We develop a novel modification of the classic Kermack–McKendrick SIR (Susceptible–Infectious–Recovered) model. The new SIR model with ”intervention polynomial factor” (SIR-IPF) can be used successfully to model and play different scenarios for the infectious disease spread. A generalized ”reproduction number” is introduced. A similar modifications are proposed for the classic SEIR and G-SEIR models. The specialists working in the field of ”reaction-kinetic mechanisms” have the
word.
Numerical examples, illustrating our results are given using CAS Mathematica.Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014-2020) and co-financed by the European Union through the European structural and Investment funds

Iliev, Anton

Kyurkchiev, Nikolay

Kyurkchiev, Vesselin

Rahnev, Asen

International Journal of Differential Equations and Applications

Bulgarian Digital Mathematics Library at IMI-BAS

International Journal of Differential Equations and ApplicationsVolume 20, No. 1 (2021), pages: 15-30ISSN (Print): 1311-2872; ISSN (Online): 1314-6084;url: https://www.ijdea.euPAijdea.euA NEW MODIFICATIONS OF THE SIR/SEIR MODELSWITH ”INTERVENTION POLYNOMIAL FACTOR”.METHODOLOGICAL ASPECTSNikolay Kyurkchiev1,2, Vesselin Kyurkchiev1,Anton Iliev1,2, Asen Rahnev11Faculty of Mathematics and InformaticsUniversity of Plovdiv Paisii Hilendarski24, Tzar Asen Str., 4000 Plovdiv, BULGARIA2Institute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 8, 1113 Sofia, BULGARIAABSTRACT: We develop a novel modification of the classic Kermack–McKendrickSIR (Susceptible–Infectious–Recovered) model. The new SIR model with ”interventionpolynomial factor” (SIR-IPF) can be used successfully to model and play differentscenarios for the infectious disease spread. A generalized ”reproduction number” isintroduced. A similar modifications are proposed for the classic SEIR and G-SEIRmodels. The specialists working in the field of ”reaction-kinetic mechanisms” have theword.Numerical examples, illustrating our results are given using CAS Mathematica.AMS Subject Classification: 41A46Key Words: reaction equations, mass action kinetics, Kermack–McKendrick’s SIRmodel, new SIR model with ”intervention polynomial factor” (SIR-IPF), (SEIR-IPF)model, (G-SEIR-IPF) modelReceived: February 5, 2021 Revised: May 20, 2021Published: May 27, 2021 doi: 10.12732/ijdea.v20i1.2Academic Publications, Ltd. https://acadpubl.eu16 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. Rahnev1. INTRODUCTIONDynamical models consisting of a systems of ”reaction” differential equations are com-monly used in chemistry, there the differential equations are called reaction equations.In chemistry reaction differential equations are induced by chemical reactions networksvia reaction kinetic principles, such as mass action kinetics [1]–[2]. In [3]–[4] the au-thor focuses his attention on growth functions (models) formulated as solutions todifferential equations or systems of differential equation. For some modifications andextensions, see [5]–[11]. In the field of epidemiology [12]–[13], several mathematicalapproaches aiming to describe the infectious disease spread have been employed. Theclassic Kermack–McKendrick [14]–[16] SIR (Susceptible–Infectious–Recovered) modelcan be formulated as a chemical reaction network by [17]:S + Ik1K−→ 2IIk2−→ R.