 Research
 Open Access
Modelling infectious diseases with relapse: a case study of HSV2
 Jinliang Wang^{1},
 Xiaoqing Yu^{1},
 Heidi L. Tessmer^{2},
 Toshikazu Kuniya^{3}Email author and
 Ryosuke Omori^{2, 4}
https://doi.org/10.1186/s1297601700594
© The Author(s) 2017
Received: 9 April 2017
Accepted: 28 June 2017
Published: 17 July 2017
Abstract
Background
Herpes Simplex Virus Type 2 (HSV2) is one of the most common sexually transmitted diseases. Although there is still no licensed vaccine for HSV2, a theoretical investigation of the potential effects of a vaccine is considered important and has recently been conducted by several researchers. Although compartmental mathematical models were considered for each special case in the previous studies, as yet there are few global stability results.
Results
In this paper, we formulate a multigroup SVIRI epidemic model for HSV2, which enables us to consider the effects of vaccination, of waning vaccine immunity, and of infection relapse. Since the number of groups is arbitrary, our model can be applied to various structures such as risk, sex, and age group structures. For our model, we define the basic reproduction number ℜ_{0} and prove that if ℜ_{0}≤1, then the diseasefree equilibrium is globally asymptotically stable, whereas if ℜ_{0}>1, then the endemic equilibrium is so. Based on this global stability result, we estimate ℜ_{0} for HSV2 by applying our model to the risk group structure and using US data from 2001 to 2014. Through sensitivity analysis, we find that ℜ_{0} is approximately in the range of 23. Moreover, using the estimated parameters, we discuss the optimal vaccination strategy for the eradication of HSV2.
Conclusions
Through discussion of the optimal vaccination strategy, we come to the following conclusions. (1) Improving vaccine efficacy is more effective than increasing the number of vaccines. (2) Although the transmission risk in female individuals is higher than that in male individuals, distributing the available vaccines almost equally between female and male individuals is more effective than concentrating them within the female population.
Keywords
 Multigroup SVIRI epidemic model
 Relapse
 Basic reproduction number
 Global asymptotic stability
 Herpes Simplex Virus Type 2
 Vaccination
Background
Herpes Simplex Virus Type 2 (HSV2) is one of the most common sexually transmitted diseases, and has infected about 417 million people aged 1549 worldwide [1]. Although there is still no licensed vaccine for HSV2, a theoretical investigation of the potential effects of a vaccine is considered important and has recently been conducted by several researchers (see [2–4]). In [2, 3], compartmental epidemic models with vaccination for HSV2 were considered and the effectiveness of the vaccination was discussed in connection with the basic reproduction number ℜ_{0} (see [5]) through numerical simulations. However, there was little discussion about the stability of each equilibrium. As observed in several papers on epidemic models with vaccination (see, for instance, [6–8]), backward bifurcation can occur at ℜ_{0}=1 for some special models and ℜ_{0}<1 does not necessarily imply the global asymptotic stability of the diseasefree equilibrium, that is, the eradication of the disease. In that case, the vaccination effort solely to make ℜ_{0}<1 has less significance. Therefore, a global stability analysis is critical for theoretically justifying the epidemiological discussion.
In [4], Lou et al. considered a compartmental epidemic model for HSV2 with age and risk group structures and discussed the effectiveness of the vaccination together with the global stability analysis of each equilibrium. In their study, the vaccination was limited to female individuals, who are known to be the highrisk group for HSV2, and it was concluded that such a vaccination strategy can reduce the total infections in both females and males. However, to support their conclusion, we need to consider a more general model in which male individuals can also benefit from the vaccination and show that the optimal distribution ratio of the vaccines is 1 to 0 for female and male individuals. In this paper, we consider such a general model and investigate the optimal distribution ratio of the vaccines. As opposed to their conclusion, our result shows that distributing the vaccines almost equally to females and males is more effective for the eradication of HSV2 than concentrating them within the female population.
