To Cure or Not to Cure: Consequences of Immunological Interactions in CML Treatment

Abstract

Recent clinical findings in chronic myeloid leukemia (CML) patients suggest that the number and function of immune effector cells are modulated by tyrosine kinase inhibitors (TKI) treatment. There is further evidence that the success or failure of treatment cessation at least partly depends on the patients immunological constitution. Here, we propose a general ODE model to functionally describe the interactions between immune effector cells with leukemic cells during the TKI treatment of CML. In total, we consider 20 different sub-models, which assume different functional interactions between immune effector and leukemic cells. We show that quantitative criteria, which are purely based on the quality of model fitting, are not able to identify optimal models. On the other hand, the application of qualitative criteria based on a dynamical system framework allowed us to identify nine of those models as more suitable than the others to describe clinically observed patterns and, thereby, to derive conclusion about the underlying mechanisms. Additionally, including aspects of early CML onset, we can demonstrate that certain critical parameters, such as the strength of immune response or leukemia proliferation rate, need to change during CML growth prior to diagnosis, leading to bifurcations that alter the attractor landscape. Finally, we show that the crucial parameters determining the outcome of treatment cessation are not identifiable with tumor load data only, thereby highlighting the need to measure immune cell number and function to properly derive mathematical models with predictive power.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Basu S, Pollack R, Coste-Roy MF (2007) Algorithms in real algebraic geometry, vol 10. Springer, Berlin

    Google Scholar 

  2. Besse A, Clapp GD, Bernard S, Nicolini FE, Levy D, Lepoutre T (2018) Stability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemia. Bull Math Biol 80(5):1084–1110

    MathSciNet  Article  MATH  Google Scholar 

  3. Bower H, Bjrkholm M, Dickman PW, Hglund M, Lambert PC, Andersson TML (2016) Life expectancy of patients with chronic myeloid leukemia approaches the life expectancy of the general population. J Clin Oncol 34:2851–2857. https://doi.org/10.1200/JCO.2015.66.2866

    Article  Google Scholar 

  4. Catlin SN, Guttorp P, Abkowitz JL (2005) The kinetics of clonal dominance in myeloproliferative disorders. Blood 106(8):2688–2692. https://doi.org/10.1182/blood-2005-03-1240

    Article  Google Scholar 

  5. Chereda B, Melo JV (2015) Natural course and biology of CML. Ann Hematol 94(Suppl 2):S107–S121. https://doi.org/10.1007/s00277-015-2325-z

    Article  Google Scholar 

  6. Clapp G, Levy D (2015) A review of mathematical models for leukemia and lymphoma. Drug Discov Today Dis Models 16:1–6

    Article  Google Scholar 

  7. Clapp GD, Lepoutre T, El Cheikh R, Bernard S, Ruby J, Labussire-Wallet H, Nicolini FE, Levy D (2015) Implication of the autologous immune system in BCR-ABL transcript variations in chronic myelogenous leukemia patients treated with imatinib. Cancer Res 75:4053–4062. https://doi.org/10.1158/0008-5472.CAN-15-0611

    Article  Google Scholar 

  8. Clark RE, Polydoros F, Apperley JF, Milojkovic D, Pocock C, Smith G, Byrne JL, de Lavallade H, O’Brien SG, Coffey T, Foroni L, Copland M (2017) De-escalation of tyrosine kinase inhibitor dose in patients with chronic myeloid leukaemia with stable major molecular response (destiny): an interim analysis of a non-randomised, phase 2 trial. Lancet Haematol 4:e310–e316. https://doi.org/10.1016/S2352-3026(17)30066-2

    Article  Google Scholar 

  9. DeConde R, Kim PS, Levy D, Lee PP (2005) Post-transplantation dynamics of the immune response to chronic myelogenous leukemia. J Theor Biol 236(1):39–59

    MathSciNet  Article  Google Scholar 

  10. Deininger MW, Obrien SG, Ford JM, Druker BJ (2003) Practical management of patients with chronic myeloid leukemia receiving imatinib. J Clin Oncol 21(8):1637–1647

    Article  Google Scholar 

  11. Dingli D, Traulsen A, Lenaerts T, Pacheco JM (2010) Evolutionary dynamics of chronic myeloid leukemia. Genes Cancer 1:309–315. https://doi.org/10.1177/1947601910371122

    Article  Google Scholar 

  12. Druker BJ, Guilhot F, O’Brien SG, Gathmann I, Kantarjian H, Gattermann N, Deininger MWN, Silver RT, Goldman JM, Stone RM, Cervantes F, Hochhaus A, Powell BL, Gabrilove JL, Rousselot P, Reiffers J, Cornelissen JJ, Hughes T, Agis H, Fischer T, Verhoef G, Shepherd J, Saglio G, Gratwohl A, Nielsen JL, Radich JP, Simonsson B, Taylor K, Baccarani M, So C, Letvak L, Larson RA, Investigators I (2006) Five-year follow-up of patients receiving imatinib for chronic myeloid leukemia. N Engl J Med 355:2408–2417. https://doi.org/10.1056/NEJMoa062867

    Article  Google Scholar 

  13. Eftimie R, Bramson JL, Earn DJ (2011) Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull Math Biol 73(1):2–32

    MathSciNet  Article  MATH  Google Scholar 

  14. Fassoni AC, Baldow C, Roeder I, Glauche I (2018) Reduced tyrosine kinase inhibitor dose is predicted to be as effective as standard dose in chronic myeloid leukemia: a simulation study based on phase III trial data. Haematologica 103:1825–1834. https://doi.org/10.3324/haematol.2018.194522

    Article  Google Scholar 

  15. Hehlmann R, Hochhaus A, Baccarani M, LeukemiaNet E (2007) Chronic myeloid leukaemia. Lancet (London, England) 370:342–350. https://doi.org/10.1016/S0140-6736(07)61165-9

    Article  Google Scholar 

  16. Hochhaus A, Larson RA, Guilhot F, Radich JP, Branford S, Hughes TP, Baccarani M, Deininger MW, Cervantes F, Fujihara S, Ortmann CE, Menssen HD, Kantarjian H, O’Brien SG, Druker BJ, Investigators I (2017) Long-term outcomes of imatinib treatment for chronic myeloid leukemia. N Engl J Med 376:917–927. https://doi.org/10.1056/NEJMoa1609324

    Article  Google Scholar 

  17. Hoffbrand A, Pettit J, Moss P (2001) Essential haematology. Essentials. Wiley, New York

    Google Scholar 

  18. Holling CS (1959) The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can Entomol 91(5):293–320

    Article  Google Scholar 

  19. Holyoake TL, Vetrie D (2017) The chronic myeloid leukemia stem cell: stemming the tide of persistence. Blood 129(12):1595–1606

    Article  Google Scholar 

  20. Hughes A, Yong AS (2017) Immune effector recovery in chronic myeloid leukemia and treatment-free remission. Front Immunol 8:469

    Article  Google Scholar 

  21. Hughes A, Clarson J, Tang C, Vidovic L, White DL, Hughes TP, Yong ASM (2017) CML patients with deep molecular responses to TKI have restored immune effectors and decreased PD-1 and immune suppressors. Blood 129:1166–1176. https://doi.org/10.1182/blood-2016-10-745992

    Article  Google Scholar 

  22. Ilander M, Olsson-Strmberg U, Schlums H, Guilhot J, Brck O, Lhteenmki H, Kasanen T, Koskenvesa P, Sderlund S, Hglund M, Markevrn B, Sjlander A, Lotfi K, Dreimane A, Lbking A, Holm E, Bjreman M, Lehmann S, Stenke L, Ohm L, Gedde-Dahl T, Majeed W, Ehrencrona H, Koskela S, Saussele S, Mahon FX, Porkka K, Hjorth-Hansen H, Bryceson YT, Richter J, Mustjoki S (2017) Increased proportion of mature NK cells is associated with successful imatinib discontinuation in chronic myeloid leukemia. Leukemia 31:1108–1116. https://doi.org/10.1038/leu.2016.360

    Article  Google Scholar 

  23. Jabbour E, Kantarjian H (2016) Chronic myeloid leukemia: 2016 update on diagnosis, therapy, and monitoring. Am J Hematol 91:252–265. https://doi.org/10.1002/ajh.24275

    Article  Google Scholar 

  24. Kennedy J (2001) Some polynomial theorems. https://monks.scranton.edu/files/courses/ProblemSolving/POLYTHEOREMS.pdf Online. Accessed 6 Dec 2018

  25. Kim PS, Lee PP, Levy D (2008) Dynamics and potential impact of the immune response to chronic myelogenous leukemia. PLoS Comput Biol 4:e1000095. https://doi.org/10.1371/journal.pcbi.1000095

    MathSciNet  Article  Google Scholar 

  26. Komarova NL, Wodarz D (2009) Combination therapies against chronic myeloid leukemia: short-term versus long-term strategies. Cancer Res 69:4904–4910. https://doi.org/10.1158/0008-5472.CAN-08-1959

    Article  Google Scholar 

  27. Kumagai T, Nakaseko C, Nishiwaki K, Yoshida C, Ohashi K, Takezako N, Takano H, Kouzai Y, Murase T, Matsue K et al (2018) Dasatinib cessation after deep molecular response exceeding 2 years and natural killer cell transition during dasatinib consolidation. Cancer Sci 109(1):182–192

    Article  Google Scholar 

  28. Kuznetsov VA, Makalkin IA, Taylor MA, Perelson AS (1994) Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull Math Biol 56(2):295–321

    Article  MATH  Google Scholar 

  29. La Rosee P, Johnson K, Corbin AS, Stoffregen EP, Moseson EM, Willis S, Mauro MM, Melo JV, Deininger MW, Druker BJ (2004) In vitro efficacy of combined treatment depends on the underlying mechanism of resistance in imatinib-resistant Bcr-Abl-positive cell lines. Blood 103(1):208–215

    Article  Google Scholar 

  30. Mahon FX, Ra D, Guilhot J, Guilhot F, Huguet F, Nicolini F, Legros L, Charbonnier A, Guerci A, Varet B, Etienne G, Reiffers J, Rousselot P, des Leucmies Mylodes Chroniques IF (2010) Discontinuation of imatinib in patients with chronic myeloid leukaemia who have maintained complete molecular remission for at least 2 years: the prospective, multicentre stop imatinib (STIM) trial. Lancet Oncol 11:1029–1035. https://doi.org/10.1016/S1470-2045(10)70233-3

  31. Michor F, Hughes T, Iwasa Y, Branford S, Shah N, Sawyers C, Nowak M (2005) Dynamics of chronic myeloid leukaemia. Nature 435(7046):1267–70

    Article  Google Scholar 

  32. Moore H, Li NK (2004) A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J Theor Biol 227(4):513–523

    MathSciNet  Article  Google Scholar 

  33. Paquin D, Kim PS, Lee PP, Levy D (2011) Strategic treatment interruptions during imatinib treatment of chronic myelogenous leukemia. Bull Math Biol 73(5):1082–1100

    MathSciNet  Article  MATH  Google Scholar 

  34. Rea D, Henry G, Khaznadar Z, Etienne G, Guilhot F, Nicolini F, Guilhot J, Rousselot P, Huguet F, Legros L, Gardembas M, Dubruille V, Guerci-Bresler A, Charbonnier A, Maloisel F, Ianotto JC, Villemagne B, Mahon FX, Moins-Teisserenc H, Dulphy N, Toubert A (2017a) Natural killer-cell counts are associated with molecular relapse-free survival after imatinib discontinuation in chronic myeloid leukemia: the immunostim study. Haematologica 102:1368–1377. https://doi.org/10.3324/haematol.2017.165001

    Article  Google Scholar 

  35. Rea D, Nicolini FE, Tulliez M, Guilhot F, Guilhot J, Guerci-Bresler A, Gardembas M, Coiteux V, Guillerm G, Legros L, Etienne G, Pignon JM, Villemagne B, Escoffre-Barbe M, Ianotto JC, Charbonnier A, Johnson-Ansah H, Noel MP, Rousselot P, Mahon FX, des Leucmies Mylodes Chroniques FI (2017b) Discontinuation of dasatinib or nilotinib in chronic myeloid leukemia: interim analysis of the STOP 2G-TKI study. Blood 129:846–854. https://doi.org/10.1182/blood-2016-09-742205

  36. Roeder I, Horn M, Glauche I, Hochhaus A, Mueller MC, Loeffler M (2006) Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat Med 12(10):1181–1184

    Article  Google Scholar 

  37. Rousselot P, Charbonnier A, Cony-Makhoul P, Agape P, Nicolini FE, Varet B, Gardembas M, Etienne G, Ra D, Roy L, Escoffre-Barbe M, Guerci-Bresler A, Tulliez M, Prost S, Spentchian M, Cayuela JM, Reiffers J, Chomel JC, Turhan A, Guilhot J, Guilhot F, Mahon FX (2014) Loss of major molecular response as a trigger for restarting tyrosine kinase inhibitor therapy in patients with chronic-phase chronic myelogenous leukemia who have stopped imatinib after durable undetectable disease. J Clin Oncol 32:424–430. https://doi.org/10.1200/JCO.2012.48.5797

    Article  Google Scholar 

  38. Saussele S, Richter J, Hochhaus A, Mahon FX (2016) The concept of treatment-free remission in chronic myeloid leukemia. Leukemia 30:1638–1647. https://doi.org/10.1038/leu.2016.115

    Article  Google Scholar 

  39. Saussele S, Richter J, Guilhot J, Gruber FX, Hjorth-Hansen H, Almeida A, Janssen JJWM, Mayer J, Koskenvesa P, Panayiotidis P, Olsson-Strmberg U, Martinez-Lopez J, Rousselot P, Vestergaard H, Ehrencrona H, Kairisto V, Machov Polkov K, Mller MC, Mustjoki S, Berger MG, Fabarius A, Hofmann WK, Hochhaus A, Pfirrmann M, Mahon FX, Investigators ES (2018) Discontinuation of tyrosine kinase inhibitor therapy in chronic myeloid leukaemia (EURO-SKI): a prespecified interim analysis of a prospective, multicentre, non-randomised, trial. Lancet Oncol 19:747–757. https://doi.org/10.1016/S1470-2045(18)30192-X

  40. Schütz C, Inselmann S, Sausslele S, Dietz C, Müller M, Eigendorff E, Brendel C, Metzelder S, Brümmendorf T, Waller C et al (2017) Expression of the CTLA-4 ligand CD86 on plasmacytoid dendritic cells (PDC) predicts risk of disease recurrence after treatment discontinuation in CML. Leukemia 31(4):829

    Article  Google Scholar 

  41. Sopper S, Mustjoki S, White D, Hughes T, Valent P, Burchert A, Gjertsen BT, Gastl G, Baldauf M, Trajanoski Z et al (2016) Reduced CD62l expression on T cells and increased soluble CD62l levels predict molecular response to tyrosine kinase inhibitor therapy in early chronic-phase chronic myelogenous leukemia. J Clin Oncol 35(2):175–184

    Article  Google Scholar 

  42. Stein AM, Bottino D, Modur V, Branford S, Kaeda J, Goldman JM, Hughes TP, Radich JP, Hochhaus A (2011) BCR-ABL transcript dynamics support the hypothesis that leukemic stem cells are reduced during imatinib treatment. Clin Cancer Res 17:6812–6821. https://doi.org/10.1158/1078-0432.CCR-11-0396

    Article  Google Scholar 

  43. Wodarz D (2010) Heterogeneity in chronic myeloid leukaemia dynamics during imatinib treatment: role of immune responses. Proc Biol Sci 277:1875–1880. https://doi.org/10.1098/rspb.2009.2179

    Article  Google Scholar 

  44. Woywod C, Gruber FX, Engh RA, Fl T (2017) Dynamical models of mutated chronic myelogenous leukemia cells for a post-imatinib treatment scenario: response to dasatinib or nilotinib therapy. PloS ONE 12:e0179700. https://doi.org/10.1371/journal.pone.0179700

    Article  Google Scholar 

  45. Zitvogel L, Rusakiewicz S, Routy B, Ayyoub M, Kroemer G (2016) Immunological off-target effects of imatinib. Nat Rev Clin Oncol 13(7):431

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to thank Christoph Baldow and Tom Haehnel for their comments and suggestions that contributed to this work. The authors also thank to the anonymous reviewer, whose suggestions were very helpful to improve the quality of this work.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Artur César Fassoni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of ACF was partially supported by the Excellence Initiative of the German Federal and State Governments (Dresden Junior Fellowship) and by CAPES/Pós-Doutorado no Exterior Grant No. 88881.119037/2016-01. This work was further supported by the German Federal Ministry of Education and Research (www.bmbf.de/en/), Grant No, 031A424 “HaematoOpt” to IR and Grant No. 031A315 “MessAge” to IG, as well as the ERA-Net ERACoSysMed JTC-2 project “prediCt” (Project No. 031L0136A) to IR.

Appendices

Mathematical Analysis

The general dynamics of system (1) depend on the interactions between leukemic and immune cells: Four options for F(Y) and five options for G(Y) account for a total of 20 different combinations (sub-models). In this appendix, we identify which scenarios (phase portraits \(\emptyset \), I, II, III, IV and V) are exhibited by each combination. Understanding that the TKI treatment acts as a temporary force moving the system trajectories in the phase space, we analyze the asymptotic behavior of the permanent system, i.e., system (1) with \(e_{\mathrm{TKI}}=0\):

$$\begin{aligned} \dfrac{\mathrm{d}X}{\mathrm{d}t}= & {} -p_{XY} X + p_{YX} Y,\nonumber \\ \dfrac{\mathrm{d}Y}{\mathrm{d}t}= & {} p_{XY} X - p_{YX} Y +p_Y Y\left( 1-\dfrac{Y}{T_Y}\right) - m_K \; Z \; F(Y),\nonumber \\ \dfrac{\mathrm{d}Z}{\mathrm{d}t}= & {} p_Z -d_Z Z + m_R \; Z\; G(Y). \end{aligned}$$
(3)

Non-dimensionalization

To reduce the number of parameters in the analytic study, we introduce the non-dimensional variables

$$\begin{aligned} \tau = t d_Z, \quad x=\dfrac{X}{T_Y}, \quad y=\dfrac{Y}{T_Y}, \quad z=\dfrac{Z}{p_Z/d_Z}, \end{aligned}$$
(4)

and obtain a non-dimensional model, mathematically equivalent to (3), given by

$$\begin{aligned} \dfrac{\mathrm{d}x}{\mathrm{d}\tau }= & {} - \sigma _{xy} x +\sigma _{yx} y, \nonumber \\ \dfrac{\mathrm{d}y}{\mathrm{d}\tau }= & {} \sigma _{xy} x- \sigma _{yx}y + \sigma _y y(1-y) - \gamma _k \; z \; f(y), \nonumber \\ \dfrac{\mathrm{d}z}{\mathrm{d}\tau }= & {} 1-z+\gamma _r \; z\; g(y), \end{aligned}$$
(5)

where the non-dimensional parameters are

$$\begin{aligned}&\sigma _{xy}=\dfrac{p_{XY}}{d_Z}, \quad \sigma _{yx}=\dfrac{p_{YX}}{d_Z}, \quad \sigma _y=\dfrac{p_Y}{d_Z}, \quad \gamma _k=\dfrac{p_Z m_K}{d_Z^2 T_Y}, \nonumber \\&\quad \gamma _r=\dfrac{m_R }{d_Z}, \quad \xi _k = \dfrac{C_K}{T_Y}, \quad \xi _r = \dfrac{C_R}{T_Y}, \end{aligned}$$
(6)

and the non-dimensional functional responses f(y) and g(y) are given, in each case \(f_i(y)=F_i(T_y y )\), \(i\in \{A,B,C,D\}\) and \(g_j(y)=G_j(T_y y)\), \(j\in \{1,2,3,4,5 \}\), by

$$\begin{aligned} f_A(y)= & {} \dfrac{y}{\xi _k}, \ \ f_B(y)=\dfrac{y}{\xi _k+y}, \ \ f_{{C}}(y)= \dfrac{y^2}{\xi _k^2+y^2},\ \ f_D(y)= \dfrac{2\xi _k y}{\xi _k^2+y^2}, \end{aligned}$$
(7)
$$\begin{aligned} g_1(y)= & {} \dfrac{y}{\xi _r}, \ g_2(y)=\dfrac{y}{\xi _r+y}, \ g_3(y)=\dfrac{y^2}{\xi _r^2+y^2},\nonumber \\ g_4(y)= & {} \dfrac{2\xi _r y}{\xi _r^2+y^2}, \ g_5(y)=\dfrac{-y^2}{\xi _r^2+y^2}.\nonumber \\ \end{aligned}$$
(8)

Conditions on Model Parameters

We may impose biologically plausible conditions on some model parameters. As stated in Sect. 2, parameters \(C_K\) and \(C_R\) define the levels of leukemic cells at which the functions \(F_i(Y)\) and \(G_j(Y)\) exert their action. Thus, it is reasonable to assume that all these effects occur on a population level lower than the carrying capacity \(T_Y\) of the leukemic cells. Thus, we assume that \(C_K,C_R<T_Y\). Further, we note that the expression \(1/d_Z\) corresponds to the mean life time of immune effector cells, while \(1/m_R\) can be understood as the mean time of individual immune recruitment when the recruitment is at maximum, i.e., \(G(Y)\approx 1\). In other words, \(1/m_R\) is the mean time spent by one immune cell between the last contact with leukemic cells and the arrival of a new immune cell, in a situation where the recruitment is at maximum. It is reasonable to assume that the former time (\(1/d_Z\)) is greater than the latter (\(1/m_R\)), i.e, \(d_Z<m_R\). Taken together, these conditions on the dimensional parameters correspond to conditions on the non-dimensional parameters:

$$\begin{aligned} \xi _k<1, \ \ \xi _r<1, \ \gamma _r>1. \end{aligned}$$
(9)

Analysis of the Trivial Steady State \(E_0\)

All models for immune response f(y) and recruitment g(y) satisfy \(f(0)=g(0)=0\). Thus, the point \(E_0=(0,0,1)\) (with coordinates (xyz)) is the trivial steady state for system (5). \(E_0\) is the “cure steady state” The Jacobian matrix of system (5) evaluated at \(E_0\) is

$$\begin{aligned} J(E_0) = \left[ \begin{matrix} -\,\sigma _{xy} &{}\quad \sigma _{yx} &{}\quad 0 \\ \sigma _{xy} &{}\quad \sigma _y-\sigma _{yx}-\gamma _kf'(0) &{}\quad 0 \\ 0 &{}\quad \gamma _r g'(0) &{}\quad -\,1 \end{matrix} \right] . \end{aligned}$$
(10)

The characteristic polynomial of \(J(E_0)\) is \(p_0(\lambda )=-(\lambda +1) ( \lambda ^2 + a_1 \lambda + a_0 )\), where \(a_1=\sigma _{xy}+\sigma _{yx}+\gamma _k f'(0)-\sigma _{y}\) and \(a_0=\sigma _{xy} \left( \gamma _k f'(0)-\sigma _{y} \right) \). Thus, one eigenvalue of \(J(E_0)\) is \(\lambda _1=-1<0\), while the other two satisfy \(\lambda _2+\lambda _3=-a_1\) and \(\lambda _2\lambda _3=a_0\). We have the following cases:

  1. 1.

    If \(\gamma _k f'(0)>\sigma _{y}\) then \(a_0>0\) and \(a_1>0\). Thus, \(\lambda _2,\lambda _3<0\) and \(E_0\) is a stable node.

  2. 2.

    If \(\gamma _k f'(0)<\sigma _{y}\) then \(a_0<0\). Thus, \(\lambda _2<0<\lambda _3\) and \(E_0\) is a saddle-point, with two negative eigenvalues.

The above results are summarized in the following proposition.

Proposition 1

Let

$$\begin{aligned} \phi =\dfrac{\gamma _k}{\sigma _y}f'(0). \end{aligned}$$
(11)

If \(\phi >1\), then \(E_0\) is a locally asymptotically stable steady state for system (5). If \(\phi <1\), then \(E_0\) is a saddle-point for system (5), with one positive eigenvalue and two eigenvalues with negative real part.

Remark 1

In terms of the dimensional parameters, parameter \(\phi \) is written as

$$\begin{aligned} \phi =\dfrac{p_Z}{d_Z} \dfrac{m_K}{p_Y} F'(0), \end{aligned}$$
(12)

(note that \(f'(0)=F'(0)T_Y\)). Thus, the cure-equilibrium \(E_0\) is stable if, and only if,

$$\begin{aligned} \dfrac{p_Z }{d_Z} m_K F'(0)>p_Y. \end{aligned}$$
(13)

Analysis of Nontrivial Steady States

Besides the trivial steady state, system (5) can have multiple other nontrivial steady states, which we study in the following. Setting \(\mathrm{d}x/\mathrm{d}\tau =0\), we obtain \(x=({\sigma _{yx}}/{\sigma _{xy}})y\). Substituting this expression in \(\mathrm{d}y/\mathrm{d}\tau =0\) and solving for z, we obtain

$$\begin{aligned} z=h_1(y):=\dfrac{\sigma _y y(1-y)}{\gamma _k f(y)}. \end{aligned}$$
(14)

On the other hand, setting \(\mathrm{d}z/\mathrm{d}\tau =0\) leads to

$$\begin{aligned} z=h_2(y):=\dfrac{1}{1-\gamma _rg(y)}. \end{aligned}$$
(15)

Taken together, (14) and (15) imply that the coordinate \(y\ne 0\) of a nontrivial steady state should satisfy

$$\begin{aligned} h_1(y)=h_2(y), \end{aligned}$$
(16)

Therefore, the nontrivial steady states of system (5) are the points \(E=(x^*(y),y,z^*(y))\) where

$$\begin{aligned} x^*(y) = \dfrac{\sigma _{yx}}{\sigma _{xy}}y, \ \ \mathrm{and} \ \ z^*(y) = h_2(y). \end{aligned}$$
(17)

and y is a nonzero root of (16), which will be refereed as the equation for the nontrivial steady states.

Existence of Biologically Feasible Nontrivial Steady States

A nontrivial steady state is biologically feasible if its coordinates are positive, i.e., \(y>0\), \(x^*(y) > 0\) and \(z^*(y)\ge 0\). Checking expressions in (14) and (15) and using the fact that \(f(y)\ge 0\) for all choices of f(y), we conclude that a root y of \(h_1(y)=h_2(y)\) is feasible if, and only if, \(0<y\le 1\) and \(h_2(y) > 0\). These results are summarized in the following proposition.

Proposition 2

The feasible nontrivial steady states of system (5) are the equilibrium points \(E=(x^*(y), y,\)\( z^*(y))\) where \(x^*(y)\) and \(z^*(y)\) are defined in (17) and y is a root of (16) such that \(0<y\le 1\) and \(h_2(y) > 0\).

It is our goal to describe which qualitatively distinct phase portraits (classified as 5 different scenarios, defined in Fig. 4) are exhibited by each sub-model and which are not. This question may be approached by assessing, for each sub-model, the number of roots of (16) which satisfy the feasibility conditions (note that scenario I corresponds to 1 feasible nontrivial steady state, scenario II corresponds to 2 feasible nontrivial steady states, and so on; see Fig. 4).

To do so, we first present the expressions of \(h_1\) and \(h_2\) for each choice of f(y) and g(y). We will express \(h_1(y)\) in terms of \(\phi \), as this parameter is a threshold for the stability of \(E_0\) and so is related to changes between scenarios. Remembering that \(\phi = (\gamma _k/\sigma _y)f'(0)\) and using the formulas for \(f_i(y)\), \(i\in \{ A,B,C,D\}\) in Eq. (7), we obtain:

$$\begin{aligned} f_A'(0)=\dfrac{1}{\xi _k}, \ \ f_B'(0)=\dfrac{1}{\xi _k}, \ \ f_{{C}}'(0)=0, \ \ f_D'(0)=\dfrac{2}{\xi _k}. \end{aligned}$$
(18)

Then, \(\phi = \gamma _k/(\sigma _y \xi _k) \) if \( f= f_A\), \(\phi =\gamma _k/(\sigma _y \xi _k)\) if \( f= f_B\), \(\phi = 0\) if \( f= f_{{C}}\), and \(\phi = 2\gamma _k/(\sigma _y \xi _k)\) if \( f= f_D\). Thus, in each case, the formula for \(h_1(y)\) can be expressed in terms of \(\phi \) as

$$\begin{aligned} h_1 (y) = \left\{ \begin{array}{lll} \dfrac{1-y}{\phi }&{}\quad \mathrm{if}\,\,f=f_A, \\ \\ \dfrac{(1-y)(1+y/\xi _k)}{\phi }&{}\quad \mathrm{if} \,\, f=f_B, \\ \\ \dfrac{\sigma _y}{\xi _k} \dfrac{(1-y)(\xi _k^2+y^2)}{y}&{}\quad \mathrm{if} \,\, f=f_{{C}}, \\ \\ \dfrac{(1-y)(1+(y/\xi _k)^2)}{\phi } &{}\quad \mathrm{if}\,\, f=f_D. \end{array} \right. \end{aligned}$$
(19)

In the same way, the expressions for \(h_2(y)\) in each case \(g (y)=g_j(y)\), \(j\in \{ 1,2,3,4,5\}\), are

$$\begin{aligned} h_2 (y) = \left\{ \begin{array}{lllll} \dfrac{1}{1-\gamma _r (y/\xi _r)}&{}\quad \mathrm{if} \ g=g_1, \\ \\ \dfrac{1+(y/\xi _r)}{1-(\gamma _r-1) (y/\xi _r)} &{}\quad \mathrm{if} \ g=g_2, \\ \\ \dfrac{1+(y/\xi _r)^2}{1-(\gamma _r-1) (y/\xi _r)^2} &{}\quad \mathrm{if} \ g=g_3, \\ \\ \dfrac{1+(y/\xi _r)^2}{1-2 \gamma _r (y/\xi _r) + (y/\xi _r)^2} &{}\quad \mathrm{if} \ g=g_4, \\ \\ \dfrac{1+(y/\xi _r)^2}{1+(\gamma _r+1) (y/\xi _r)^2} &{}\quad \mathrm{if} \ g=g_5. \end{array} \right. \end{aligned}$$
(20)

To analyze the sub-models, we will also study a polynomial equation equivalent to \(h_1=h_2.\) From Eqs. (19) and (20), we note that \(h_1\) and \(h_2\) are rational functions, i.e., quotient of polynomial functions. Therefore, equation \(h_1(y)=h_2(y)\), is equivalent to the polynomial equation

$$\begin{aligned} q(y):=\mathrm{num}_{h_1}(y)\mathrm{den}_{h_2}(y)-\mathrm{num}_{h_2}(y)\mathrm{den}_{h_1}(y)=0, \end{aligned}$$
(21)

where \(\mathrm{num}_{h_i}\) and \(\mathrm{den}_{h_i}\) are the numerator and denominator of \(h_i\), respectively, and are polynomial functions. For each choice \((f_i,g_j)\) of the sub-models, a different polynomial equation will be obtained. From the expressions of \(h_1\) and \(h_2\) above, we observe that the maximum degree for q(y) is 5, which means that the maximum number of feasible nontrivial steady states is 5.

The number of positive roots of Eq. (21) will be studied by using the Descartes’ Rule of Sign. If these roots satisfy the feasibility conditions, \(0<y\le 1\) and \(h_2(y)>0\), then, their quantity correspond to the number of feasible nontrivial steady states. The Descartes’ Rule of Sign states that the number of positive roots of a polynomial equation is equal to the number of sign variations between consecutive nonzero coefficients or less than it by some even number (Kennedy 2001). For instance, if q is a fourth-degree polynomial and its coefficients satisfy \(c_0>0\), \(c_1>0\), \(c_2<0\), \(c_3>0\) and \(c_4<0\), then the sign sequence is of q is \(S_q=(++-+-)\) (S for sequence) and the number of sign variations is \(V_q=3\) (V for variation). Therefore, such polynomial has 1 or 3 positive roots. The notation introduced in this paragraph will be used throughout the subsequent analysis. Further, we denote by ± the sign of a coefficient which can be positive or negative. To facilitate the referencing, we summarize the above results in the following lemma.

Lemma 1

Let \(q(y)=0\) the polynomial equation for the nontrivial roots, with coefficients \(c_i\). The number of roots of q satisfying \(y>0\) is given by \(V_q-2l\), where l is a positive integer such that \( V_q \ge 2 l\).

Now, we introduce a way to easily verify which positive roots of q satisfy the feasibility condition \(y\le 1\). Substituting \(y=1+u\) in \(q(y)=0\), we obtain another polynomial equation, \(r(u)=0\), where \(r(u)=q(1+u)\). Each root \(u>0\) of \(r(u)=0\) corresponds to a root \(y=1+u>1\) of \(q(y)=0\). Thus, using the Descartes’ Rule of Sign as above, we may count the number of positive roots of r(u), which is equal to the number of positive roots of q which do not satisfy the feasibility condition \(y\le 1\). Further, we note that r(u) has the same degree as q and, if \(q(y)=c_0+c_1y+\cdots + c_n y^n\) and \(r(u)=d_0+d_1u+\cdots + d_nu^n\) have degree n, then the coefficients \(d_0\) and \(d_n\) of r(u) are related to the coefficients \(c_i\) of q as

$$\begin{aligned} d_0=c_0+c_1+\cdots + c_n \ \ \ \ \mathrm{and} \ \ \ \ d_n = c_n. \end{aligned}$$
(22)

Furthermore, we conclude that, if \(d_0\) and \(d_n\) have different signs, then the sequence \(S_r\) of signs of r(u) has at least one sign variation and the number of sign variations \(V_r\) is an odd number. This is true because every sign change between \(d_0\) and \(d_n\) needs to occur in pairs and \(V_r=1+2k\) for some integer \(k\ge 0\), i.e., \(V_r\) is an odd number if \(d_0\) and \(d_n\) have different signs. Using this result, we obtain a simple criterion for the existence of positive but non-feasible roots for q, which is summarized in the following lemma.

Lemma 2

Let \(q(y)=0\) the polynomial equation for the nontrivial roots, with coefficients \(c_i\), and \(r(u)=q(1+u)\), with coefficients \(d_i\). Then, the formulas in (22) hold. Furthermore, if \(d_0\) and \(d_n\) have different signs, then the number of the positive roots of q which satisfy the condition \(y>1\) is a positive odd number, less or equal the degree of q.

Using the above tools, we analyzed the number of feasible nontrivial steady states for each sub-model. The detailed analysis for each case is given in “Analysis of Number of Feasible Nontrivial Steady States for Each Sub-Model” of appendix.

Stability of the Biologically Feasible Nontrivial Steady States

The analysis on the existence of feasible nontrivial steady states is further complemented by an analytic approach to address their stability. Theorem 1 asserts that, together with the trivial steady state \(E_0\), such nontrivial steady states have alternated stabilities, i.e., denoting by \(y_i\) the y coordinates of the steady states \(E_i\), with \(0=y_0<y_1<\cdots <y_n\), we show that if \(E_i\) is stable then \(E_{i+1}\) and \(E_{i-1}\) are unstable. This guarantees that the maximum number of stable steady states (and of basins of attraction) is three [since the maximum number of feasible nontrivial steady states is five, which is the highest possible degree for Eq. (21)].

Theorem 1

Consider system (5) under conditions (9) and a given choice of \((f,g)=(f_i,g_j)\), \(i\in \{A,B,C,D\}\), \(j\in \{1,2,3,4,5 \}\). Denote the trivial steady state as \(E_0=(0,y_0,1)\) with \(y_0=0.\) Suppose that Eq. (16) admits n nontrivial roots \(y_i\), \(i=1,\ldots , n\) satisfying the feasibility conditions of Proposition 2, ordered as \(0=y_0<y_1<\cdots <y_n\). Denote by \(E_i=(x^*(y_i),y_i,z^*(y_i))\) the corresponding feasible nontrivial steady state and \(J(E_i)\) the Jacobian matrix of system (5) evaluated at \(E_i\), \(i=1,\ldots , n\). Then, for all \(i=0,1,\ldots ,n\), we have

$$\begin{aligned} \mathrm{sign } \left( \det \left( J(E_i) \right) \right) = \mathrm{sign }\left( 1-\phi \right) (-1)^i, \ \ \ i=1,\ldots ,n. \end{aligned}$$
(23)

Therefore, if \(i<n\) and \(E_i\) is a stable steady state, then \(E_{i+1}\) is an unstable steady state (unstable focus or node, or saddle-point). In the same way, if \(i>0\) and \(E_i\) is a stable steady state, then \(E_{i-1}\) is an unstable steady state.

Proof

From expressions (10) and (7), we obtain that \(\det (J(E_0)) = \sigma _y \sigma _{xy}(1-\phi )\), which shows that Eq. (23) holds for \(i=0\). To proof it for the nontrivial equilibria, we observe that the Jacobian matrix of system (5) is

$$\begin{aligned} J= \left[ \begin{matrix} -\,\sigma _{xy} &{}\quad \sigma _{yx} &{}\quad 0 \\ \sigma _{xy} &{}\quad -\,\sigma _{yx} +u_1 &{}\quad -\,\gamma _k f(y) \\ 0 &{}\quad u_2 &{}\quad -\,1+\gamma _r g(y) \end{matrix} \right] . \end{aligned}$$
(24)

where

$$\begin{aligned} u_1=\sigma _y(1-2y) - \gamma _k z f'(y), \ \ \mathrm{and} \ \ u_2 = \gamma _r z g'(y). \end{aligned}$$
(25)

Therefore, the determinant of J is

$$\begin{aligned} \det J = \sigma _{xy}\left( u_1(1-\gamma _rg(y))-u_2\gamma _kf(y)\right) . \end{aligned}$$
(26)

To simplify such expression, we first calculate the derivatives of \(h_1(y)\) and \(h_2(y)\) in (14) and (15):

$$\begin{aligned} h_1'(y)=\dfrac{\sigma _y(1-2y)\gamma _kf(y)-\sigma _y y (1-y) \gamma _kf'(y)}{(\gamma _kf(y))^2}, \ \ \ h_2'(y)=\dfrac{\gamma _r g'(y)}{(1-\gamma _rg(y))^2}. \end{aligned}$$
(27)

Now, we observe that the coordinates of the nontrivial steady state satisfy \(z=h_1(y)=h_2(y)\). Substituting \(z=h_1(y)\) and \(z=h_2(y)\) into the expressions for \(u_1\) and \(u_2\), respectively, and comparing the resulting expressions with (27), we obtain that

$$\begin{aligned} u_1= & {} \dfrac{\sigma _y(1-2y)\gamma _kf(y)-\sigma _y y (1-y) \gamma _kf'(y)}{\gamma _kf(y)}= h_1'(y)\gamma _kf(y), \nonumber \\ u_2= & {} \dfrac{\gamma _r g'(y)}{1-\gamma _rg(y)} = h_2'(y)(1-\gamma _rg(y)). \end{aligned}$$
(28)

From (26) and (28), we obtain that the Jacobian determinant evaluated at the \(E_i=(x_i,y_i,z_i)\) is

$$\begin{aligned} \det J(E_i)&= \sigma _{xy}\gamma _kf(y_i)(1-\gamma _rg(y_i))\left( h_1'(y_i)-h_2'(y_i)\right) \nonumber \\&= \sigma _{xy}\gamma _kf(y_i)(1-\gamma _rg(y_i))h'(y_i), \end{aligned}$$
(29)

where \(h(y)=h_1(y)-h_2(y)\). The feasibility conditions guarantee that \(f(y_i)(1-\gamma _rg(y_i))>0\). Therefore, the sign of \(\det J(E_i)\) is equal to the sign of \(h'(y_i)\). Note that the roots of h(y) are the nontrivial roots \(y_i \) of (16), defining the nontrivial steady states. Since h(y) is differentiable, we conclude that the sign of \(h'(y)\) changes at two consecutive roots \(y_i\) and \(y_{i+1}\), i.e., \(h'(y_i)h'(y_{i+1})<0\). (Otherwise, there would be another root between \(y_i\) and \(y_{i+1}\).) Therefore, from expression (29), we conclude that the Jacobian determinant alternate its signs at consecutive nontrivial equilibria.

It remains to show that the sign of \(\det J(E_1)\) is opposite to the sign of \((1-\phi )\). To do so, we consider the function \(r(y)=yh(y)\). Note that h(y) is defined at \(y=0\) only when \(f=f_A,f_B,f_D\) [because \(h_1(0)\) is not defined when \(f=f_{{C}}\), see (14)]. For these cases, the roots of \(r(y)=0\) are exactly \(y_0=0\) and the nontrivial roots \(y_i\) of (16). Since \(y_0=0\) and \(y_1>0\) are consecutive roots of r(y), which is differentiable, we conclude that \(r'(0)\) and \(r'(y_1)\) have opposite signs. On the other hand, in these cases we have \(h_1'(0)=1/\phi \) [see (14)], and so \(r'(0)=h'(0)=(1-\phi )/\phi \) and \(r'(y_1)=y_1h'(y_1)\). Since \(y_1>0\), we conclude \(h'(y_1)\) have the same sign of \(r'(y_1)\) which is opposite to the sign \(r'(0)=(1-\phi )\). This shows that the sign of \(\det (J(E_1))\) is opposite to the sign of \((1-\phi )\) when \(f=f_A,f_B,f_D\). For \(f=f_{{C}}\), note that \(\phi =0\) and so we need to show that the sign of \(\det (J(E_1))\) is negative. In this case, from (14) and from the fact that \(h_2(0)=1\), we have \(\lim _{y\rightarrow 0^+} h(y) =+\infty \), which implies that h(y) is positive for \(y>0\) sufficiently small. Since \(y_1\) is the first root of h(y), we conclude that the sign of h(y) changes from positive to negative at \(y_1\), which means that \(h'(y_1)<0\) and so the sign of \(\det (J(E_1))\) is negative.

The last claim of Theorem 1 is an immediate consequence that the product of the three eigenvalues of \(J(E_i)\) is given by its determinant, and so a stable steady state leads to a negative Jacobian determinant, while a positive determinant implies at least one positive eigenvalue. This concludes the proof. \(\square \)

Remark 2

Since an unstable steady state may also have a negative Jacobian determinant, it follows that Theorem 1 does not exclude the possibility of having three consecutive unstable steady states. We numerically verified that some sub-models indeed present such configuration. In such cases, we found that there may be an unstable focus between two saddle-points. This is probably originating from a Hopf bifurcation, and we found certain parameter intervals where numerical results indeed point to such a bifurcation and to the existence of a stable limit cycle around such unstable focus (see Fig. 9). In scenarios III, IV and V, we would still have two, three and three basins of attraction, respectively, but one basin would be the one of the limit cycle. Thus, those cases would correspond to modified scenarios which we denote as III\(^\mathrm{o}\), IV\(^\mathrm{o}\) and V\(^\mathrm{o}\), where one of the attractors is a limit cycle instead of steady state. The precise investigation of such phenomena is outside the scope of this study and will be considered in a future work.

Fig. 9
figure9

Occurrence of three consecutive unstable steady states. a Bifurcation diagram of steady states of system (5) for sub-model \((f,g)=(f_4,g_D)\). Starting in scenario IV, with high values of the bifurcation parameter \(m_k\), the steady state \(E_L\) is a stable focus (light blue line). Decreasing the parameter value, \(E_L\) becomes a unstable focus (purple line), indicating the occurrence of a Hopf bifurcation, and leading to scenario V\(^\mathrm{o}\) (o for oscillation), and thus, we have three consecutive unstable steady states (the other are two saddle-points, indicated by the orange lines). For smaller values of the bifurcation parameter, scenario III\(^\mathrm{o}\) is observed, b resulting phase portrait of scenario V\(^\mathrm{o}\) for a parameter value in the range indicated in a. The steady state \(E_L\) is a unstable focus (indicated by \(\odot \)). The orange solution oscillates around \(E_L\) and provides an idea of the limit cycle size, c model time courses showing different asymptotic behaviors after treatment cessation in different time points. The red solution is attracted to \(E_H\), the blue solution is attracted to \(E_D\), and the orange solution is attracted to the limit cycle around \(E_L\) (these solutions correspond to the trajectories showed in b) (Color figure online)

Summary of Analysis of Nontrivial Steady States

With the above results, we were be able to conclude that some scenarios are (1) not exhibited by a given sub-model, due to the number of feasible nontrivial roots admitted by Eq. (16), and (2) possibly exhibited by such sub-model. Then, in order to confirm the occurrence of such possible scenarios, we numerically investigated each sub-model and obtained parameter values for which the possible scenarios are indeed exhibited by each sub-model. All results are summarized in Theorem 2.

Theorem 2

Consider system (5) under conditions (9), and scenarios \(\emptyset \), I, II, III, IV and V, defined by their number of feasible nontrivial steady-states, 0, 1, 2, 3, 4 and 5, respectively, with their local stability as indicated in Fig. 4. Then, for each choice \((f,g)=(f_i,g_j)\), \(i\in \{A,B,C,D\}\), \(j\in \{1,2,3,4,5 \}\), given in (7) and (8), a given subset of these scenarios cannot be exhibited by such sub-model, while the remaining scenarios do. Such subsets and the parameter values which provide the feasible scenarios are indicated in Tables 2 and 3.

Remark 3

Theorem 2 is a “non-existence result” which exclude the possibility of some sub-models to exhibit some scenarios. The affirmative answer is obtained by setting some model parameters (parameters \(p_Y\), \(C_K\), \(C_R\), \(m_K\) and \(m_R\)) to a certain set of values in such way the scenario is observed when we numerically calculate the steady states and their local stability (through the numerical calculation of the Jacobian matrix and its eigenvalues). The global behavior of system (5) (existence of limit cycles or other nontrivial attractors) is not approached by Theorem 2. Table 1 summarizes the results of Tables 2 and 3.

Table 2 Summary of analysis of models \(F=F_A,F_B\) and \(G=G_1,G_2,G_3,G_4,G_5\)
Table 3 Summary of analysis of models \(F=F_{{C}},F_D\) and \(G=G_1,G_2,G_3,G_4,G_5\)

Analysis of Number of Feasible Nontrivial Steady States for Each Sub-Model

In the following, we analyze each sub-model in detail. Conditions (9) are assumed to hold.

Analysis for\(f=f_A\)and\(g=g_1,g_2,g_3\) For \(f(y)=f_A(y)\), from (19), we have that

$$\begin{aligned} h_1'(y)=-\dfrac{1}{\phi }. \end{aligned}$$

Therefore, \(h_1(y)\) is a decreasing function. Furthermore, \(h_1(y)\) is nonnegative only in the interval \(0\le y <1\) with \(h_1(0)=1/\phi \) and \(h_1(1)=0\).

Now, we study \(h_2(y)\) for the choices \(g=g_1,g_2,g_3\). From the expressions of \(h_2(y)\) in these cases, we conclude the denominator of \(h_2(y)\) vanishes and change its sign at an unique \(y_d>0\), where \(y_d=\xi _r/\gamma _r\) for \(g=g_1\), \(y_d=\xi _r/(\gamma _r-1)>0\) for \(g=g_2\), and \(y_d=\xi _r/\sqrt{(\gamma _r-1)}\) for \(g=g_3\) (since \(\gamma _r>1\)). Analyzing the sign of the denominator, we conclude that \(h_2(y)\) is positive for \(0\le y<y_d\) and is negative for \(y>y_d\). Therefore, the feasibility condition \(h_2(y) > 0\) restricts the analysis to the interval \(0 \le y < y_d\). Now we show that \(h_2(y)\) is increasing in this interval. From the general equation for \(h_2(y)\) (15), we have

$$\begin{aligned} h_2'(y)=\dfrac{\gamma _r g'(y)}{(1-\gamma _r g(y))^2}. \end{aligned}$$

Since g(y) is an increasing function when \(g=g_1,g_2,g_3\), we have \(g'(y)\ge 0\). Thus, \(h_2'(y)\ge 0\) and \(h_2(y)\) is an increasing function for these cases. Summarizing, in the interval \(0 \le y < y_d\), we have \(h_2(0)=1\), \(h_2(y)\) is increasing, and \(h_2(y)\rightarrow +\infty \) when \(y\rightarrow y_d\), \(y<y_d\).

Combining the properties of \(h_1\) and \(h_2\), we conclude the following. If \(h_1(0)>h_2(0)\), then \(h_1\) and \(h_2\) intersect in the interval \(0<y<y_d\) exactly once, at some \(y^*\) such that \(0<y^*<1\) and \(h_2(y^*)>0\). Thus, there is an unique nontrivial steady state E. If \(h_1(0)<h_2(0)\), then \(h_1\) and \(h_2\) do not intersect and there is no feasible nontrivial steady state. Since \(h_1(0)=1/\phi \) and \(h_2(0)=1\), we have established the following result: if \(\phi >1\), system (5) has an unique nontrivial steady state E; If \(\phi <1\), system (5) does not admit nontrivial steady states. This result states that sub-models with \(f=f_A\) and \(g=g_1,g_2,g_3\) do not exhibit scenarios II, III, IV and V. The only possible scenarios exhibited by these sub-models are I and \(\emptyset \). We numerically verified that such scenarios are indeed exhibited by these sub-models (see Table 2).

Analysis for\(f=f_A\)and\(g=g_4\) For \(f=f_A\), we have \(h_1(y)=(1-y)/\phi \). Thus, \(h_1(y)\) is a decreasing function with \(h_1(0)=1/\phi> h_1(y) > h_1(1)=0\) for \(0<y<1\). On the other hand, since \(g_4(y)\ge 0\)\(h_2(y)=1/(1-\gamma _r g_4(y))\), we conclude that \(h_2(y)\ge 1\) if \(g_4(y)<1/\gamma _r\) and, \(h_2(y) < 0\) if \(g_4(y)>1/\gamma _r\). Since one of the feasibility conditions is \(h_2(y)>0\), we conclude that if a feasible root \(y^*\) occurs, then \(h_2(y^* )\ge 1\). Hence, no feasible root occurs for \(\phi >1\), since, in this case, \(1>h_1(y)>0\) for \(0<y<1\). Therefore, only scenario \(\emptyset \) occurs if \(\phi >1\). The case \(\phi <1\) is studied with the polynomial equation q, which has degree \(\partial (q) = 3\) and coefficients

$$\begin{aligned} c_0 = 1-\phi>0, \ c_1 = -1-2 \dfrac{\gamma _r}{\xi _r}<0, \ c_2 = \dfrac{1-\phi +2\gamma _r \xi _r}{\xi _r^2} >0, \ c_3 = -\dfrac{1}{\xi _r^2} <0. \end{aligned}$$

Thus, the sign sequence of q is \(S_q=(+-+-)\) and the number of sign variations is \(V_q=3\). Therefore, q admits 1 or 3 positive roots. Thus, scenarios I and III are possible for model \((f_A,g_4)\) if \(\phi <1\), while the other scenarios are excluded. We numerically verified the occurrence of scenarios I and III (see Table 2).

Analysis for\(f=f_A\)and\(g=g_5\). In this case, q is a third-degree polynomial with coefficients

$$\begin{aligned} c_0 = 1-\phi , \ c_1 = -1<0, \ c_2 = \dfrac{1-\phi +\gamma _r}{\xi _r^2}, \ c_3 = -\dfrac{1+\gamma _r}{\xi _r^2} <0, \end{aligned}$$

[remember that the non-dimensional parameters are positive and satisfy conditions (9)]. If \(\phi <1\), the sign sequence is \(S_q= (+-+- )\) and the number of sign variations is \(V_q=3\). Thus, such equation admits 1 or 3 positive roots and scenarios I and III are possible for model \((f_A,g_5)\). If \(\phi >1\), the sign sequence is \(S_q= (--\pm - )\) and the number of sign variations is \(V_q=0\) or 2. If we additionally have \(\phi <1 + \gamma _r\), then \(c_2>0\) and \(V_q=2\). In this case, equation \(q(y)=0\) admits 0 or 2 positive roots and scenarios \(\emptyset \) and II are possible for model \((f_A,g_5)\). We numerically verified the occurrence of scenarios I and III, and \(\emptyset \) and II, under the respective conditions above (see Table 2).

Analysis for\(f=f_B\)and\(g=g_1\). In this case, q is a third-degree polynomial with coefficients

$$\begin{aligned} c_0 = 1-\phi , \ c_1 = \dfrac{1-\xi _k}{\xi _k} -\dfrac{\gamma _r}{\xi _r}, \ c_2 = - \dfrac{ \gamma _r (1-\xi _k)+\xi _r }{\xi _r^2} <0, \ c_3 = \dfrac{\gamma _r}{\xi _r \xi _k} >0, \end{aligned}$$

(remember that \(\xi _k<1\)). By Lemma 2, we obtain that the coefficients \(d_0\) and \(d_3\) of r(u) are

$$\begin{aligned} d_0 = -\phi< 0 < d_3 = \dfrac{\gamma _r}{\xi _r \xi _k}. \end{aligned}$$

Thus, it follows also by Lemma 2 that 1 or 3 positive roots of q are non-feasible, i.e., satisfy \(y>1\). Now we analyze the sign sequence of q. If \(\phi < 1\), the sign sequence is \(S_q= (+\pm -+)\) and the number of sign variations is \(V_q=2\). Thus, such equation admits 0 or 2 positive roots. Since at least one is non-feasible, equation q has 2 positive roots but an unique feasible root. Thus, only scenario I is possible if \(\phi <1\). If \(\phi >1\), the sign sequence is \(S_q= (-\pm -+)\) and the number of sign variations is \(V_q=1\) or 3. Thus, equation \(q(y)=0\) admits 1 or 3 positive roots. Since 1 or 3 of such roots are non-feasible, we have 0 or 2 feasible roots. Hence, only scenarios \(\emptyset \) and II are possible if \(\phi >1\). We numerically verified the occurrence of scenarios I, and \(\emptyset \) and II, under the respective conditions above (Table 2).

Analysis for\(f=f_B\)and\(g=g_2\). In this case, q is a third-degree polynomial with coefficients

$$\begin{aligned} c_0 = 1-\phi , \ c_1 = \dfrac{1-\xi _k}{\xi _k} -\dfrac{\gamma _r}{\xi _r}, \ c_2 = - \dfrac{ \gamma _r (1-\xi _k)+\xi _r }{\xi _r^2} <0, \ c_3 = \dfrac{\gamma _r-1}{\xi _r \xi _k} >0. \end{aligned}$$

By Lemma 2, the coefficients \(d_0\) and \(d_3\) of r(u) are

$$\begin{aligned} d_0 = -\phi -\dfrac{\phi }{\xi _r}< 0 < d_3 = \dfrac{\gamma _r-1}{\xi _r \xi _k}. \end{aligned}$$

Thus, it follows that 1 or 3 positive roots of q satisfy \(y>1\). Now we analyze the sign sequence of q. If \(\phi < 1\), we have \(S_q= (+\pm -+)\) and \(V_q=2\). Thus, q admits 0 or 2 positive roots. As at least one satisfies \(y>1\), we conclude that q has 2 positive roots but an unique root in the interval \(0<y<1\). Hence, only scenario I is possible if \(\phi <1\). If \(\phi >1\), we have \(S_q= (-\pm -+)\) and \(V_q=1\) or 3. Thus, q admits 1 or 3 positive roots. Since 1 or 3 of such roots are non-feasible, we have 0 or 2 feasible roots. Hence, only scenarios \(\emptyset \) and II are possible if \(\phi >1\). We numerically verified the occurrence of scenarios I, and \(\emptyset \) and II, under the respective conditions above (see Table 2).

Analysis for\(f=f_B\)and\(g=g_3\). In this case, q is a fourth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} 1-\phi , c_1 = \dfrac{1-\xi _k}{\xi _k}>0, c_2 = -\dfrac{1}{\xi _k} - \dfrac{ \gamma _r - 1}{\xi _r^2} - \dfrac{\phi }{\xi _r^2}<0, \\ c_3= & {} \dfrac{\gamma _r-1}{\xi _r^2} \dfrac{\xi _k-1}{\xi _k} <0,\quad c_4 = \dfrac{\gamma _r-1}{\xi _k \xi _r^2} >0. \end{aligned}$$

The coefficients \(d_0\) and \(d_4\) of r(u) are

$$\begin{aligned} d_0 = -\dfrac{\phi }{\xi _r^2}(1+\xi _r^2)< 0 < d_4 = \dfrac{\gamma _r-1}{\xi _k \xi _r^2}. \end{aligned}$$

Thus, by Lemma 2, 1 or 3 positive roots of q satisfy \(y>1\). Let us analyse the sign sequence of q. If \(\phi < 1\), then \(S_q= (++--+)\) and \(V_q=2\). Thus, q admits 0 or 2 positive roots. As at least one satisfies \(y>1\), it follows that q has 2 positive roots but an unique root in the interval \(0<y<1\). Hence, only scenario I is possible if \(\phi <1\). If \(\phi >1\), then \(S_q= (-+--+)\) and \(V_q=3\). Thus, q admits 1 or 3 positive roots. Since 1 or 3 of such roots satisfy \(y>1\), we have 0 or 2 feasible roots. Hence, only scenarios \(\emptyset \) and II are possible if \(\phi >1\). We numerically verified the occurrence of scenarios I, and \(\emptyset \) and II, under the respective conditions above (see Table 2).

Analysis for\(f=f_B\)and\(g=g_4\). In this case, q is a fourth-degree polynomial. A priori, q may admit up to 4 positive roots and scenarios \(\emptyset \), I, II, III and IV would be possible, while scenario V is not possible, since it corresponds to five positive roots for q. Indeed, we numerically verified the occurrence of all these scenarios (see Table 2).

Analysis for\(f=f_B\)and\(g=g_5\). In this case, the next lemma guarantees that equation \(h_1=h_2\) does not admit more than two different positive roots. This implies that only scenarios \(\emptyset \), I and II are possible. We numerically verified the occurrence of these scenarios (see Table 2).

Lemma 3

If \((f,g)=(f_B,g_5)\), then equation \(h_1=h_2\) admits at most two different positive roots.

Proof

Suppose by contradiction that \(h_1(y)=h_2(y)\) admits three or more different positive roots. Since \(h_1\) and \(h_2\) are differentiable functions for all \(y\ge 0\), it follows from the Rolle theorem that the difference of their derivatives, \(w(y)=h_1'(y)-h_2'(y)\), has at least two different positive roots, say \(0< \hat{y}_1 < \hat{y}_2\). The expressions for \(h_1'(y)\) and \(h_2'(y)\) are

$$\begin{aligned} h_1'(y) = \dfrac{1-\xi _k-2y}{\xi _k\phi } \ \ \mathrm{and} \ \ h_2'(y) = \dfrac{-2\gamma _r\xi _r^2 y}{(\xi _r^2+(1+\gamma _r)y^2)^2}. \end{aligned}$$

Thus, \(\lim _{y\rightarrow \infty }{w(y)}=-\infty \), which implies that there exists \(M>0\) such that \(w(y)<0\) for all \(y\ge M\). Hence, \( \hat{y}_1 \) and \( \hat{y}_2\) are inside the interval (0, M). From the Rolle theorem, it follows that \(w'(y)\) has a root \(\tilde{y}_1 \in (\hat{y}_1,\hat{y}_2) \subset (0,M)\).

We shall show that \(w'(y)\) admits a second root \(\tilde{y}_2 \ne \tilde{y}_1\) within (0, M). If \(w'(\hat{y}_i)=0\) for some \(i_0 \in \lbrace 1,2 \rbrace \), then just take \(\tilde{y}_2=\hat{y}_{i_0}\). Now, suppose \(w'(\hat{y}_i)\ne 0\) for \(i=1,2\). Since \(w(0)=(1-\xi _k)/(\xi _k \phi )>0>w(M)\), we conclude that w(y) changes its sign an odd number of times within the interval [0, M]. As \(w'(\hat{y}_i)\ne 0\) for \(i=1,2\), it follows that w(y) changes it sign at the roots \(\hat{y}_i\) and necessarily w(y) admits a third root \(\hat{y}_3 \in [0,M]\), distinct from \(\hat{y}_1\) and \(\hat{y}_2\). Suppose without loss of generality that \(\hat{y}_3>\hat{y}_2\). Thus, it follows from the Rolle theorem that \(w'(y)\) has another root \(\tilde{y}_2\) within the interval (0, M). Therefore, we’ve proved that \(w'(y)=h_1''(y)-h_2''(y)\) has two distinct roots \(\tilde{y}_1\) and \(\tilde{y}_2\) within the interval (0, M).

Thus, the curves of \(h_1''\) and \(h_2''\) have at least two intersections for \(y>0\). Since

$$\begin{aligned} h_2''(y)=\dfrac{2\gamma _r \xi _r^2}{(\xi _r^2+(1+\gamma _r)y^2)^3}\left( 3y^2(1+\gamma _r)-\xi _r^2 \right) , \end{aligned}$$

the interval of positive values of y where \(h_2''(y)\) is negative is \(I_*=[0,u_1)\), where \(u_1=\xi _r/(\sqrt{3}\sqrt{1+\gamma _r})\). On the other hand, we have that \(h_1''(y)=-2/(\xi _k \phi )\) is negative for \(y\ge 0\). Thus, the two roots of \(h_1''=h_2''\) should occur within \(I_*\). The derivative of \(h_2''\),

$$\begin{aligned} h_2'''(y) =\dfrac{24 \gamma _r (1+\gamma _r) \xi _r^2 y}{\xi _r^2+(1+\gamma _r)y^2)^4}\left( \xi _r^2-(1+\gamma _r)y^2 \right) \end{aligned}$$

has the following properties: if \(u_2=\xi _r/\sqrt{1+\gamma _r}\), then \(h_2'''(y)>0\) for \(0\le y<u_2\), \(h_2'''(u_2)=0\), \(h_2'''(y)<0\) for \(y>u_2\), and \(u_2>u_1\). Thus, \(h_2''(y)\) is an increasing function within the interval \(I_*\), while \(h_1''(y)\) is constant. Therefore, the curves of \(h_1''(y)\) and \(h_2''(y)\) do not have two intersections for \(y>0\), which is a contradiction. Therefore, the equation \(h_1(y)=h_2(y)\) do not have more than two positive roots. \(\square \)

Analysis for\(f=f_{{C}}\)and\(g=g_1\). In this case, q is a fourth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} \sigma _y \xi _k^2, c_1 = -\gamma _k - \dfrac{\sigma _y \xi _k^2 (\gamma _r + \xi _r)}{\xi _r}<0, c_2 = \sigma _y + \dfrac{\gamma _r \sigma _y \xi _k^2}{\xi _r} >0, \\ c_3= & {} - \dfrac{\sigma _y (\gamma _r + \xi _r)}{\xi _r} < 0, c_4 = \dfrac{\gamma _r \sigma _y}{\xi _r}. \end{aligned}$$

Thus, \(S_q= (+-+-+)\), \(V_q=4\) and q admits 0 or 2 or 4 positive roots. The coefficients \(d_0\) and \(d_4\) or r are

$$\begin{aligned} d_0 = - \gamma _k< 0 < d_4 = \dfrac{\gamma _r \sigma _y}{\xi _r}. \end{aligned}$$

By Lemma 2, 1 or 3 positive roots of q satisfy \(y>1\). Therefore, q admits 1 or 3 feasible roots. Hence, only scenarios I and III are possible. We numerically verified the occurrence of such scenarios (see Table 3).

Analysis for\(f=f_{{C}}\)and\(g=g_2\). In this case, q is a fourth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} \sigma _y \xi _k^2>0, \ c_1 = -\gamma _k - \sigma _y \xi _k^2 - \dfrac{\sigma _y \xi _k^2 (\gamma _r-1 )}{\xi _r}<0, \\ \ c_2= & {} \sigma _y -\dfrac{\gamma _k }{\xi _r} + \dfrac{(\gamma _r-1) \sigma _y \xi _k^2}{\xi _r},\\ c_3= & {} - \sigma _y - \dfrac{\sigma _y (\gamma _r - 1)}{\xi _r} < 0, \ c_4 = \dfrac{(\gamma _r-1) \sigma _y}{\xi _r} >0. \end{aligned}$$

Thus, \(S_q= (+-\pm -+)\), \(V_q=2\) or 4, and q admits 0 or 2 or 4 positive roots. The coefficients \(d_0\) and \(d_4\) are

$$\begin{aligned} d_0 = - \dfrac{\gamma _k (\xi _r+1)}{\xi _r}< 0 < d_4 = \dfrac{\sigma _y (\gamma _r -1)}{\xi _r}. \end{aligned}$$

By Lemma 2, 1 or 3 positive roots of q satisfy \(y>1\). Therefore, q admits 1 or 3 feasible roots. Hence, only scenarios I and III are possible. We numerically verified the occurrence of such scenarios (see Table 3).

Analysis for\(f=f_{{C}}\)and\(g=g_3\). In this case, q is a fifth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} \sigma _y \xi _k^2>0, \ c_1 = -\gamma _k - \sigma _y \xi _k^2<0, \ c_2 = \sigma _y - \dfrac{(\gamma _r-1) \sigma _y \xi _k^2}{\xi _r} ,\ c_3 = -c_2-\dfrac{\gamma _k}{\xi _r^2},\\ c_4= & {} - \dfrac{(\gamma _r-1) \sigma _y}{\xi _r^2} <0, \ c_5 = \dfrac{(\gamma _r-1) \sigma _y}{\xi _r^2}>0. \end{aligned}$$

If \(c_2>0\), then \(c_3<0\), \(S_q= (+-+--+)\), \(V_q=4\), and q admits 0 or 2 or 4 positive roots. If \(c_2<0\), then \(S_q= (+--\pm -+)\), \(V_q=2\) or 4, and q admits 0 or 2 or 4 positive roots. Thus, both cases result in the same possibilities. The coefficients \(d_0\) and \(d_5\) of r(u) are

$$\begin{aligned} d_0 = - \gamma _k\left( 1+\dfrac{1}{\xi _r^2}\right)< 0 < d_5 = \dfrac{\sigma _y (\gamma _r -1)}{\xi _r^2}. \end{aligned}$$

By Lemma 2, q has 1 or 3 or 5 roots \(y>1\). Therefore, q admits 1 or 3 feasible roots. Hence, only scenarios I and III are possible. We numerically verified the occurrence of such scenarios (see Table 3).

Analysis for\(f=f_{{C}}\)and\(g=g_4\). In this case, q is a fifth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} \sigma _y \xi _k^2>0, \ c_1 = -\gamma _k - \sigma _y \xi _k^2 - \dfrac{2 \gamma _r \sigma _y \xi _k^2 }{\xi _r}<0, \ c_2 = \sigma _y + \dfrac{\sigma _y \xi _k^2}{\xi _r^2} + \dfrac{2 \gamma _r \sigma _y \xi _k^2 }{\xi _r},\\ c_3= & {} -\dfrac{\gamma _k}{\xi _r^2} - \sigma _y - \dfrac{\sigma _y \xi _k^2}{\xi _r^2} - \dfrac{2 \gamma _r \sigma _y }{\xi _r}< 0, \ c_4 = \dfrac{\sigma _y}{\xi _r^2} + \dfrac{2 \gamma _r \sigma _y }{\xi _r} >0 , \ c_5 = -\dfrac{\sigma _y}{\xi _r^2}<0. \end{aligned}$$

Thus, \(S_q= (+-+-+-)\), \(V_q=5\), and q admits 1 or 3 or 5 positive roots. Hence, only scenarios I, III and V are possible. We numerically verified the occurrence of such scenarios (see Table 3).

Analysis for\(f=f_{{C}}\)and\(g=g_5\). In this case, q is a fifth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} \sigma _y \xi _k^2>0, \ c_1 = -\gamma _k - \sigma _y \xi _k^2< 0, \ c_2 = \sigma _y + \dfrac{(1+\gamma _r)\sigma _y \xi _k^2}{\xi _r^2} ,\\ c_3= & {} -\dfrac{\gamma _k}{\xi _r^2} - \sigma _y - \dfrac{(1+\gamma _r)\sigma _y \xi _k^2}{\xi _r^2}< 0, \\ c_4= & {} \dfrac{\sigma _y (1+\gamma _r)}{\xi _r^2} >0, \ c_5 = -\dfrac{\sigma _y(1+\gamma _r)}{\xi _r^2}<0. \end{aligned}$$

Thus, \(S_q= (+-+-+-)\), \(V_q=5\), and q admits 1 or 3 or 5 positive roots. Hence, only scenarios I, III and V are possible. We numerically verified the occurrence of such scenarios (see Table 3).

Analysis for\(f=f_D\)and\(g=g_1\). In this case, q is a fourth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} 1-\phi , c_1 = -\dfrac{\gamma _r+\xi _r}{\xi _r}< 0, c_2 = \dfrac{1}{\xi _k^2} + \dfrac{\gamma _r}{\xi _r}> 0,\\ c_3= & {} - \dfrac{\gamma _r+\xi _r}{\xi _k^2 \xi _r} < 0, \ c_4 = \dfrac{\gamma _r}{\xi _k^2\xi _r} >0. \end{aligned}$$

The coefficients \(d_0\) and \(d_4\) of r(u) are

$$\begin{aligned} d_0 = - \phi< 0 < d_4 = \dfrac{\gamma _r}{\xi _k^2\xi _r}. \end{aligned}$$

By Lemma 2, q has 1 or 3 positive roots which satisfy \(y>1\). Analyzing the sign sequence of q, we conclude the following. If \(\phi <1\), then \(S_q= (+-+-+)\), \(V_q=4\), and q admits 0 or 2 or 4 positive roots. Since 1 or 3 are non-feasible, it follows that q has 1 or 3 feasible roots and only scenarios I and III are possible in this case. If \(\phi >1\), then \(S_q= (--+-+)\), \(V_q=3\), and q admits 1 or 3 positive roots. Since 1 or 3 are non-feasible, it follows that q has 0 or 2 feasible roots and only scenarios \(\emptyset \) and II are possible in this case. We numerically verified the occurrence of scenarios \(\emptyset \), I, II and III (see Table 3).

Analysis for\(f=f_D\)and\(g=g_2\). In this case, q is a fourth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} 1-\phi , c_1 = \dfrac{1-\gamma _r-1-\xi _r-\phi }{\xi _r}< 0, c_2 = \dfrac{1}{\xi _k^2} + \dfrac{\gamma _r-1}{\xi _r}> 0, \\ c_3= & {} \dfrac{1-\gamma _r-\xi _r}{\xi _k^2 \xi _r} < 0, c_4 = \dfrac{\gamma _r-1}{\xi _k^2\xi _r} >0. \end{aligned}$$

The coefficients \(d_0\) and \(d_4\) of r(u) are

$$\begin{aligned} d_0 = - \dfrac{\phi (1+\xi _r) }{\xi _r}< 0 < d_4 = \dfrac{\gamma _r-1}{\xi _k^2\xi _r}. \end{aligned}$$

By Lemma 2, q has 1 or 3 positive roots which satisfy \(y>1\). Analyzing the sign sequence of q, we conclude the following. If \(\phi <1\), then \(S_q= (+-+-+)\), \(V_q=4\), and q admits 0 or 2 or 4 positive roots. Since 1 or 3 are non-feasible, it follows that q has 1 or 3 feasible roots and only scenarios I and III are possible in this case. If \(\phi >1\), then \(S_q= (--+-+)\), \(V_q=3\), and q admits 1 or 3 positive roots. Since 1 or 3 are non-feasible, it follows that q has 0 or 2 feasible roots and only scenarios \(\emptyset \) and II are possible in this case. We numerically verified the occurrence of scenarios \(\emptyset \), I, II and III (see Table 3).

Analysis for\(f=f_D\)and\(g=g_3\). In this case, q is a fifth-degree polynomial with coefficients

$$\begin{aligned} c_0= & {} 1-\phi , c_1 = -1< 0, c_2 = \dfrac{1}{\xi _k^2} - \dfrac{\gamma _r-1 + \phi }{\xi _r^2}, c_3 = \dfrac{\gamma _r-1}{\xi _r^2}- \dfrac{1}{\xi _k^2}, \\ c_4= & {} \dfrac{1-\gamma _r}{\xi _k^2\xi _r^2} < 0, \ c_5 = \dfrac{\gamma _r-1}{\xi _k^2\xi _r^2} >0. \end{aligned}$$

The coefficients \(d_0\) and \(d_5\) of r(u) are

$$\begin{aligned} d_0 = -\phi \dfrac{(1+\xi _r^2) }{\xi _r^2}< 0 < d_5 = \dfrac{\gamma _r-1}{\xi _k^2\xi _r^2}. \end{aligned}$$

By Lemma 2, q has 1 or 3 positive roots which satisfy \(y>1\). Analyzing the sign sequence of q, we conclude the following. If \(\phi <1\), then \(S_q= (+-\pm \pm - +)\), \(V_q=2\) or 4, and q admits 0 or 2 or 4 positive roots. Since 1 or 3 are non-feasible, it follows that q has 1 or 3 feasible roots and only scenarios I and III are possible in this case. If \(\phi >1\), then \(S_q= (--\pm \pm - +)\), \(V_q=1\) or 3, and q admits 1 or 3 positive roots. Since 1 or 3 are non-feasible, it follows that q has 0 or 2 feasible roots and only scenarios \(\emptyset \) and II are possible in this case. We numerically verified the occurrence of scenarios \(\emptyset \), I, II and III (see Table 3).

Analysis for\(f=f_D\)and\(g=g_4\). In this case, q is a fifth-degree polynomial and all scenarios are possible. We numerically verified the occurrence of each scenario (see Table 3).

Analysis for\(f=f_D\)and\(g=g_5\). In this case, q is a fifth-degree polynomial and all scenarios are possible. We numerically verified the occurrence of scenarios \(\emptyset \), I, II, III and IV (see Table 3), while scenario V is excluded due to the following lemma.

Lemma 4

If \((f,g)=(f_D,g_5)\), then q(y) does not admit five different roots in the interval \(y\in [0,1]\).

Proof

For \(f=f_D\) and \(g=g_5\), we have

$$\begin{aligned} q(y)=c_0+c_1y+c_2y^2+c_3y^3+c_4y^4+c_5y^5 \end{aligned}$$
(30)

with coefficients

$$\begin{aligned} c_0= & {} 1-\phi , c_1 = -1, c_2 = \frac{\gamma _r+1-\phi }{\xi _r^2}+\frac{1}{\xi _k^2}, c_3 = -\frac{\gamma _r+1}{\xi _r^2}-\frac{1}{\xi _k^2},\nonumber \\ c_4= & {} \frac{\gamma _r+1}{\xi _k^2 \xi _r^2}, c_5 = - \frac{\gamma _r+1}{\xi _k^2 \xi _r^2}. \end{aligned}$$
(31)

If \(\phi >1\), then the sign sequence of q is \((--+-+-)\), and thus \(V_q=4\). Hence, q(y) has at most 4 positive roots and our claim is valid.

Now, assume that \(\phi <1\), and suppose by contradiction that q(y) has 5 different roots within the interval [0, 1]. Then, by the Rolle theorem, it follows that \(q'(y)\) has 4 roots within (0, 1) and then \(q''(y)\) has three roots within (0, 1). The function \(q''(y)\) is a third-degree polynomial, and we will count its roots by calculating its Sturm sequence and applying the Sturm’s theorem (see Basu et al. 2007, section 2.2.2).

The Sturm sequence of a polynomial S(y) is a sequence \((S_0,S_1,S_2,\ldots )\) of polynomials defined as follows:

$$\begin{aligned} \begin{array}{l} S_0=S, \\ S_1=S', \\ S_{n+1} = -\mathrm{rem}\;(S_{n-1},S_n), \ \ n\ge 1, \end{array} \end{aligned}$$

where \(\mathrm{rem}(S_{n-1},S_n )\) is the remainder of the Euclidean division of \(S_{n-1}\) by \(S_n\) (see Basu et al. 2007, section 2.2.2). The number of sign variations of the Sturm sequence evaluated at \(c\in \mathbb {R}\) is denoted by \(V_S(c)\) and defined as the number of sign variations in the sequence of numbers

$$\begin{aligned} (S_0(c), \ S_1(c), \ S_2(c), \ldots ). \end{aligned}$$

Let \(a,b\in \mathbb {R}\). The Sturm’s Theorem states that the number of roots of S(y) within the interval [ab] is equal to the difference \(V_S(a)-V_S(b)\).

We will apply the Sturm’s Theorem to \(S(y)=q''(y)\) and count its roots inside the interval \([a,b]=[0,1]\). The Sturm sequence of a third-degree polynomial

$$\begin{aligned} S(y)=C_0+C_1y+C_2y^2+C_3 y^3 \end{aligned}$$

is given by

$$\begin{aligned} \begin{array}{l} S_0(y) = S(y)=C_0+C_1y+C_2y^2+C_3 y^3, \\ \\ S_1(y) = S'(y)=C_1+2 C_2y+3C_3 y^2, \\ \\ S_2(y) = -\mathrm{rem}\;(S_{0},S_1) = \dfrac{C_1 C_2}{9 C_3}-C_0+\left( \frac{2 C_2^2}{9 C_3}-\frac{2 C_1}{3}\right) y, \\ \\ S_3(y) = -\mathrm{rem}\;(S_{1},S_2) \\ \quad \qquad \,\,= \dfrac{9 C_3 \left( 18 C_0 C_1 C_2 C_3+C_1^2 C_2^2-4 C_3 C_1^3-C_0 \left( 4 C_2^3+27 C_0 C_3^2\right) \right) }{4 \left( C_2^2-3 C_1 C_3\right) {}^2}. \end{array} \end{aligned}$$

Thus, the Sturm sequence of S evaluated at \(y=0\) is

$$\begin{aligned}&\left( S_0(0),S_1(0),S_2(0),S_3(0)\right) \nonumber \\&\quad = \left( C_0,C_1,\frac{C_1 C_2}{9 C_3}-C_0, \dfrac{9 C_3 \left( 18 C_0 C_1 C_2 C_3+C_1^2 C_2^2-4 C_3 C_1^3-C_0 \left( 4 C_2^3+27 C_0 C_3^2\right) \right) }{4 \left( C_2^2-3 C_1 C_3\right) }^2 \right) ,\nonumber \\ \end{aligned}$$
(32)

and the Sturm sequence of S evaluated at \(y=1\) is

$$\begin{aligned}&\left( S_0(1),S_1(1),S_2(1),S_3(1)\right) \nonumber \\&\quad = \left( C_0+C_1+C_2+C_3, C_1+2C_2+3C_3, \dfrac{C_1 C_2+2C_2^2}{9 C_3}-C_0\right. \nonumber \\&\qquad \left. -\,\dfrac{2C_1}{3}, \dfrac{9 C_3 \left( 18 C_0 C_1 C_2 C_3+C_1^2 C_2^2-4 C_3 C_1^3-C_0 \left( 4 C_2^3+27 C_0 C_3^2\right) \right) }{4 \left( C_2^2-3 C_1 C_3\right) }^2 \right) .\nonumber \\ \end{aligned}$$
(33)

From (30), we have \(q''(y)=2c_2+6 c_3y+12 c_4y^2+20 c_5y^3\) with \(c_i\) given in (31). Writing \(S(y)=q''(y)=C_0+C_1y+C_2y^2+C_3 y^3\), we obtain the coefficients

$$\begin{aligned} C_0 = 2c_2, \ \ C_1 = 6c_3, \ \ C_2 = 12 c_4, \ \ C_3 = 20 c_5. \end{aligned}$$
(34)

Hence, using formulas (32), (34) and (31), and the fact that \(\phi<1<\gamma _r\), we obtain that the first three terms of the Sturm sequence of \(q''\) evaluated at \(y=0\) are

$$\begin{aligned} \begin{array}{l} S_0(0)= 2 \frac{\gamma _r+1-\phi }{\xi _r^2}+\frac{2}{\xi _k^2} >0 , \\ S_1(0)= -6\frac{\gamma _r+1}{\xi _r^2}-\frac{6}{\xi _k^2}<0 , \\ S_2(0)=-\frac{8}{5 \xi _k^2 }-2 \frac{4(1-\phi ) +3 \gamma _r + (\gamma _r-\phi )}{5\xi _r^2}<0, \end{array} \end{aligned}$$

while \(S_3(0)\) has a complicated expression with an undetermined sign. However, we can conclude that \(V_{q''}(0)=2\) if \(S_3(0)>0\) and \(V_{q''}(0)=1\) if \(S_3(0)<0\).

Now, again using the fact that \(\phi<1<\gamma _r\) and formulas (33), (34) and (31), we obtain that the first three terms of the Sturm sequence of \(q''\) evaluated at \(y=1\) are

$$\begin{aligned} \begin{array}{l} S_0(1)= -2 \dfrac{(2\gamma _r(2+\xi _k^2) + 2 (2+\xi _r^2 ) +\xi _k^2 (2+\phi ) )}{\xi _k^2 \xi _r^2}<0, \\ S_1(1)= -6 \dfrac{(\xi _r^2 + (1+\gamma _r)(6+\xi _k^2))}{\xi _k^2 \xi _r^2} <0, \\ S_2(1)= 2\dfrac{(-4 +\gamma _r(6\xi _k^2-4)+6\xi _r^2+\xi _k^2(6+5\phi ))}{5 \xi _k^2 \xi _r^2}, \end{array} \end{aligned}$$

which has an undetermined sign, and \(S_3(1)\), which has a complicated expression with an undetermined sign. However, notice from (32) and (33) that \(S_3(1)=S_3(0)\). Thus, we can conclude that \(V_{q''}(1)=1\) if \(S_3(0)>0\) and \(V_{q''}(1)=0\) or 2 if \(S_3(0)<0\).

Therefore, the number of roots of \(q''(y)\) within the interval [0, 1], which, by the Sturm’s Theorem, is equal to the difference \(V_{q''}(0)-V_{q''}(1)\), is always less than 3, which is a contradiction. Hence, the equation q(y) cannot have five roots within the interval [0, 1]. \(\square \)

Model Parameterization and Parameter Estimation

Here we report the parameter values used for the model simulations. To describe the different scenarios with each sub-model (Tables 2, 3), we started with the following set of basic values. Parameters regarding proliferating and quiescent LSCs were set to values used in a previous publication and correspond to the median values of a cohort of 122 CML patients (Fassoni et al. 2018). The values are \(p_{XY}=0.05,\)\( p_{YX}=0.001\), \(p_Y = 0.2\) and \( T_Y = 10^6\). For immune cells, we adopted the values \(d_Z=1\) month\(^{-1}\) and \(p_Z = 10^3\) cells/month so that the normal level of immune cells is \(p_Z/d_Z = 10^3\) cells. Parameters \(C_K\) and \(C_R\) were set to values defined according to the specific functional responses used in each sub-model. For linear functional responses (\(F=F_A\) or \(G=G_1\)), we adopted \(C_K,C_R= 5 \times 10^5\) meaning that F and G reach their maximum values for BCR-ABL1/ABL1 ratios around 50%. For the Holling type II and III functional responses (\(F=F_B,F_{{C}}\) or \(G=G_2,G_3\)), we adopted \(C_K,C_R= 10^4\) meaning that F and G reach half of their maximum values for BCR-ABL1/ABL1 ratios around 1%. For immune window (\(F=F_D\) or \(G=G_4\)) and the immune suppression (\(G=G_5\)) functional responses, we adopted \(C_K,C_R= 10^3\) meaning that F and G reach their maximum values for BCR-ABL1/ABL1 ratios around 0.1%, i.e., MR3. All these basic values above were used as starting values for searching the possible scenarios for each sub-model. Parameters \(m_K\) and \(m_R\) did not have a specific starting value. By varying the least possible number of parameters (starting with \(m_K\), \(m_R\), then \(C_K\), \(C_R\) and then \(p_Y\)), we obtained the parameter values shown in Tables 2 and 3, leading to different scenarios for each possible sub-model.

Table 4 Estimated values for \(p_Y\), \(p_{YX}\), \(m_K\) and \(m_R\), used in Fig. 3
Table 5 Values for \(m_K\) and \(m_R\) used to generate the different outcomes in the simulations shown in Fig. 8: relapse (solid lines) and remission/cure (dashed lines)

To estimate the model parameter values corresponding to the fits in Fig. 3, we allowed parameters \(p_Y\), \(p_{YX}\), \(m_K\) and \(m_R\) to vary and used a minimization algorithm to find those parameter values that minimize the quadratic error between the model solution and the patient data, defined as

$$\begin{aligned} E(p_Y,p_{YX}, m_K, m_R)=\sum _{i=1}^{n_d} \left( \mathrm{log}_{10}(L_{\mathrm{OBS}}(t_i)) - \mathrm{log}_{10}(L_{\mathrm{MOD}}(t_I)) \right) ^2, \end{aligned}$$
(35)

where \(L_{\mathrm{OBS}}(t_i)\) are the observed BCR-ABL1/ABL1 ratios at times \(t_i\) in the patient time course with \(n_d=30\) data points, and \(L_{\mathrm{MOD}}(t_i)=100Y(t_i)/T_Y\) are the simulated BCR-ABL1/ABL1 ratios at the same time points. The following intervals for parameter searches were used: \(0.05\le p_Y\le 1\), \(0\le m_K\le 1000\), \(d_Z< m_R\le 1000\) and \(0\le p_{YX} \le 0.1\). The other parameters remained constant, and we assumed the following values: regarding leukemic cells, we used the values obtained in our previous model for such specific patient: \(p_{XY}=0.0451256\), \( T_Y = 10^6\) and \(e_{\mathrm{TKI}}=0.493541 + p_Y\) (Fassoni et al. 2018). For parameters \(d_Z\), \(p_Z\), \(C_K\), \(C_R\), we adopted the same values as above (values for \(C_K\) and \(C_R\) varied according to each sub-model). The results with the estimated values are given in Table 4.

To generate Fig. 8, we fixed all parameters, with exception of \(m_K\) and \(m_R\), which were allowed to vary in order to lead to different outcomes after cessation for each selected sub-model. The values for \(m_K\) and \(m_R\) were manually selected and are given in Table 5. Regarding the other parameters, in all simulations we used the fixed values \(p_{YX}=0.00353003\), \(p_Y=0.2\), \(e_{\mathrm{TKI}}=0.693541\), \(p_{XY}=0.0451256\), while \(T_Y\)\(d_Z\), \(p_Z\), \(C_K\), and \(C_R\) are the same as above (values for \(C_K\) and \(C_R\) varied according to each sub-model).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

César Fassoni, A., Roeder, I. & Glauche, I. To Cure or Not to Cure: Consequences of Immunological Interactions in CML Treatment. Bull Math Biol 81, 2345–2395 (2019). https://doi.org/10.1007/s11538-019-00608-x

Download citation

Keywords

  • Mathematical modeling
  • Ordinary differential equations
  • Bifurcations
  • Chronic myeloid leukemia
  • Immunological control
  • Treatment-free remission

Mathematics Subject Classification

  • 92B05
  • 37N25
  • 34C60
  • 37G35