(1)The reaction network (1) induces the following differential system:dsdt= −k1Ks(t)i(t)didt= k1Ks(t)i(t) − k2i(t)drdt= k2i(t)(2)where s(t) - number of individuals susceptible but not yet to infected with an infectiousdisease at time t; i(t) - number of infected individuals at time t and can spread theinfectious disease to susceptible individuals; r(t) - number of recovered (or deceased)individuals at time t and assumed to be immune for life; k1 - transmission rate throughthe exposure of the infectious disease; k2 - rate of recovering from the infectious diseaseand 1k2is the mean period during which an infected individual can pass it on. In theSIR model, the determination of the basic reproduction numberR̃ =k1k2(3)is a challenging task for COVID-19 infectious disease. In [18] the authors discussed thefollowing SIR model with ”intervention factor”:dsdt= − (1−ρ)k1Ks(t)i(t)didt= (1−ρ)k1Ks(t)i(t)− k2i(t)drdt= k2i(t)(4)A NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 17where ρ ∈ [0, 1] is ”intervention parameter”. For some models, see [19]–[23].In this paper we consider a new modification of the SIR model with ”interventionpolynomial factor”. A new reproduction number is introduced. If desired, this newmodel can be assigned to the ”class of dynamic models” for the analysis of epidemio-logical diseases. A similar modification is proposed for the classic SEIR and general-ized G-SEIR models. Numerical examples, illustrating our results are given using CASMathematica.2. MAIN RESULTS2.1. A NEW MODIFICATION OF THE SIR MODEL WITH”INTERVENTION POLYNOMIAL FACTOR” (SIR-IPF)We consider the following reaction network:S + If(t)K−→ 2IIg(t)−→ R.(5)The reaction network (5) induces the following differential system:ds(t)dt= − f(t)Ks(t)i(t)di(t)dt= f(t)Ks(t)i(t)− g(t)i(t)dr(t)dt= g(t)i(t)(6)wheref(t) =m∑i=1k2i−1ti−1; g(t) =m∑i=1k2iti−1.From the relationlimt→∞f(t)g(t)=k2m−1k2m(7)we see, that the new reproduction number has the form:R̃new =k2m−1k2m. (8)Evidently, for m = 1, we get the classic Kermack–McKendrick SIR model (2).Our main goal was to offer to the attention of specialists and other tools for possiblesimulation and playback of scenarios in the analysis of infectious disease spread using18 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. RahnevFigure 1: K = 1000; k1 = 0.2; k2 = 0.071; s0 = s(0) = 995; i0 = i(0) = 5;r0 = r(0) = 0 [18].a new technique - ”intervention polynomial factor” and the resulting ”reproductionnumbers”.2.2. SIMULATIONS USING (SIR-IPF) MODELThe new SIR model with ”intervention polynomial factor” (SIR-IPF) can be usedsuccessfully to model and play different scenarios for the distribution of COVID-19. Forthe analysis of COVID-19 cases in India, in [18] the authors solved system of differentialequations (2) for K = 1000; k1 = 0.2; k2 = 0.071; s0 = s(0) = 995; i0 = i(0) = 5;r0 = r(0) = 0 (see, Fig. 1).Consider the model (SIR–IFP) for fixed m = 3 and K = 1000; k1 = 0.2; k3 = 0.05;k5 = 0.005; k2 = 0.071; k4 = 0.005; k6 = 0.002; s0 = s(0) = 995; i0 = i(0) = 5;r0 = r(0) = 0. The solutions are visualized on Fig. 2 in the interval (0, 200). For fixedm = 2 and K = 1000; k1 = 0.2; k3 = 0.01; k2 = 0.071; k4 = 0.01; s0 = s(0) = 995;i0 = i(0) = 5; r0 = r(0) = 0 see, Fig. 3.If desired, this new model can be assigned to the ”class of dynamic models” for theanalysis of epidemiological diseases.A NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 19Figure 2: The model (SIR–IFP) for fixed m = 3 and K = 1000; k1 = 0.2;k3 = 0.05; k5 = 0.005; k2 = 0.071; k4 = 0.005; k6 = 0.002; s0 = s(0) = 995;i0 = i(0) = 5; r0 = r(0) = 0.Figure 3: The model (SIR–IFP) for fixed m = 2 and K = 1000; k1 = 0.2;k3 = 0.01; k2 = 0.071; k4 = 0.01; s0 = s(0) = 995; i0 = i(0) = 5; r0 = r(0) = 0.20 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. Rahnev2.3. A NEW MODIFICATION OF THE SEIR MODEL WITH”INTERVENTION POLYNOMIAL FACTOR” (SEIR-IPF)Other mathematical models can be used to investigate the spread of diseases, such asSEIR. MSEIR, SEIRS and MSEIRS [19]–[23]. For the SEIR (Susceptible–Exposed–Infectious–Recovered) model, the corresponding system of differential equations is:ds(t)dt= − f(t)Ks(t)i(t)de(t)dt= f(t)Ks(t)i(t)− σe(t)di(t)dt= σe(t)− g(t)i(t)dr(t)dt= g(t)i(t).