To consider the effect of vaccination with imperfect immunity, SVIR epidemic models are often formulated, in which the total population is subdivided into the susceptible (S), vaccinated (V), infective (I) and recovered (R) populations (see, for instance, [2, 6–10]). However, to take into account the relapse of HSV2 (see [2, 11]), it is necessary to also consider a direct transition from R to I. Thus, in this paper, we formulate a multigroup SVIRI epidemic model for HSV2, which enables us to consider the effects of vaccination, of waning vaccine immunity, and of infection relapse. Since the number of groups is arbitrary, our model can be applied to various structures such as risk, sex, and age group structures. In the empirical portion of this paper, we apply our model to the risk group structure and estimate the basic reproduction number ℜ_{0} for HSV2 by using data from the US from 2001 to 2014. Since the infective population of HSV2 seems to be in endemic equilibrium, the estimation of ℜ_{0} must be carried out under the global asymptotic stability of the endemic equilibrium. However, in general, the global asymptotic stability of the endemic equilibrium is not trivial.
Recently, multigroup epidemic models have been studied by many authors [10, 12–24]. One of the most effective approaches for global stability analysis of multigroup epidemic models is the graphtheoretic approach developed by Guo et al. [14]. Since our model has a quite complex form with the paths from V to S (the waning of vaccineinduced immunity), R to I (relapse) and distributed time delay, the global asymptotic stability analysis is challenging. In this paper, by applying the graphtheoretic approach as in [14] together with an approach of max function as in [10], we prove that if ℜ_{0}>1, then the endemic equilibrium is globally asymptotically stable, whereas if ℜ_{0}≤1, then the diseasefree equilibrium is so. Based on this theoretical result, we estimate ℜ_{0} for HSV2 by using US data from 2001 to 2014. By using the estimated parameters, we discuss the optimal vaccination strategy for the eradication of HSV2.
Methods
The general multigroup SVIRI epidemic model
are the forces of infection to susceptible and vaccinated individuals in group \(i \in {\mathcal {N}}\) at time t≥0, respectively. Here we assume standard incidence. (A4) The per capita vaccination rate for susceptible individuals in group \(i\in {\mathcal {N}}\) is v _{ i }>0. The per capita rate for the waning of vaccineinduced immunity for vaccinated individuals in group \(i \in {\mathcal {N}}\) is ω _{ i }≥0. (A5) The per capita recovery rate of infective individuals in group \(i \in {\mathcal {N}}\) is γ _{ i }>0. (A6) The survival probability for recovered individuals in group \(i \in {\mathcal {N}}\), who spent time t in the recovered class, is \(P_{i}(t):=\exp (\int _{0}^{t} \delta _{i}(\eta) \mathrm {d}\eta)\), where δ _{ i }(η) denotes the relapse risk for individuals who spent time η in the recovered class in group i. For each \(i \in {\mathcal {N}}\), \(\delta _{i} \in L_{\text {loc}, +}^{1} (0,+\infty)\) and \(\int _{0}^{+\infty }\delta _{i}(\eta)\mathrm {d}\eta = +\infty \).
Note that the differential equation of \(R_{i}(t), \ i \in {\mathcal {N}}\) can be omitted since it does not appear in the above three equations.
Hence, we have μ _{ i }+γ _{ i }−Q _{ i }>0 for all \(i \in {\mathcal {N}}\).
where ρ(·) denotes the spectral radius of a matrix. We will obtain the global stability results for (6) in connection with ℜ_{0} (see the “Results” section).
The special multigroup SVIRI epidemic model for HSV2
where Q _{ i }=δ _{ i } γ _{ i }/(μ _{ i }+δ _{ i }) and we write β _{ ij } as β _{ i,j } for improved readability.

