Ratio Mathematica Volume 46, 2023 Theoretical analysis on the growth kinetics of SARS-CoV (within host) Pavithra Sivasamy* Vanthana Ramesh Kumar† Abstract A mathematical model is investigated to analyze the biological inter- actions between the immune system and SARS-CoV (within host). Homotopy Perturbation Method is executed to obtain an analytical solution to the non-linear system of ordinary differential equations. Graphical illustration to these solutions is also presented. The reli- ability and the simplicity of the aforementioned method is studied through the comparison between the numerical and graphical results. This comparison aids the better understanding of the disease dynam- ics and also the establishment of probable strategies for the treatment of COVID-19. Keywords: Mathematical Modeling, COVID-19, Non -linear initial value problem, Homotopy Perturbation Method. 2020 AMS subject classifications: Find your subject classification at https://mathscinet.ams.org/msnhtml/msc2020.pdf. 1 *The Standard Fireworks Rajaratnam College for Women, Sivakasi – 626123, India. pavithra- mat@sfrcollege.edu.in †The Standard Fireworks Rajaratnam College for Women, Sivakasi – 626123, India. vanthana- mat@sfrcollege.edu.in 1Received on September 15, 2022. Accepted on December 15, 2022.Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1074. ISSN: 1592-7415. eISSN: 2282-8214. ©Pavithra Sivasamy et al. This paper is published under the CC-BY licence agreement. 178 Pavithra Sivasamy and Vanthana Ramesh Kumar 1 Introduction The World Health Organization declared the SARS-CoV as a potential threat to the human society Sharma et al. [2020]Ramadan and Shaib [2019]Ouassou et al. [2020]. Since then, researchers have been working really hard on various strategies to control the disease Ahmed et al. [2020]Nie et al. [2020]. A key factor to control the disease is by studying the disease severity and progression in the host. However, the quantitative analysis of the growth kinetics of the virus has not been able to study the severity in the host. It is therefore important to analyze the interactions within the host between SARS-CoV and the immune system Du and Yuan [2020]Hernandez-Vargas and Velasco-Hernandez [2020]Wang et al. [2020]. And this can be achieved by developing a mathematical model as they serve as a tool for characterizing the disease dynamics and in the forecast of severity of the disease. A mathematical model is therefore developed by S.M.E.K Chowdhury et.al Chowdhury et al. [2022]in the context of immune surveillance. We study this model analytically. The goal of our work is to derive a closed form analyti- cal solution for the COVID-19 model for the susceptible, infected, recovered and exposed. This is done by implementing the technique of Homotopy Perturbation Method [HPM] He [1999]He [2004a]He [2004b]He [2004c] which is the cou- pling of homotopy and perturbation technique. The derived analytical results are beneficial in two fronts: first, it would contribute to a better comprehension of the disease dynamics which assist in designing effective treatment strategies and sec- ondly, would enable the scientist in modifying the COVID-19 model to study the correlation between various model parameters and their impacts. This approach in its implementation is novel to this non-linear system of COVID-19 model. 