Global stability analysis of an SVEIR epidemic model with general incidence rate
 591 Downloads
Abstract
In this paper, a susceptiblevaccinatedexposedinfectiousrecovered (SVEIR) epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated, assuming that the horizontal transmission is governed by an unspecified function \(f(S,I)\). The role that temporary immunity (vaccinatedinduced) and treatment of infected people play in the spread of disease, is incorporated in the model. The basic reproduction number \(\mathcal{R}_{0}\) is found, under certain conditions on the incidence rate and treatment function. It is shown that the model exhibits two equilibria, namely, the diseasefree equilibrium and the endemic equilibrium. By constructing a suitable Lyapunov function, it is observed that the global asymptotic stability of the diseasefree equilibrium depends on \(\mathcal{R}_{0}\) as well as on the treatment rate. If \(\mathcal{R}_{0}>1\), then the endemic equilibrium is globally asymptotically stable with the help of the Li and Muldowney geometric approach applied to four dimensional systems. Numerical simulations are also presented to illustrate our main results.
Keywords
Epidemic model Reproduction number Lyapunov function Geometric approach Global stability Susceptible–Vaccinated–Exposed–Infectious–RecoveredAbbreviations
 (SVEIR)
Susceptible–Vaccinated–Exposed–Infectious–Recovered
MSC
92D25 92D30 34D23 37B251 Introduction
Mathematical modeling enjoys popularity in both preventing and controlling infectious diseases such as severe acute respiratory syndrome (SARS) [1], human immunodeficiency virus infection/acquired immune deficiency syndrome (HIV/AIDS) [2], H5N1 (avian flu) [3] and H1N1 (swine flu)[4]. In recent years, a lot of efforts have been made to develop realistic diseases and further study the asymptotic behavior of such epidemic models [5]. In the field of studying epidemic model behavior, one of the most important parts is to analyze steady states together with their stability [6]. In general, there are two distinct techniques named Lyapunov’s direct method and Li–Muldowney’s geometric approach to give sufficient conditions of global stability for the equilibrium states (see, for example, [7, 8, 9, 10, 11, 12, 13, 14]). We would like to mention some related work concerned with the existence of positive solutions for the discrete fractional boundary value problem [15], the sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions [16] and the nonexistence of global in time solution of the mixed problem for the nonlinear evolution equation with memory generalizing the Voigt–Kelvin rheological model [17].
It is well known that the rate of incidence plays the main part in modeling infectious diseases. The rise and fall of epidemics can be influenced by some factors, such as density of population and life style [18, 19]. Many researchers have adopted different nonlinear incidence rates in their works. For more details, we refer the reader to [8, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] and the references therein. When it comes to control of a disease, it is generally known that the spread of many diseases can be prevented by vaccinating. When massive vaccination is impossible, the second stage of defensive mechanism could be medical treatment. Individuals need to bear in mind that the treatment is an indispensable way to take precautions for some diseases (for instance, measles, phthisis and influenza). In recent years, many treatment functions have been introduced by several authors to study some epidemic models under different conditions (see, for instance, [12, 14, 27, 31, 35, 36, 37, 38]).
As pointed out by Liu and Yang [11], due to the high similarity between computer virus and biological virus, it is acceptable to establish dynamical models describing biological virus among a population by modifying an eepidemic model. Thus, it is interesting and important to extend model (1.2) to study the biological virus in the infectious disease.
 A:

the rate at which new individuals (including newborns and immigrants) enter the susceptible population,
 \(\delta_{0}\):

natural death rate of population all classes,
 η:

the rate at which the vaccinated population lose their immunity and join the susceptible class,
 μ:

vaccination rate coefficient,
 \(\delta_{1}\):

the rate at which exposed population become infective,
 \(\delta_{2}\):

natural recovery rate of infective population,
 \(\delta_{3}\):