\(\rho _{x_{i} y_{i}}\) denotes the HSV2 infection risk for the risk group i. The risk group is stratified by sex and the number of partners within the last 12 months, the risk group i denotes the individuals whose number of partners within the last 12 months is x _{ i } and sex is y _{ i }. \(\rho _{x_{i} y_{i}}\) is given by;Here, similar to previous modelling studies of sexually transmitted infections, we modeled the relationship between infection risk and sexual behavior by a power law function [26].$$\rho_{x_{i} y_{i}} = c_{y_{i}} (x_{i} + 1)^{\phi}. $$

c denotes the sex specific HSV2 transmission coefficient.

ϕ denotes the exponent parameter describing the heterogeneity of the infection risk between different sexual behaviors.

R denotes the mixing matrix between the risk groups defined by sexual behavior, x;This is the classical oneparameter ‘preferred mixing’ formulation, proposed by [27].$$\mathbf{R}_{x_{i} x_{j}} = q \delta_{x_{i} x_{j}} + (1q) \frac{\sum_{y} \rho_{xy} N_{xy}}{\sum_{x} \sum_{y} \rho_{xy} N_{xy}}. $$

δ denotes Kronecker’s delta.

q denotes assortative coefficient. When q=0, the mixing between risk groups defined by sexual behavior is ‘proportionately mixing’, and the mixing is ‘fully assortative mixing’ when q=1.

S denotes the mixing matrix between sexes;$$\mathbf{S} = \left(\begin{array}{cc} a & 1 a \\ 1a & a \end{array} \right). $$

a denotes the proportion of homosexual behavior.
We will use the special model (11) with transmission rate (12) to estimate the basic reproduction number ℜ_{0} for HSV2 (see the “Results” section), and (10) with (12) to discuss the effectiveness of vaccination strategy (see the “Discussion” section).
Results
The main theorem
The following proposition is proved:
Proposition 1
Ω is positively invariant for system (6).
The main theorem of this paper is as follows.
Theorem 1
Let ℜ_{0} and Ω be defined by (8) and (14), respectively. Let \(\bar {\Omega }\) denote the closure of Ω. (i) If ℜ_{0}≤1, then the diseasefree equilibrium \(E^{0} \in \bar {\Omega }\) of system (6) is globally asymptotically stable in Ω and there exists no endemic equilibrium E ^{∗} in \(\bar {\Omega }\). (ii) If ℜ_{0}>1, then the system (6) has the unique endemic equilibrium E ^{∗} in Ω and it is globally asymptotically stable in Ω.
For the proofs of Proposition 1 and Theorem 1, see the Appendix.
Theorem 1 still works for (10) since it is a special case of (6). In particular, although (10) does not include the integrated time delay, to our knowledge, there is no previous study on the global asymptotic stability of the endemic equilibrium of model (10). From this viewpoint, our main theorem can be regarded as valuable for the empirical study in the subsequent sections.
Estimation of ℜ_{0} for HSV2
Based on Theorem 1, we estimate the basic reproduction number ℜ_{0} for HSV2 in the US from 2001 to 2014. For the estimation of ℜ_{0}, we use the special model (11) with transmission rate (12). Note that (11) corresponds to the case where v _{ i }=σ _{ i }=ω _{ i }=0 for all i∈{1,2,⋯,12}. Although the case where v _{ i }=0 for all i∈{1,2,⋯,12} is excluded under assumption (A4), it is easy to check in a completely similar way as in the Appendix that the global stability result similar to Theorem 1 holds.
The model parameters and related estimates
Parameter  Meaning  Value  Reference 

δ _{ i } (i=1,2,⋯,12)  Relapse risk  1/78.5  [30] 
γ _{ i } (i=1,2,⋯,12)  Recovery rate  1/13  [30] 
μ _{ i } (i=1,2,⋯,12)  Rate of removal from sexual activity  0.0231  [31] 
q  Assortative coefficient  0.3  [32] 
a  Proportion of homosexual behavior  0.02  [33] 
c _{1}  Transmission coefficient for male  0.228  Estimated 
c _{2}  Transmission coefficient for female  1.78  Estimated 
ϕ  Exponent parameter  0.700  Estimated 
ℜ_{0}  Basic reproduction number  2.07  Estimated 
Discussion
Using the demographic and epidemiological parameters obtained above, we discuss the effectiveness of each vaccination strategy. We investigate the sensitivity of the basic reproduction number ℜ_{0} to the vaccination parameters, that is, the vaccination rate among susceptible population v and the vaccination efficacy σ. Here we have assumed that vaccination is conducted with the same rate v for the susceptible population over time. For simplicity, we assume that the efficacy of vaccine σ is the same for all risk groups.
Conclusion
In this paper, we have formulated the multigroup SVIRI epidemic model (6), which enables us to consider the effects of vaccination, the waning of vaccineinduced immunity, and relapse. We have defined the basic reproduction number ℜ_{0} and proved Theorem 1, which states that if ℜ_{0}≤1, then the diseasefree equilibrium E ^{0} is globally asymptotically stable, whereas if ℜ_{0}>1, then the endemic equilibrium E ^{∗} is so. Based on Theorem 1, we have estimated the basic reproduction number ℜ_{0} for HSV2 as 2.07 (95% CI 2.03 to 2.11) by using US HSV2 data from 2001 to 2014. Through the sensitivity analysis for uncertain parameters on sexual behavior, we have found that ℜ_{0} is approximately in the range of 23. Furthermore, using sensitivity analysis for vaccination parameters, we have discussed the effectiveness of the vaccination. As a result, we have come to the following conclusions. (1) Improving vaccine efficacy is more effective than increasing the number of vaccines. (2) Although the transmission risk in female individuals is higher than that in male individuals, distributing vaccines almost equally to females and males is more effective than concentrating them within the female population.
Appendix
Proof of Proposition 1
which is a contradiction. Hence, we see that S _{ i }(t)>0 and V _{ i }(t)>0 for all t>0 and \(i \in {\mathcal {N}}\).
where \(h_{i} (t) := \int _{0}^{+\infty } \delta _{i}(\xi) \gamma _{i} I_{i}(t\xi)e^{\mu _{i} \xi } e^{\int _{0}^{\xi } \delta _{i}(\eta) \mathrm {d}\eta } \mathrm {d}\xi \). We see that h _{ i }(t)≥0 for all \(i \in {\mathcal {N}}\) and t∈ [ 0,t _{1}). Hence, from (15), we have \(I_{\tilde {i}}(t_{2}) > 0\), which is a contradiction. Hence, we see that I _{ i }(t)>0 for all t>0 and \(i \in {\mathcal {N}}\).
The boundedness of the solution of system (6) immediately follows from the fact that N i′(t)=b _{ i }−μ _{ i } N _{ i }(t), S i′(t)≤b _{ i }−(μ _{ i }+v _{ i })S _{ i }(t)+ω _{ i } V _{ i }(t) and V i′(t)≤v _{ i } S _{ i }(t)−(μ _{ i }+ω _{ i })V _{ i }(t) for all t>0 and \(i \in {\mathcal {N}}\). This completes the proof.
Proof of (i) of Theorem 1
In fact, it is easy to see that \(\rho (M^{0}) = \rho (\mathcal {K}) =\Re _{0}\).
Suppose that \((S_{1},V_{1},\cdots,S_{n},V_{n}) \neq \left (S_{1}^{0},V_{1}^{0},\cdots,S_{n}^{0},V_{n}^{0}\right)\). Then (ℓ _{1},⋯,ℓ _{ n })·M(S _{1},V _{1},⋯,S _{ n },V _{ n })<(ℓ _{1},⋯,ℓ _{ n })·M ^{0}=ρ(M ^{0})(ℓ _{1},⋯,ℓ _{ n })=(ℓ _{1},⋯,ℓ _{ n }). Hence, (18) has only the trivial solution such that I _{ i }=0 for all \(i \in {\mathcal {N}}\). This implies that \(\mathcal {L}_{DFE}' = 0\) holds only in the diseasefree equilibrium \(E^{0} \in \bar {\Omega }\). Consequently, from the LaSalle’s invariance principle (see [39]), we can conclude that the diseasefree equilibrium E ^{0} is globally asymptotically stable.
Proof of (ii) of Theorem 1
If ℜ_{0}>1, then \(\left (\ell _{1}, \cdots, \ell _{n} \right) \cdot \left (M \left (S_{1}^{0},V_{1}^{0}, \cdots, S_{n}^{0}, V_{n}^{0} \right)  E_{n} \right) \cdot (I_{1},\cdots, I_{n})^{T} = \left (\rho (M^{0})  1 \right) \left (\ell _{1}, \cdots, \ell _{n} \right) \cdot (I_{1},\cdots, I_{n})^{T} \ > \ 0\). Hence, we see from the third equality in (17) that in a neighborhood of \(\left (S_{1}^{0},V_{1}^{0},\cdots, S_{n}^{0}, V_{n}^{0}\right)\), \(\mathcal {L}_{DFE}' > 0\). This implies the instability of the diseasefree equilibrium E ^{0}.
In order to make this function welldefined, without loss of generality, we can restrict our attention to the solution such that \(I_{i}(s)=\varphi _{i} (s), \ i \in \mathcal {N}\) on (−∞,0], where φ _{ i }(0)=I _{ i }(0) and \(0 < m_{i} < \varphi _{i}(s) < M_{i} < +\infty, \ s \in (\infty, 0], \ i \in \mathcal {N}\) for positive constants m _{ i } and M _{ i }, \(i \in \mathcal {N}\). Then, from the positive invariance of set Ω and the uniform persistence of system (6), we see that the Lyapunov functional \(\mathcal {L}_{EE}\) is welldefined.
We see that all elements in the max in the last expression of the above formula are non positive because of the inequality of arithmetic and geometric means. Similarly, we can easily check that for all unicycles CG with at most n vertices, the second sum in the last expression of (27) are nonpositive (see [10, Proof of Theorem 4.1]). Hence, \(\mathcal {L}_{EE}'\) is nonpositive and it is easy to check that the equality \(\mathcal {L}_{EE}' = 0\) holds if and only if \(\left (S_{1},V_{1},I_{1},\cdots, S_{n},V_{n},I_{n} \right) = \left (S_{1}^{*},V_{1}^{*},I_{1}^{*},\cdots, S_{n}^{*},V_{n}^{*},I_{n}^{*} \right) \). This implies, from the LaSalle’s invariance principle, that the endemic equilibrium E ^{∗} is globally asymptotically stable.
Declarations
Acknowledgements
We would like to thank the editor and anonymous reviewers for their helpful comments to the earlier version of this paper. We would like to thank Dr. Akihiro Ishii for helpful discussions regarding the serological test for HSV2.
Funding
JW was supported by National Natural Science Foundation of China (nos. 11401182, 11471089), Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No. 2014TD005). HT was supported by GrantinAid for JSPS Fellows from the Ministry of Education, Culture, Sports, Science, and Technology in Japan. TK was supported by GrantinAid for Young Scientists (B) of Japan Society for the Promotion of Science (No. 15K17585) and the program of the Japan Initiative for Global Research Network on Infectious Diseases (JGRID); from Japan Agency for Medical Research and Development, AMED. RO was supported by GrantinAid for Young Scientists (B) of Japan Society for the Promotion of Science (No. 15K19217) and Precursory Research for Embryonic Science and Technology (PRESTO) grant number JPMJPR15E1 from Japan Science and Technology Agency (JST). The authors were supported by JSPS Bilateral Joint Research Project (Open Partnership).
Availability of data and materials
The data that support the findings of this study are available in the National Health and Nutrition Examination Survey, “http://www.cdc.gov/nchs/nhanes/”.
Authors’ contributions
JW and XY formulated the model. HT improved the whole manuscript. TK carried out the theoretical analysis of the model. RO carried out the epidemiological study including the estimation of ℜ_{0} for HSV2. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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