2 Model formulation The mathematical framework of the developed model is built on the interplay between the lymphocytes and virus particles within the host. Taking into account this interaction the model is proposed as Chowdhury et al. [2022] dE dt = a1 − a2E − a3EV (1) dEi dt = a3EV − b1Ei (2) dV dt = c1E i − c2V − −c3V L (3) dN dt = d1 − d2N (4) 179 Theoretical analysis on the growth kinetics of SARS-CoV (within host) Parameters Physical Meaning a1 Regenration rate of epithelial cells a2 Rate at which epithelial cells die a3 Rate of infection of epithelial cells b1 Rate at which the infected epithelial cells die c1 Production rate of virus from infected epithelial cells c2 Rate at which the virus dies e1 Rate of proliferation of T-lymphocytes e2 Rate at which the T-lymphocytes die e3 Regeneration rate of T-lymphocytes c3 Rate at which the Natural Killer cells kills the virus d1 External influx rate of Natural Killer cells d2 Rate at which the Natural Killer cells die c4 Rate at which the T-lymphocytes kills the virus n Half saturation constant of T-lymphocytes Table 1: Nomenclature dL dt = e1LV (n + V ) − e2L + e3 (5) Where denotes the count of susceptible epithelial cells E(t) , infected epithelial cells Ei(t), viral load V (t) , natural killer cells N(t) and T-lymphocytes L(t) re- spectively. The production rate of the virus is c1 where as the infection rate is denoted by a3 . the natural death rate of susceptible epithelial cells, infected ep- ithelial cells, viral load , natural killer cells and T-lymphocytes are a2, b1,c2,d2,e−2 respectively. a1 denotes the regeneration rate of the epithelial cells. The degener- ation of virus particles occur when they are interacted with natural killer cells and T-lymphocytes at the rate d1 and d3 respectively. The NK cell’s constant external source is denoted as a2 . refers to the T-lymphocytes natural recruitment rate c4 . In the presence of virus particles, the T-lymphocytes proliferate at a rate n. 3 Homotopy perturbation method Epidemiological modeling of the diseases using nonlinear dynamical equa- tions gives deeper insights into the behavioral patterns and the transmission dy- namics of the disease. As solutions to these non-linear problems are a bit more complex, a substantial amount of work has been dedicated by researchers in de- veloping a solution method to these models. Such methods include ADM , VIM , HAM , HPM He [1999]He [2004c]He [2005]Abbasbandy [2006]Rafei and Ganji [2006]Yıldırım and Öziş [2007]Sivasamy and Kumar [2021] etc. In this paper, 180 Pavithra Sivasamy and Vanthana Ramesh Kumar we execute HPM in finding an analytical solution to the COVID-19 model. The advantage of HPM over other method is its ability to reduce a complex non-linear problem into a serious of linear equation and finding the solution the same. 3.1 Basic idea of homotopy perturbtion method [HPM] Consider the following function Do(u) − f(r) = 0, rϵΩ (6) with the boundary conditions as Bo(u, ∂u ∂n ) = 0, rϵΓ (7) Where D0 is a general differential operator, B0 is a boundary operator, f(r) is a known analytical function and Γ is the boundary of the domain Ω . In general, the operator D0 can be divided into a linear part L and a non-linear part N. We can rewrite the eqn (6) as L(u) + N(u) − f(r) = 0 (8) We now construct a homotopy as v(r, p) :→ Ω × [0, 1] × ℜ by the homotopy technique which statisfies H(v, p) = (1 − p)[L(v) − L(v0)] + p[D0 − f(r)] = 0 (9) H(v, p) = L(v) − L(v0) + p[N(v) − f(r)] = 0 (10) Here p is the embedding parameter and belongs to the interval [0, 1] and the initial approximation of eqn.