one vector $ \textbf{R} \in \mathbb{R}^{M} $
    6: Constant total population size $ N : = \textbf{S}_{1} + \textbf{I}_{1} + \textbf{R}_{1} $
    7: Initialize vectors $ {\alpha} : = \textbf{0} \in \mathbb{R}^{M} $ and $ {\beta} : = \textbf{0} \in \mathbb{R}^{M} $
    8: Parameter Identification Algorithm
    9: for $ j \in \left\{ 1, \ldots, M - 1 \right\} $ do
    10:  $ h_{j + 1} : = t_{j + 1} - t_{j} $
    11:  $ {\alpha}_{j + 1} : = \dfrac{N}{\textbf{S}_{h, j + 1} \cdot \textbf{I}_{h, j}} \cdot \dfrac{\left(\textbf{S}_{h, j} - \textbf{S}_{h, j + 1} \right)}{h_{j + 1}} $
    12:  $ {\beta}_{j + 1} = \dfrac{1}{\textbf{I}_{h, j + 1}} \cdot \dfrac{\left(\textbf{R}_{h, j + 1} - \textbf{R}_{h, j} \right)}{h_{j + 1}} $
    13: end for
    14: Outputs
    15: Calculated vectors $ {\alpha}, {\beta} \in \mathbb{R}^{M} $

    In this section, we want to strengthen our theoretical findings by two numerical examples. We apply GNU OCTAVE for our implementation [43].

    The initial conditions for our first example read $ S \left(0 \right) = 5000 $, $ I \left(0 \right) = 5000 $, $ R \left(0 \right) = 0 $, $ N = 10,000 $, $ \alpha = 0.5 $, $ \beta = 0.1 $ and our simulation interval is given by $ \left[0, T \right] $ with $ T = 70 $. We compare three numerical algorithms, namely our proposed non-standard finite-difference-method (NSFDM), the explicit Eulerian scheme (EE) and an explicit Runge-Kutta scheme with two stages (RK2) and use equidistant time step sizes $ h $ for our computations.

    As it can be clearly seen in Figures 1 and 2, only our novel non-standard finite-difference-method is unconditionally non-negativity-preserving and stable. Both explicit standard time-discretizations suffer from time-step restrictions to obtain non-negativity preservation. In Figure 3, we observe that all three numerical algorithms possess same properties for smaller time step sizes $ h $. On the positive side for all algorithms, they conserve total population size $ N $ as depicted in Figure 4.

    Figure 1.  Example 1: Instabilities for time step size $ h = 7.5 $ for (EE, not shown) and (RK2, shown).
    Figure 2.  Example 1: Instabilities and negativity for time step size $ h = 5 $ for (EE) and (RK2).
    Figure 3.  Example 1: Similar behavior of all three algorithms for time step size $ h = 0.1 $.
    Figure 4.  Example 1: Total population size conservation of all three algorithms for time step size $ h = 0.1 $.

    Finally, underlining our convergence result from Theorem 6, we compare results of our non-standard finite-difference-method (NSFDM) for different time step sizes $ h $ with a fine scale Runge-Kutta scheme of order 4 (RK4) at $ T = 70 $ where a fine time step size of $ 0.0001 $ is applied for (RK4). The errors

    ● Error for $ S \left(t \right) $: $ \left| S_{ \text{NSFDM}, M} - S_{ \text{RK4}, M} \right| $;

    ● Error for $ I \left(t \right) $: $ \left| I_{ \text{NSFDM}, M} - I_{ \text{RK4}, M} \right| $;

    ● Error for $ R \left(t \right) $: $ \left| R_{ \text{NSFDM}, M} - R_{ \text{RK4}, M} \right| $

    are used for this purpose. It can be clearly seen from Table 1 that the theoretical convergence order is achieved by our non-standard finite-difference-method.

    Table 1.  Errors for example 1 with (NSFDM).
    $ h $ $ 0.03125 $ $ 0.0625 $ $ 0.125 $ $ 0.25 $ $ 0.5 $ $ 1.0 $
    Error for $ S \left(t \right) $ $ 0.27673 $ $ 0.55049 $ $ 1.0889 $ $ 2.1307 $ $ 4.0817 $ $ 7.5129 $
    Error for $ I \left(t \right) $ $ 0.14805 $ $ 0.2971 $ $ 0.59806 $ $ 1.2115 $ $ 2.4843 $ $ 5.2145 $
    Error for $ R \left(t \right) $ $ 0.41436 $ $ 0.83717 $ $ 1.6766 $ $ 3.3317 $ $ 6.5556 $ $ 12.717 $

     | Show Table
    DownLoad: CSV

    We consider real-world data from Spain [44]. $ N = 47,370,000 $ is the total population size of Spain. The cumulative cases of infected people are modified such

    $ S(t)=NI(t)R(t);I(t)=Nconfirmed(t)Ndead(t)Nrecovered(t);R(t)=Ndead(t)+Nrecovered(t)
    $

    hold from the given data. We take $ \alpha \left(t \right) = 0.52 \cdot \text{e}^{-0.03 \cdot t} $ and $ \beta \left(t \right) = 0.045 $. For our simulation, we additionally assume $ h = 0.75 $ as an equidistant time step size.