diseaserelated death rate of infective population.
 (i)
The new individuals enter the population with a constant rate and all the new individuals are susceptible.
 (ii)
Susceptible individuals move to exposed class by adequate contact with infective individuals and after some time (i.e., latency period), they become infectious and move to infectious class.
 (iii)
The infectious individuals are assumed to leave the infectious class as a result of natural death and diseaserelated death as well as recovery of infected individuals.
 (iv)
After recovery the individuals become immunized and hence they are no longer susceptible to it.
 (v)
It is assumed that a fraction of susceptible individuals get vaccinated and join the vaccinated class. A part of vaccinated individuals may lose their immunity and rejoin the susceptible class.
It is easy to see that system (1.3) includes (1.1) and (1.2) as special cases and so model (1.3) provides a uniform setting for the computer virus and biological virus studies. Following the classical assumptions [27, 40], it is reasonable to suppose that the transmission of the infection is governed by an incidence rate \(f(S,I)\) in model (1.3). Moreover, as pointed out by Wang [31], the recovery rate is naturally dependent on the number of infected individuals provided the health care resources are constrained and so it is natural to use the nonlinear function \(g(I)\) as the treatment function in model (1.3).
The main purpose of this paper is to derive the expression for the basic reproduction number and further show the global stability of diseasefree as well as endemic equilibria by the aid of Lyapunov function and Li–Muldowney geometric approach applied to four dimensional systems. This paper is organized as follows. In Sect. 2, some elementary assumptions on the functions f and g will be given, and the basic reproduction number \(R_{0}\) is provided. Also the equilibrium points are discussed. The global stability of diseasefree equilibrium and endemic equilibrium are analyzed in Sects. 3 and 4, respectively. All our important analytical findings are numerically verified with the help of Mathlab in Sect. 5. Finally, a brief conclusion is given in Sect. 6.
2 Basic reproduction number and equilibrium
 (H1)\(f:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}_{+}\) is differentiable such that

\(f(S,0)=f(0,I)=0\) for all \(S,I\geq0\);

\(f(S,I)>0\) for all \(S,I>0\);

\(\frac{\partial f(S,I)}{\partial S}>0\) for all \(S\geq0\) and \(I>0\);

\(\frac{\partial f(S,I)}{\partial I}\geq0\) for all \(S,I\geq0\);

\(I\frac{\partial f(S,I)}{\partial I}f(S,I)\leq0\) for all \(S,I\geq0\).