(6) is u0 which satisfies the boundary condition. Now,eqn (9) and (10) leads to H(v, 0) = L(v) − L(u0) = 0 (11) H(v, 1) = D0 − f(r) = 0 (12) Setting p=0 makes the eqns. (9) and (10) as linear and seeting p=1 makes them non-linear.. This process is presented as L(v) − L(u0) = 0 to D0 − f(r) = 0. We use p, the embedding parameter as a small parameter and assume that the solutions of eqns. (9) and (10) can be written as a power series in p: v = v0 + pv1 + p 2v2 + ... (13) Fixing p=1 leads to the approximation of eqn (6) as v = v0 + pv1 + p 2v2 + ... (14) This is the basic idea of the HPM. 181 Theoretical analysis on the growth kinetics of SARS-CoV (within host) 4 Analytical solution to the COVID-19 model using homotopy perturbation method (1 − p) ( dE dt = a1 − a2E ) + p ( dE dt = a1 − a2E − a3EV ) = 0 (15) (1 − p) ( dEi dt + b1E i ) = p ( ( dEi dt = a3EV − b1Ei ) = (16) (1 − p) ( dV dt + c2V ) + p ( ( dV dt = c1E i − c2V − −c3V L ) = 0 (17) (1 − p) ( dL dt + e2L − e3 ) + p ( ( dL dt = e1LV (n + V ) − e2L + e3 ) = 0 (18) Supposing the approximate solutions of Eq. (1-5) have the form E = E0 + pE1 + p 2E2 + ... (19) Ei = Ei0 + pE i 1 + p 2Ei2 + ... (20) V = V0 + pV1 + p 2V2 + ... (21) L = L0 + pL1 + p 2L2 + ... (22) Substituting the Eq. (15-18) respectively into Eq. (1-5) (1 − p) ( d(E0 + pE1 + p 2E2 + ...) dt − a1 + a2(E0 + pE1 + p2E2 + ...) ) + p ( d(E0 + pE1 + p 2E2 + ...) dt − a1 + a2(E0 + pE1 + p2E2 + ...) + a3(E0 + pE1 + p 2E2 + ...)(V0 + pV1 + p 2V2 + ...) = 0 (23) (1 − p) ( d(Ei0 + pE i 1 + p 2Ei2 + ...) dt + b1(E i 0 + pE i 1 + p 2Ei2 + ...) ) + p ( d(Ei0 + pE i 1 + p 2Ei2 + ...) dt − a3(E0 + pE1 + p2E2 + ...)(V0 + pV1 + p2V2 + ...) − b1(Ei0 + pE i 1 + p 2Ei2 + ...) = 0 (24) 182 Pavithra Sivasamy and Vanthana Ramesh Kumar (1 − p) ( d(V = V0 + pV1 + p 2V2 + ...) dt + c2(V = V0 + pV1 + p 2V2 + ...) ) + p ( ( d(V = V0 + pV1 + p 2V2 + ...) dt − c1(Ei0 + pE i 1 + p 2Ei2 + ...) − c2(V = V0 + pV1 + p2V2 + ...) − c3(V = V0 + pV1 + p2V2 + ...)(L0 + pL1 + p2L2 + ...) = 0 (25) (1 − p) ( d(L0 + pL1 + p 2L2 + ...) dt + e2(L0 + pL1 + p 2L2 + ...) − e3 ) + p ( ( dL dt = e1(L0 + pL1 + p 2L2 + ...)(V0 + pV1 + p 2V2 + ...) (n + (V0 + pV1 + p2V2 + ...)) − e2(L0 + pL1 + p2L2 + ...) + e3 = 0 (26) Equating the terms of Eq (23-26) with the identical powers of p, we obtain p0 : dE0 dt + a2E − a1 = 0 (27) p0 : dEi0 dt + b1E i 0 = 0 (28) p0 : dV0 dt + c2V0 = 0 (29) p0 : dL0 dt + e2L0 − e3 = 0 (30) dE1 dt + a2E1 + a3E0V0 = 0 (31) dEi1 dt − a3E0V0 + b1Ei0 = 0 (32) dV1 dt − c1Ei0 − c2V1 + c3V0L0 = 0 (33) dL dt − e1L0V0 (n + Vi) + e2L1 = 0 (34) Number of susceptible epithelial cells are given by E(t) = a1 a2 + (Ei − a1 a2 )e−a2t − a3a1Vie −c2t a2(a2 − c2) + a3(Ei − a1a2 )Vie (−a2+c2)t c2 + a3a1Vie −a2t a2(a2 − c2) − a3(Ei − a1a2 )Vie −a2t c2 (35) 183 Theoretical analysis on the growth kinetics of SARS-CoV (within host) Infected epithelial cells are given by Ei(t) = Ei(t)eb1t + a3a1Vie −c2t a2(b1 − c2) + a3(Ei − a1a2 )Vie (−a2−c2)t b1 − a2 − c2 + a3a1Vie −b1t a2(b1 − c2) − a3(Ei − a1a2 )Vie −b1t b1 − a2 − c2 (36) Viral load cells are given by V (t) = Vi(t)e −c2t + c1Eie −b1t c2b1 + ( c3d1Vi d2 − c4e3Vi e2 )te−c2t + c3(Ni − d1d2 )Vie (−(d2+c2)t d2 − c1Eie −c2t c2(b1 − c2) − c3(Ni − d1d2 )Vie (−c2)t d2 + c3(Ni − d1d2 )Lie (−(e2+c2)t e2 − c3(Ni − d1d2 )Lie (−e2)t e2 (37) Natural killer cells are given by N(t) = d1 d2 − ( Ni − d1 d2 ) e−d2t (38) T-lymphocytes cells are given by L(t) = e3 e2 + (Li − e3 e2 )e−e2t + e3e1Vie −c2t e3(n + Vi)(−c2 + e2) − e1(Li − e3e2 )Vie (−a2+e2)t c2(n + Vi) − e3e1Vie −e2t e3(n + Vi)(−c2 + e2) + e1(Li − e3e2 )Vie (−e2+e2)t c2(n + Vi) (39) 5 Numerical simulation An analytical expression for the time-dependent non-linear COVID-19 model is derived by executing the method of Homotopy Perturbation for the equations (1-6). The numerical solutions to these equations are obtained using MATLAB software. In order to check the efficiency of HPM in solving the COVID-19 model, comparison between the numerical and analytical results are carried out which is illustrated in the figures 1-12. 184 Pavithra Sivasamy and Vanthana Ramesh Kumar Figure 1: Illustration of numerical and analytical results for the populations of susceptible epithelial cells , infected epithelial cells , Viral Load , Natural Killer Cells and T-lymphocytes against time t. Figure 2: Illustration of analytical and graphical results for the population of sus- ceptible against time t. 185 Theoretical analysis on the growth kinetics of SARS-CoV (within host) Figure 3: Illustration of analytical and graphical results for the population of sus- ceptible against time t. Figure 4: Illustration of analytical and graphical results for the population of sus- ceptible against time t. 186 Pavithra Sivasamy and Vanthana Ramesh Kumar Figure 5: Illustration of analytical and graphical results for the population of sus- ceptible against time t. Figure 6: Illustration of analytical and graphical results for the population of in- fected epithelial against time t. 187 Theoretical analysis on the growth kinetics of SARS-CoV (within host) Figure 7: Illustration of analytical and graphical results for the population of in- fected epithelial against time t. Figure 8: Illustration of analytical and graphical results for the population of nat- ural killer cells against time t. 188 Pavithra Sivasamy and Vanthana Ramesh Kumar Figure 9: Illustration of analytical and graphical results for the population of nat- ural killer cells against time t. Figure 10: Illustration of analytical and graphical results for the population of T- lymphocytes against time t. 189 Theoretical analysis on the growth kinetics of SARS-CoV (within host) Figure 11: Illustration of analytical and graphical results for the population of T- lymphocytes against time t. Figure 12: Illustration of analytical and graphical results for the population of T- lymphocytes against time t. 190 Pavithra Sivasamy and Vanthana Ramesh Kumar 5.1 Results and discussion Figure 1 illustrates the comparison between the analytical and numerical re- sults for the populations of susceptible epithelial cells , infected epithelial cells , Viral Load , Natural Killer Cells and T-lymphocytes against time t for the pa- rameter values n = 0.1, d1 = 5, a1 = 10, a2 = 0.02, a3 = 0.10, b1 = 0.10, c1 = 0.24, c2 = 5.36, c3 = 0.231, c4 = 0.431, e1 = 0.0041, e2 = 0.0796, e3 = 0.0146, d2 = 0.02. Figure 2-4 presents the plot of susceptible epithelial cells against time t. Figure 5-7 presents the plot of infected epithelial cells against time t. Figure 8-9 indicated the plot of Natural Killer Cells against time t. Figure 10-12 represents the plot of T-lymphocytes against time t. Figure 2 depicts that the ratio of suscep- tible cells increases with time as the regeneration rate increases. Figure 3 indi- cates that the growth of epithelial cells is exponential when the rate of infection is higher. Figure 4 presents that even when the death of the infected cells is higher the growth of susceptible cells increases as there is an increase in the production rate of infected cells. Figure 5 depicts that the there is a steady decline in the rate of infected cells as the death rate of the infected cells increases. Figure 6 presents that the rate of growth of infected cells decreases when the rate of infection of the epithelial cells is lower and also a steady decline in the death rate of infected ep- ithelial cells. Figure 7 indicates that the growth rate of the infected epithelial cells also depends on the death rate of the virus. Figure 8 and 9 represents that rate of natural killer cells increases in spite of the infection when there is an increase in external influx. Figure 10-12 presents that proliferation rate, regeneration are and the death rate of the T-lymphocytes has an impact on the growth of T-lymphocytes. 6 Conclusions A theoretical model outling the interactions between the SARS-Cov and the immune system within the host has been investigated by combining the functions of NK cells and T-lymphocytes. Homotopy Perturbation method is executed to solve the non-linear equations and a closed form analytical solution is obtained.. Graphical illustration of the analyical and numerical solution is performed. A sound agreement between these results is noted. This comparison shows that HPM is an effective approach for solving such models. References S. Abbasbandy. Application of he’s homotopy perturbation method for laplace transform. Chaos, Solitons & Fractals, 30(5):1206–1212, 2006. 191 Theoretical analysis on the growth kinetics of SARS-CoV (within host) S. Ahmed, A. Quadeer, and M. Mckay. Preliminary identification of potential vaccine targets for the covid-19 coronavirus (sars-cov-2) based on sars-cov. Im- munol Stud, pages 1–15, 2020. S. Chowdhury, J. Chowdhury, S. F. Ahmed, P. Agarwal, I. A. Badruddin, and S. Kamangar. Mathematical modelling of covid-19 disease dynamics: Interac- tion between immune system and sars-cov-2 within host. AIMS Mathematics, 7(2):2618–2633, 2022. S. Q. Du and W. Yuan. Mathematical modeling of interaction between innate and adaptive immune responses in covid-19 and implications for viral pathogenesis. Journal of medical virology, 92(9):1615–1628, 2020. J.-H. He. Homotopy perturbation technique. Computer methods in applied me- chanics and engineering, 178(3-4):257–262, 1999. J.-H. He. The homotopy perturbation method for nonlinear oscillators with dis- continuities. Applied mathematics and computation, 151(1):287–292, 2004a. J.-H. He. Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation, 156(2):527–539, 2004b. J.-H. He. The homotopy perturbation method for nonlinear oscillators with dis- continuities. Applied mathematics and computation, 151(1):287–292, 2004c. J.-H. He. Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Sciences and Numerical Simulation, 6(2): 207–208, 2005. E. A. Hernandez-Vargas and J. X. Velasco-Hernandez. In-host modelling of covid- 19 kinetics in humans. medrxiv, page 20044487, 2020. Q. Nie, X. Li, W. Chen, D. Liu, Y. Chen, H. Li, D. Li, M. Tian, W. Tan, and J. Zai. Phylogenetic and phylodynamic analyses of sars-cov-2. Virus research, 287: 198098, 2020. H. Ouassou, L. Kharchoufa, M. Bouhrim, N. E. Daoudi, H. Imtara, N. Bencheikh, A. ELbouzidi, and M. Bnouham. The pathogenesis of coronavirus disease 2019 (covid-19): evaluation and prevention. Journal of immunology research, 2020, 2020. M. Rafei and D. Ganji. Explicit solutions of helmholtz equation and fifth-order kdv equation using homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation, 7(3):321–328, 2006. 192 Pavithra Sivasamy and Vanthana Ramesh Kumar N. Ramadan and H. Shaib. Middle east respiratory syndrome coronavirus (mers- cov): A review. Germs, 9(1):35, 2019. A. Sharma, S. Tiwari, M. K. Deb, and J. L. Marty. Severe acute respiratory syn- drome coronavirus-2 (sars-cov-2): a global pandemic and treatment strategies. International journal of antimicrobial agents, 56(2):106054, 2020. P. Sivasamy and V. R. Kumar. Approximate analytical solution of relapsing remit- ting multiple sclerosis using homotopy perturbation method. NVEO-NATURAL VOLATILES & ESSENTIAL OILS Journal— NVEO, pages 2613–2624, 2021. S. Wang, Y. Pan, Q. Wang, H. Miao, A. N. Brown, and L. Rong. Modeling the viral dynamics of sars-cov-2 infection. Mathematical biosciences, 328:108438, 2020. A. Yıldırım and T. Öziş. Solutions of singular ivps of lane–emden type by homo- topy perturbation method. Physics Letters A, 369(1-2):70–76, 2007. 193