    Our estimated transmission rate $ \alpha \left(t \right) $ and constant recovery rate $ \beta \left(t \right) $ are user-chosen and adapted to the data in Figures 6 and 7, while the modified data for infected and recovered people are depicted in Figure 5. We solely consider the beginning of the pandemic because, later, vaccines have been available and reinfections have occurred after virus modifications. In Figure 8, we see results of our numerical simulations with $ h = 0.75 $. Results are relatively sensitive with respect to time step sizes and estimates rates, e.g., time-varying transmission rates. Hence, we can only conclude that good data acquisition and different predictions in different settings are crucial for future pandemics to make predictions for courses of such future epidemics.

    Figure 5.  Example 2: Data for infected ($ I \left(t \right) $) and recovered ($ R \left(t \right) $) from Spain.
    Figure 6.  Example 2: Estimated time-varying transmission rate ($ \alpha \left(t \right) $) from data.
    Figure 7.  Example 2: Estimated time-varying recovery rate ($ \beta \left(t \right) $) from data.
    Figure 8.  Example 2: Numerical simulation for estimated rates.

    In this article, we first summarized some analytical properties of a time-continuous epidemiological model (1.1) as given in [15]. Afterwards, we designed a new non-standard finite-difference-method (1.3) as a time-discrete version of the continuous model. In Section 3, we contributed our main results on our proposed non-standard finite-difference-method (1.3). Namely, we investigated unconditionally non-negativity preservation, conservation of total population size, convergence towards a disease-free equilibrium point and linear convergence by letting the time size $ h \to 0 $. Conclusively for our theoretical findings, we developped a simple parameter identification algorithm in Section (4). In Section 5, we gave two numerical examples to underline our theoretical findings. Comparing Algorithms 1 and 2, we conclude that the latter one is definitely easier to implement and it can serve as a starting point for constructing higher order time discretization schemes for (1.1). Regarding practical applications, our proposed time-discretization by a non-standard finite-difference-method preserves all desired properties from the time-continuous model which are expected from our real-word intuition. Additionally, the simple parameter identification algorithm produce fast graphs from given data where trends in time-varying transmission rates such as contact restrictions might be seen from.

    There might be different directions to extend our work:

    ● It seems straightforward to integrate further compartments into our model such that we can additionally investigate influences such as, for example, vaccines or the simple extension to the SEIR model. For example, we readily can construct

    $ {Sh,j+1Sh,jhj+1=αj+1Sh,j+1Eh,jN,Eh,j+1Eh,jhj+1=αj+1Sh,j+1Eh,jNγj+1Eh,j+1,Ih,j+1Ih,jhj+1=γj+1Eh,j+1βj+1Ih,j+1,Rh,j+1Rh,jhj+1=βj+1Ih,j+1,Sh,1=S0,Eh,1=E0,Ih,1=I0,Rh,1=R0
    $
    (6.1)

    as one variant of a non-standard finite-difference-method for the SEIR model and it follows directly that this scheme conserves total population size $ N $ and can be written in an explicit fashion as our proposed non-standard finite-difference-method for our considered SIR model by

    $ {Sh,j+1=Sh,j1+hj+1αj+1Eh,jN,Eh,j+1=Eh,j+hj+1αj+1Sh,j+1Eh,jN1+hj+1γj+1,Ih,j+1=Ih,j+hj+1γj+1Eh,j+11+hj+1βj+1,Rh,j+1=Rh,j+hj+1βj+1Ih,j+1
    $
    (6.2)

    to see that it also preserves non-negativity assuming positive initial conditions. One disadvantage is that there are different possibilities to construct such non-standard finite-difference-methods by non-local approximations and a suitable choice depends on the time-continuous properties which should be transferred to the time-discrete case. However, as seen by this example, it is flexible to construct different possibilities of algorithms and they are easy to implement;

    ● It might be of interest to construct higher order non-standard finite-difference-methods from our first-order non-standard finite-difference-method in order to get higher accuracy although, especially, our second example demonstrated that predictions of such models need to be considered with care. One straightforward way to improve accuracy, and hence, the convergence order of numerical solution schemes is application of Richardson's extrapolation [45];

    ● One can use our discretization model for the discretization of optimal control problems for minimizing certain epidemic aspect such as eradication time [46].