 (H2)
\(g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) is differentiable such that \(g(0)=0\), \(g'(I)>0\) and \(g''(I)\leq0\) for \(I\geq0\).
 (H3)
\(\frac{d}{dI}(\log\frac{g(I)}{f_{S}(I)})\geq0\) holds for all \(S,I>0\), where \(f_{S}(I):=f(S,I)\) for \(S,I>0\).
Remark 2.1
 (1)It is easy to check that the classes of \(f(S,I)\) satisfying (H1) include incidence rates such asfor \(\beta,a,b>0\) and \(0\leq q\leq1\).$$f(S,I)=\frac{\beta SI}{1+aI^{q}}, \qquad f(S,I)=\frac{\beta SI}{1+aS+bI},\qquad f(S,I)= \frac{\beta SI}{1+aS+bI+abSI}, $$
 (2)It is straightforward to show that the classes of \(g(I)\) satisfying (H2) include removal rates such asfor \(r>0\), \(a>0\), \(b>0\).$$g(I)=\frac{rI}{1+bI}, \qquad g(I)=\frac{rI}{I+a},\qquad g(I)=r\arctan I, $$
 (3)
By hypothesis (H2), we know that \(\varPhi (I)=\frac{g(I)}{I}\) is a monotone decreasing function on \(I>0\).
 (4)The assumption (H3) is equivalent to the following inequality:which can be found in [27].$$\frac{\partial}{\partial I}f(S,I)g(I)\leq f(S,I)\frac{d}{dI}g(I), $$
 (5)By the assumptions, it is easy to find that system (1.3) always has a diseasefree equilibrium point \(P_{0}=(S_{0},0,0,0,V_{0})\), where$$S_{0}=\frac{(\delta_{0}+\eta)A}{\delta_{0}^{2}+(\mu+\eta)\delta_{0}},\qquad V_{0}=\frac {A\mu}{\delta_{0}^{2}+(\mu+\eta)\delta_{0}}. $$
We shall assume that (H1), (H2) and (H3) hold in the rest of this paper.
Suppose that there exists another positive equilibrium point \(P_{1}=(S_{1},E_{1},I_{1},R_{1},V_{1})\). Then \(F'(I_{1})\geq0\) due to the property of continuous function. This is a contradiction. Therefore, system (2.1) has a unique endemic equilibrium \(P^{*}\) when \(\mathcal{R}_{0}>1\). It can be stated as follows.
Theorem 2.1
Remark 2.2
From the proof of the existence of endemic equilibrium \(P^{*}\), it is not difficult to arrive at such a conclusion that the nonlinear treatment function \(g(I)\) has an upper bound \(\frac{A\delta_{1}}{m_{2}}\), which is reasonable for limited medical resources in our daily life.
Proposition 2.1
Proof
3 Global stability of the diseasefree equilibrium by means of Lyapunov function
In this section, we investigate the global stability of the diseasefree equilibrium \(P_{0}\) for system (1.3).
Theorem 3.1
If\(\mathcal{R}_{0}<1\frac{Ag'(0)\delta_{0}g(\frac{A}{\delta _{0}})}{A(m_{3}+g'(0))}\), then the diseasefree equilibrium\(P_{0}\)of system (1.3) is globally asymptotically stable in the feasible regionΩ. If\(\mathcal{R}_{0}>1\), then\(P_{0}\)is unstable.
Proof
4 Global stability of the endemic equilibrium by means of geometric approach
In this section, we analyze the stability of the endemic equilibrium \(P^{*}\). First, we show the local stability of the endemic equilibrium of system (1.3) around the endemic equilibrium \(P^{*}\).
Theorem 4.1
 (i)
\(\eta<\frac{a_{11}m_{4}}{\mu}\);
 (ii)
\(a_{13}<\min\{\frac{a_{11}m_{4}+(a_{11}+m_{4})(a_{33}+m_{2})+a_{33}m_{2}\mu\eta }{\delta_{1}}, \frac{a_{11}(a_{33}+m_{2})m_{4}+a_{33}(a_{11}+m_{4})m_{2}\mu\eta (a_{33}+m_{2})}{(m_{1}+m_{4})\delta_{1}}, \frac{a_{11}a_{33}m_{2}m_{4}\mu\eta a_{33}m_{2}}{(m_{1}m_{4}\mu\eta)\delta_{1}}\}\);
 (iii)
\(0< h\leq\frac{z}{a_{33}m_{4}+(a_{33}+m_{4})m_{2}}\)and\(a_{11}> \frac{(a_{33}\mu m_{2}\etaa_{21}a_{13}m_{4}\delta_{1})h^{2}a_{13}\delta _{1}h(\mu\eta h+z)+ a_{33}hz(m_{2}+m_{4})+hz(m_{2}m_{4}\mu\eta )z^{2}}{h[a_{33}m_{2}m_{4}ha_{13}m_{4}\delta_{1}h(a_{33}+m_{2}+m_{4})z]}\). Here all the parameters\(a_{11}\), \(a_{13}\), \(a_{21}\), \(a_{33}\), \(a_{43}\)are defined in (4.1). The values ofhandzequal to\(B_{1}\)and\(B_{3}\), respectively.
Proof
Here we follow the approach used in [8] for a SVEIR model of SARS epidemic spread.
We will apply the following theorem according to [44].
Lemma 4.1
If\(D_{1}\)is a compact absorbing subset in the interior ofD, and there exist\(\gamma>0\)and a Lozinskiĭ measure\(\bar{\mu}(A)\leq \gamma\)for all\(x\in D_{1}\), then every omega limit point of system (4.2) in the interior ofDis an equilibrium in\(D_{1}\).
The uniform persistence of system (4.2), incorporating the boundedness of Θ, suggests that the compact absorbing set in the interior of Θ; see [46]. Hence, Lemma 4.1 may be applied, with \(D=\varTheta \).
Theorem 4.2
Proof
The basic idea of the proof is to obtain the estimate of the right derivate \(D_{+}\\mathbf{z}\\) of the norm (4.5). For this purpose, we need to discuss sixteen cases according to the different orthants and the definition of the norm (4.5) within each orthant.
Remark 4.1
As pointed out by Buonomo and Lacitignola [7], in some real situations, different choices of the matrix Q and of the vector norm \(\\cdot\\) may lead to better sufficient conditions than those we presented here, in the sense that the assumptions on the parameters may be weakened. Thus, it is worth to note that sufficient conditions (4.6) and (4.7) in Theorem 4.2 are derived from the application of the method and numerical simulations suggest that they may be not necessary (see Example 5.1).
5 Numerical simulations
The aim of this section is to give a numerical example to illustrate our main results.
Example 5.1
6 Conclusions
Finally, we remark that there are quite a few spaces to deserve further investigation. For example, we can continue the research in this line considering the vaccination rate μ in our model (1.3) as a continuous function, and, later, a discontinuous function. On the other hand, as is well known, epidemiological models which incorporate the control strategies can be useful to both control the spread of disease and minimize the intervention costs. For our model, it is natural to consider vaccination rate coefficient as a control to reduce the disease burden. Thus, it is important and interesting to prove the existence of optimal control, characterize the optimal control, prove the uniqueness of optimal control, compute the optimal control numerically and investigate how the optimal control depends on various parameters in the models. We will devote to these questions our future work.
Notes
Acknowledgements
We would like to thank the editors and referees for their valuable comments and suggestions to improve our paper.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (11371015, 11471230, 11671282) and the Natural Science Foundation of Sichuan Provincial Education Department (Grant Nos. 18ZB0581, 14ZB0142) and the Meritocracy Research Funds of China West Normal University (17YC373).
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
References
 1.Naheed, A., Singh, M., Lucy, D.: Numerical study of SARS epidemic model with the inclusion of diffusion in the system. Appl. Math. Comput. 229, 480–498 (2014) MathSciNetzbMATHGoogle Scholar
 2.Billarda, L., Dayananda, P.W.A.: A multistage compartmental model for HIVinfected individuals: I waiting time approach. Math. Biosci. 249, 92–101 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
 3.Upadhyay, R.K., Kumari, N., Rao, V.S.H.: Modeling the spread of bird flu and predicting outbreak diversity. Nonlinear Anal., Real World Appl. 9, 1638–1648 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
 4.Pongsumpun, P., Tang, I.M.: Dynamics of a new strain of the H1N1 influenza a virus incorporating the effects of repetitive contacts. Comput. Math. Methods Med. 2014, Article ID 487974 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
 5.Zhou, X., Cui, J.: Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate. Nonlinear Dyn. 63, 639–653 (2011) MathSciNetCrossRefGoogle Scholar
 6.Kyrychko, Y., Blyuss, K.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate. Nonlinear Anal., Real World Appl. 6, 495–507 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
 7.Buonomo, B., Lacitignola, D.: Global stability for a four dimensional epidemic model. Note Mat. 30, 81–93 (2010) MathSciNetzbMATHGoogle Scholar
 8.Gumel, A.B., McCluskey, C.C., Watmough, J.: An SVEIR model for assessing potential impact of imperfect antiSARS vaccine. Math. Biosci. Eng. 3, 485–512 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
 9.Khan, M.A., Badshah, Q., Islam, S., et al.: Global dynamics of SEIRS epidemic model with nonlinear generalized incidences and preventive vaccination. Adv. Differ. Equ. 2015, 88 (2015). https://doi.org/10.1186/s1366201504293 MathSciNetCrossRefzbMATHGoogle Scholar
 10.Khan, M.A., Khan, Y., Khan, S., et al.: Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate. Int. J. Biomath. 9, 59–83 (2016). https://doi.org/10.1142/s1793524516500686 MathSciNetzbMATHGoogle Scholar
 11.Liu, X., Yang, L.: Stability analysis of an SEIQV epidemic model with saturated incidence rate. Nonlinear Anal., Real World Appl. 13, 2671–2679 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
 12.Upadhyay, R.K., Kumari, S., Misra, A.K.: Modeling the virus dynamics in computer network with SVEIR model and nonlinear incident rate. J. Appl. Math. Comput. 54, 485–509 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
 13.Yang, Y., Zhang, C.M., Jiang, X.Y.: Global stability of an SEIQV epidemic model with general incidence rate. Int. J. Biomath. 8, 103–115 (2015). https://doi.org/10.1142/S1793524515500205 MathSciNetzbMATHGoogle Scholar
 14.Zhang, X., Liu, X.N.: Backward bifunction of an epidemic model with saturated treatment. J. Math. Anal. Appl. 348, 433–443 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
 15.Chidouh, A., Torres, D.: Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunovtype inequalities. Opusc. Math. 38, 31–40 (2018) MathSciNetCrossRefGoogle Scholar
 16.Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions. Adv. Nonlinear Anal. 6, 199–235 (2017) MathSciNetzbMATHGoogle Scholar
 17.Pukach, P., Il’kiv, V., Nytrebych, Z., Vovk, M.: On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt–Kelvin rheological model. Opusc. Math. 37, 735–753 (2017) MathSciNetCrossRefGoogle Scholar
 18.Xu, R., Ma, Z.: Global stability of a delayed SEIRS epidemic model with saturation incidence rate. Nonlinear Dyn. 61, 229–239 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
 19.Alexander, M., Moghadas, S.: Bifurcation analysis of an SIRS epidemic model with generalized incidence. SIAM J. Appl. Math. 65, 1794–1816 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
 20.Bianca, C., Pennisi, M., Motta, S., et al.: Immune system network and cancer vaccine. AIP Conf. Proc. 1389, 945–948 (2011). https://doi.org/10.1063/1.3637764 CrossRefGoogle Scholar
 21.Bianca, C., Pappalardo, F., Motta, S., et al.: Persistence analysis in a Kolmogorovtype model for cancerimmune system competition. AIP Conf. Proc. 1558, 1797–1800 (2013). https://doi.org/10.1063/1.4825874 CrossRefGoogle Scholar
 22.Cai, L.M., Li, X.Z.: Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Model. 33, 2919–2926 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
 23.Capasso, V., Serio, G.: A generalization of the KermackMcKendrick deterministic epidemic model. Math. Biosci. 42, 43–61 (1978) MathSciNetCrossRefzbMATHGoogle Scholar
 24.Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 35, 274–284 (1978) MathSciNetCrossRefzbMATHGoogle Scholar
 25.Capasso, V.: Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, vol. 97. Springer, Berlin (1993) zbMATHGoogle Scholar
 26.Ghergu, M., Radulescu, V.D.: Nonlinear PDEs, Mathematical Models in Biology, Chemistry and Population Genetics. Springer Monographs in Mathematics. Springer, Heidelberg (2012) zbMATHGoogle Scholar
 27.Dénes, A., Röst, G.: Global stability for SIR and SIRS models with nonlinear incidence and removal terms via dulac functions. Discrete Contin. Dyn. Syst., Ser. B 21, 1101–1117 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
 28.Hu, Z.X., Bi, P., Ma, W.B., et al.: Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete Contin. Dyn. Syst., Ser. B 15, 93–112 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
 29.Liu, W., Levin, S.A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
 30.Ruan, S.G., Wang, W.D.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188, 135–163 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
 31.Wang, W.D.: Backward bifurcation of an epidemic model with treatment. Math. Biosci. 201, 58–71 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
 32.Wang, F., Zhang, Y., Wang, C., et al.: Stability analysis of a SEIQV epidemic model for rapid spreading worms. Comput. Secur. 29, 410–418 (2010) CrossRefGoogle Scholar
 33.Xiao, D., Ruan, S.G.: Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 208, 419–429 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
 34.Yang, Y., Xiao, D.: Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemidogical models. Discrete Contin. Dyn. Syst., Ser. B 13, 195–211 (2010) MathSciNetzbMATHGoogle Scholar
 35.Gao, D.P., Huang, N.J.: A note on global stability for a tuberculosis model. Appl. Math. Lett. 73, 163–168 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
 36.Gao, D.P., Huang, N.J.: Optimal control analysis of a tuberculosis model. Appl. Math. Model. 58, 47–64 (2018) MathSciNetCrossRefGoogle Scholar
 37.Laarabi, H., Abta, A., Hattaf, K.: Optimal control of a delayed SIRS epidemic model with vaccination and treatment. Acta Biotheor. 63, 87–97 (2015) CrossRefGoogle Scholar
 38.Wang, W.D., Ruan, S.G.: Bifurcation in an epidemic model with constant removal rate of the infectives. J. Math. Anal. Appl. 291, 775–793 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
 39.Hale, J.K.: Ordinary Differential Equations, 2nd edn. Krieger, Melbourne (1980) zbMATHGoogle Scholar
 40.Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission. Bull. Math. Biol. 30, 615–626 (2006) CrossRefzbMATHGoogle Scholar
 41.Brikhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982) Google Scholar
 42.LaSalle, J.P.: The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics. SIAM, Philadephia (1976) CrossRefzbMATHGoogle Scholar
 43.Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001) CrossRefzbMATHGoogle Scholar
 44.Li, M.Y., Muldowney, J.S.: On R.A. Smith’s autonomous convergence theorem. Rocky Mt. J. Math. 25, 365–378 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
 45.Freedman, H.I., Ruan, S.G., Tang, M.X.: Uniform persistence and flows near a closed positively invariant set. J. Differ. Equ. 6, 583–600 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
 46.Hutson, V., Schmitt, K.: Permanence and the dynamics of biological systems. Math. Biosci. 111, 1–71 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
 47.Martin, R.H.: Logarithmic norms and projections applied to linear differential systems. J. Math. Anal. Appl. 45, 432–454 (1974) MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.