(9)Here the term f(t)Ki(t) expresses the rate at which new individuals (as a proportionof the total population size) are infected by the already infectious i(t) individuals; f(t)is called transmission rate of the infection; g(t) is the removal rate. For some details, see[20]. In [20] the authors developed a novel SEIR spline model for analysis of coronavirusinfection. Here we will consider a modification of the model (9) with a special choice ofthe functions f(t) and g(t). More precisely, f(t) and g(t) are polynomials of the form:f(t) =m∑i=1k2i−1ti−1; g(t) =m∑i=1k2iti−1.We will call the new modification SEIR model with ”intervention polynomial factor” –(SEIR-IPF). Different scenarios for parameter variation and fixed degree of polynomialsf and g – (m = 2) using the (SEIR-IPF) model are given in Fig. 4.2.4. A LOOK AT THE GENERALIZED SEIR MODEL PROPOSED BYPENG, YANG ET AL. [?]A generalized SEIR model is proposed by Peng, Yang et al. in [25] (see, Fig. 5). In theconsidered system of differential equations, the functions λ(t) and k(t) are respectivelyincreasing and decreasing in the fixed time interval. A modification of the SEIR modelwith a special choice of decreasing function is considered by Anguelov et al. in [24].Here we will consider a modification of the model with a special choice of thefunctions λ(t) and k(t). More precisely, λ(t) and k(t) are polynomials of the form:λ(t) =m∑i=1k2i−1ti−1; k(t) =m∑i=1k2iti−1.A NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 21Figure 4: Different scenarios for parameter variation and fixed degree of poly-nomials f and g - (m = 2) using the (SEIR-IPF) model.We will call the new modification G-SEIR model with ”intervention polynomial factor”– (G-SEIR-IPF).We will explicitly note that in this new formulation the model has many degrees offreedom (the coefficients of the polynomials λ(t) and k(t) ), and this makes it attractivein the study and simulation of such dynamic models.For fixed m = 3 and K = 1000; k1 = 100; k3 = 1; k5 = 0.1; k2 = 100; k4 = −0.12;k6 = −0.008; s0 = 800; e0 = 40, i0 = 80; q0 = 20, r0 = 20, d0 = 20, p0 = 20, β = 0.1,α = 0.15, γ = 0.08, δ = 0.1 the simulation is depicted on Fig. 6.For fixed m = 3 and K = 800; k1 = 100; k3 = 0.2; k5 = 0.1; k2 = 120; k4 =0.1; k6 = −0.005; s0 = 500; e0 = 80; i0 = 20; q0 = 30; r0 = 100; d0 = 20; p0 = 50; β =0.2; α = 0.3; γ = 0.1; δ = 0.15 the simulation is depicted on Fig. 7.22 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. RahnevFigure 5: The generalized SEIR model by Peng, Yang et al. [25].Figure 6: The simulation using (G-SEIR-IPF) model.A NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 23Figure 7: The simulation using (G-SEIR-IPF) model.2.5. A LOOK AT THE SEIRDP (SUSCEPTIBLE-EXPOSED-INFECTED-RECOVERED-DEATH-INSUSCEPTIBLE (P)) MODEL[?]The SEIRDP (Susceptible, Exposed, Infected, Recovered, Death, Insusceptible (P))model can be described by [26] (see, Fig. 8):ds(t)dt= − βKs(t)i(t)− αs(t)de(t)dt= βKs(t)i(t)− γe(t)di(t)dt= γe(t)− λ(t)i(t) − k(t)i(t)dr(t)dt= λ(t)i(t)dd(t)dt= k(t)i(t)dp(t)dt= αs(t)(10)where- S(t): Susceptible population;- E(t): Population who are exposed to the virus, but not yet infectious in latentperiod;- I(t): Population who get laboratory positive confirmation and with infectiouscapacity;- R(t): Recovery cases;- D(t): Death number;24 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. RahnevFigure 8: The generalized SEIRDP model by Lounis and Azevedo [26].- P (t): Insusceptible cases;- α : Protection rate (include people exposed to the infectious patients and peopleexposed to the asymptomatic patients);- β : Infection rate;- γ−1 : Average latent time;- λ(t): Coefficient used in the time-dependent cure rate;- k(t): Coefficient used in the time-dependent mortality rate.In [26], the authors consider the functions (see, Fig. 9):λ(t) = λ01+e−λ1(t−λ2)k(t) = k0e−k1t(11)2.6. A NEW MODIFICATION OF THE SEIRDP MODEL WITH”INTERVENTION POLYNOMIAL FACTOR” (SEIRDP-IPF)From methodological aspects, here we will consider a modification of the model with aspecial choice of the functions λ(t) and k(t).A NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 25Figure 9: Plot of recovery and mortality rates [26].Figure 10: Plot of recovery and mortality rates (the new functions λ∗(t) andk∗(t)) .26 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. RahnevMore precisely, λ∗(t) and k∗(t) are polynomials of the form:λ∗(t) =m∑i=1k2i−1ti−1k∗(t) =m∑i=1k2iti−1.(12)We will call the new modification SEIRDP model with ”intervention polynomial factor”– (SEIRDP-IPF).For example, letλ∗(t) = 0.05 + 0.006t − 0.00015t2 − 0.0000001t3k∗(t) = 0.5 − 0.024t − 0.0008t2 + 0.00004t3(see, Fig. 10.).For fixed K = 1, λ∗(t) and k∗(t) andβ = 0.1; α = 0.15; γ = 0.08; s0 = 0.6; e0 = 0.2; i0 = 0.1;r0 = 0.05; d0 = 0.05; p0 = 0the simulation is depicted on Fig. 11.For fixed K = 1, λ∗(t) and k∗(t) andβ = 0.1; α = 0.15; γ = 0.08; s0 = 0.9; e0 = 0.06; i0 = 0.01;r0 = 0.01; d0 = 0.01; p0 = 0.01the simulation is depicted on Fig. 12.3. CONCLUSION.In the last 2 years several thousand publications have appeared, incl. and these arti-cles in arXiv.org on SIR/SEIR/GSEIR models. Of course, the SIR-IPF/SEIR-IPF/G-SEIR-IPF/SEIRDP-IPF models are very sensitive with respect to the coefficients ki ofthe polynomials. One of the good hits in our development is that in some private caseswe get some classic and newer SIR/SEIR/GSEIR models. For example, we developa novel modification of the classic Kermack–McKendrick SIR (Susceptible–Infectious–Recovered) model with new reproduction numbers ρ0, ρ1, . . . ρl in appropriate intervalsA NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 27Figure 11: The simulation using (GSEIRDP-IPF) model.Figure 12: The simulation using (GSEIRDP-IPF) model.28 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. Rahnev(t0, t1), (t1, t2), . . . (tl, tl+1). The new (SIR-IPF) can be used successfully to model andplay different scenarios for the infectious disease spread. Our research is a naturalcontinuation of previous results in approximating specific data of a strictly exponentialnature (e.g., COVID-19 Bulgaria, Cuba, China, South Korea, etc.) using modifiedlogistics and other models in which the typical reaction constants were replaced by”polynomial variable transfer” and showed good results in performing the regressionanalysis. We note that the choice of ”input functions”, especially for the GSEIR model,is quite specific and is almost subject to the requirement for these functions to be in-creasing and decreasing respectively in a fixed time interval. Only in the already citedarticle [20] the classic technique is used - spline functions! Our article opens up pos-sibilities for generating a number of ”reproduction numbers” for characteristic timeintervals and simulating and playing different scenarios. We will explicitly note thatthe article contains the explorations only of a model nature and much more complexissues related to the resistance and stability of the solution of such stratified systemsdifferential equations (especially for the GSEIR model) can be considered as open, un-til specialists working in the field of ”reaction kinetics mechanisms and models” decidethat there is reason to use our modest explorations. It is planned to upgrade the Dis-tributed Platform for e-Learning - DisPeL [27]–[28], including a specialized module forsimulation of chemical kinetics, module for solving nonlinear differential equations andverifying the mass action balance and module for simulation by the introduced newreproduction number.ACKNOWLEDGMENTSThis work has been accomplished with the financial support by the Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Pro-gram (2014-2020) and co-financed by the European Union through the European struc-tural and Investment funds.REFERENCES[1] J. D. Murray, Mathematical Biology: I. An Introduction, 3rd ed., New York BerlinHeidelberg, Springer-Verlag, (2002).[2] G. Lente, Deterministic kinetics in chemistry and systems biology, In: Briefsin Molecular Science, Cham Heidelberg New York Dordrecht London: Springer,(2015).A NEW MODIFICATIONS OF THE SIR/SEIR MODELS... 29[3] S. Markov, Reaction networks reveal new links between Gompertz and Verhulstgrowth functions, BIOMATH, 8, No. 1 (2019).[4] S. Markov, On a class of generalized Gompertz–Bateman growth–decay models,Biomath Communications (2019).[5] N. Kyurkchiev, Some new classes of growth functions generated by reaction net-works and based on ”correcting amendments” of Bateman-Gompertz and Bateman-Gompertz-Makeham-type. I., Communications in Applied Analysis, 24, No. 1(2020), 13–29[6] N. Kyurkchiev, On a Class of Growth Curves with Exponentially Variable TransferGenerated by Reaction Networks. II, International Electronic Journal of Pure andApplied Mathematics, 14, No. 1 (2020), 21–29.[7] N. Kyurkchiev, A New Class of ”Growth Functions” with Polynomial VariableTransfer Generated by Real Reaction Networks, Cyb. and Inf. Techn., 20, No 6(2020), 74–81.[8] V. Kyurkchiev, A. Iliev, A. Rahnev, N. 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Krickeberg, Modern infectious disease epidemi-ology: Concepts, Methods, Mathematical Models, and Public Health, SpringerScience and Business Media, New York, 2010.[14] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theoryof epidemics, R. Soc. Publ. (1927).30 N. Kyurkchiev, V. Kyurkchiev, A. Iliev, A. Rahnev[15] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theoryof epidemics. II. The problem of endemicity, Proc. R. Soc. Lond. A, 138 (1932),55–83.[16] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory ofepidemics. III. Further studies of the problem of endemicity, Proc. R. Soc. Lond.A, 141 (1933), 94–122.[17] M. Borisov, S. Markov, The two-step exponential decay reaction network: analy-sis of the solutions and relation to epidemiological SIR models with logistic andGompertz type infection contact patterns, J. of Math. Chemistry (2021).[18] H. Verma, A. Gupta, U. Niranja, Analysis of COVID-19 cases in India throughmachine learning: a study of intervention (Preprint).[19] M. J. Keeling, P. Rohan, Modelling infectious diseases, Princeton University Press(2008).[20] O. Kounchev, G. Simeonov, Zh. Kuncheva, Scenarios for the spread of COVID-19analyzed by the TVBG–SEIR spline model, Biomath, 10, (1) (2021), 17 pp.[21] H. W. Hethcote, SIAM Review, 42 (2000), 599.[22] N. C. Grassly, C. Fraser, Nat. Rev. Micr., 6 (2008), 477.[23] I. F. Mello, L. Squillante, G. O. Gomes et al., Epidemics, the Ising-model andpercolation theory: A comprehensive review focused on COVID-19, Physica A:Statistical Mechanics and its Applications (2021), 55 pp.[24] R. Anguelov, J. Banasiak, Ch. Bright, J. Lubuma, R. Ouifki, The big unknown:The asymptomatic spread of COVID-19, BIOMATH, 9, No 1 (2020).[25] L. Peng, W. Yang, D. Zhuge, L. Hong, Epidemic analysis of COVID-19 in Chinaby dynamical modeling, MedRxiv Epidemiol. (2020).[26] M. Lounis, J. Azevedo, Application of a generalized SEIR model for Covid-19 inAlgeria, medRxiv preprint/ doi: https://doi.org /0.1101/2020.08.10.20172155.[27] A. Rahnev, N. Pavlov, V. Kyurkchiev, Distributed Platform for e-Learning - Dis-PeL, European International Journal of Science and Technology, 3, No. 1 (2014),95–109.[28] V. Kyurkchiev, N. Pavlov, A. Rahnev, Cloud-based architecture of DisPeL, Inter-national Journal of Pure and Applied Mathematics, 120, No. 4 (2018), 573–581.

A New Modifications of the SIR/SEIR Models with ”Intervention Polynomial Factor”. Methodological Aspects

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A New Modifications of the SIR/SEIR Models with ”Intervention Polynomial Factor”. Methodological Aspects

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