    Our code and used data from [44] can be found under https://github.com/bewa87/2023-SIR-FirstOrder-NSFDM.

    We thank the editor and both anonymous reviewers for their valuable comments which definitely helped us to improve quality and readability of our manuscript.

    The authors declare there is no conflict of interest.

    [1] Lollo VD (1980) Temporal integration in visual memory.J Exp Psychol Gen 109: 75-97.
    [2] Schlangen D, Barenholtz E (2015) Intrinsic and contextual features in object recognition.J Vis 15: 1-15.
    [3] Pan JS, Bingham GP (2013) With an eye to low vision: Optic flow enables perception despite image blur.Optom Vis Sci 90: 1119-1127.
    [4] Giladgutnick S, Yovel G, Sinha P (2012) Recognizing degraded faces: The contribution of configural and featural cues.Percept 41: 1497-1511.
    [5] Gollin ES (1960) Developmental studies of visual recognition of incomplete objects.Percept Mot Skills 11: 289-298.
    [6] Gollin ES (1961) Further studies of visual recognition of incomplete objects.Percept Mot Skills 13: 307-314.
    [7] Gollin ES (1962) Factors affecting the visual recognition of incomplete objects.Percept Mot Skills 15: 583-590.
    [8] Greene E (2007) Recognition of objects displayed with incomplete sets of discrete boundary dots.Percept Mot Skills 104: 1043-1059.
    [9] Greene E (2007) Retinal encoding of ultrabrief shape recognition cues.Plos One 2: e871.
    [10] Greene E, Visani A (2015) Recognition of letters displayed as briefly flashed dot patterns.Atten Percept Psychphys 77: 1955-1969.
    [11] Greene E (2016) Information persistence evaluated with low-density dot patterns.Acta Psychol 170: 215-225.
    [12] Greene E, Hautus MJ (2017) Demonstrating invariant encoding of shapes using a matching judgment protocol.AIMS Neurosci 4: 120-147.
    [13] Green DM, Swets JA (1966) Signal detection theory and psychophysics.Robert E. Krieger 25: 1478-1481.
    [14] Hautus MJ, Hout DV, Lee HS (2009) Variants of A Not-A and 2AFC tests: Signal Detection Theory models.Food Qual Prefer 20: 222-229.
    [15] Hautus MJ (1995) Corrections for extreme proportions and their biasing effects on estimated values of d.Behav Res Methods Instrum Comput 27: 46-51.
    [16] Miller J (1996) The sampling distribution of d.Percept Psychophys 58: 65-72.
    [17] Hautus J (2012) SDT Assistant (version 1.0) [Software].Available from: http://hautus.org.
    [18] Hautus MJ (1997) Calculating estimates of sensitivity from group data: Pooled versus averaged estimators.Behav Res Methods Instrum Comput 29: 556-562.
    [19] Macmillan NA, Creelman CD (2005) Detection theory: A user's guide, 2nd Ed .
    [20] Greene E, Onwuzulike O (2017) What constitutes elemental shape information for biological vison.Trends Artif Intel 1: 22-26.
    [21] Greene E, Ogden RT (2012) Evaluating the contribution of shape attributes to recognition using the minimal transient discrete cue protocol.Behav Brain Funct 8: 53.
    [22] Phillips WA (1974) On the distinction between sensory storage and short-term visual memory.Percept Psychophys 16: 283-290.
    [23] Coltheart M (1980) Iconic memory and visible persistence.Percept Psychophys 27: 183-228.
    [24] Greene E (2007) Information persistence in the integration of partial cues for object recognition.Percept Psychophys 69: 772-784.
    [25] Greene E (2007) The integration window for shape cues is a function of ambient illumination.Behav Brain Funct 3: 15.
  • This article has been cited by:

    1. Hai-yang Xu, Heng-you Lan, Fan Zhang, General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to Stampacchia variational inequality, 2022, 41, 2238-3603, 10.1007/s40314-022-01890-7
    2. Yan Wang, Rui Wu, Shanshan Gao, The Existence Theorems of Fractional Differential Equation and Fractional Differential Inclusion with Affine Periodic Boundary Value Conditions, 2023, 15, 2073-8994, 526, 10.3390/sym15020526
    3. Sumati Kumari Panda, Thabet Abdeljawad, Fahd Jarad, Chaotic attractors and fixed point methods in piecewise fractional derivatives and multi-term fractional delay differential equations, 2023, 46, 22113797, 106313, 10.1016/j.rinp.2023.106313
    4. Philip N. A. Akuka, Baba Seidu, C. S. Bornaa, Marko Gosak, Mathematical Analysis of COVID-19 Transmission Dynamics Model in Ghana with Double-Dose Vaccination and Quarantine, 2022, 2022, 1748-6718, 1, 10.1155/2022/7493087
    5. SUMATI KUMARI PANDA, ABDON ATANGANA, THABET ABDELJAWAD, EXISTENCE RESULTS AND NUMERICAL STUDY ON NOVEL CORONAVIRUS 2019-NCOV/ SARS-COV-2 MODEL USING DIFFERENTIAL OPERATORS BASED ON THE GENERALIZED MITTAG-LEFFLER KERNEL AND FIXED POINTS, 2022, 30, 0218-348X, 10.1142/S0218348X22402149
    6. Tahair Rasham, Separate families of fuzzy dominated nonlinear operators with applications, 2024, 70, 1598-5865, 4271, 10.1007/s12190-024-02133-0
    7. Sumati Kumari Panda, Thabet Abdeljawad, A. M. Nagy, On uniform stability and numerical simulations of complex valued neural networks involving generalized Caputo fractional order, 2024, 14, 2045-2322, 10.1038/s41598-024-53670-4
    8. Sumati Kumari Panda, Kumara Swamy Kalla, A.M. Nagy, Limmaka Priyanka, Numerical simulations and complex valued fractional order neural networks via (ɛ – μ)-uniformly contractive mappings, 2023, 173, 09600779, 113738, 10.1016/j.chaos.2023.113738
    9. Hammed Anuoluwapo Abass, Maggie Aphane, Morufu Oyedunsi Olayiwola, An inertial method for solving systems of generalized mixed equilibrium and fixed point problems in reflexive Banach spaces, 2023, 16, 1793-5571, 10.1142/S1793557123502078
    10. Yassine Adjabi, Fahd Jarad, Mokhtar Bouloudene, Sumati Kumari Panda, Revisiting generalized Caputo derivatives in the context of two-point boundary value problems with the p-Laplacian operator at resonance, 2023, 2023, 1687-2770, 10.1186/s13661-023-01751-0
    11. Nihar Kumar Mahato, Bodigiri Sai Gopinadh, , 2024, Chapter 15, 978-981-99-9545-5, 339, 10.1007/978-981-99-9546-2_15
    12. Sumati Kumari Panda, Velusamy Vijayakumar, Bodigiri Sai Gopinadh, Fahd Jarad, 2024, Chapter 6, 978-981-99-9545-5, 177, 10.1007/978-981-99-9546-2_6
    13. Sumati Kumari Panda, Ilyas Khan, Vijayakumar Velusamy, Shafiullah Niazai, Enhancing automic and optimal control systems through graphical structures, 2024, 14, 2045-2322, 10.1038/s41598-024-53244-4
    14. P K Santra, G S Mahapatra, Sanjoy Basu, Stability analysis of fractional epidemic model for two infected classes incorporating hospitalization impact, 2024, 99, 0031-8949, 065237, 10.1088/1402-4896/ad4692
    15. Anupam Das, Bipan Hazarika, Mohsen Rabbani, 2024, Chapter 10, 978-981-99-9206-5, 165, 10.1007/978-981-99-9207-2_10
    16. Tahair Rasham, Sumati Kumari Panda, Ghada Ali Basendwah, Aftab Hussain, Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators, 2024, 10, 24058440, e34078, 10.1016/j.heliyon.2024.e34078
    17. Sumati Kumari Panda, Velusamy Vijayakumar, Kottakkaran Sooppy Nisar, Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings, 2023, 2023, 1687-2770, 10.1186/s13661-023-01778-3
    18. Sumati Kumari Panda, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar, Bipan Hazarika, Solving existence results in multi-term fractional differential equations via fixed points, 2023, 51, 22113797, 106612, 10.1016/j.rinp.2023.106612
    19. Sumati Kumari Panda, Velusamy Vijayakumar, Results on finite time stability of various fractional order systems, 2023, 174, 09600779, 113906, 10.1016/j.chaos.2023.113906
    20. Sumati Kumari Panda, A.M. Nagy, Velusamy Vijayakumar, Bipan Hazarika, Stability analysis for complex-valued neural networks with fractional order, 2023, 175, 09600779, 114045, 10.1016/j.chaos.2023.114045
    21. Mokhtar Bouloudene, Fahd Jarad, Yassine Adjabi, Sumati Kumari Panda, Quasilinear Coupled System in the Frame of Nonsingular ABC-Derivatives with p-Laplacian Operator at Resonance, 2024, 23, 1575-5460, 10.1007/s12346-023-00902-z
    22. Sumati Kumari Panda, Vijayakumar Velusamy, Ilyas Khan, Shafiullah Niazai, Computation and convergence of fixed-point with an RLC-electric circuit model in an extended b-suprametric space, 2024, 14, 2045-2322, 10.1038/s41598-024-59859-x
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5945) PDF downloads(1060) Cited by(3)

Figures and Tables

Figures(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog