Dynamical Analysis of an SEIRS Model With Incomplete Recovery and Relapse on Networks and Its Optimal Vaccination Control

. For some infectious diseases, such as herpes and tuberculosis, there is incomplete recovery and relapse. These phenomena make them diﬃcult to control. In consequence of this status, an SEIRS epidemic model with incomplete recovery and relapse on networks is established and the global dynamics is studied. The results show that when the basic reproduction number R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable; when R 0 > 1, the endemic equilibrium is globally asymptotically stable. In addition, in consideration of vaccination control strategy, an SVEIRS model is introduced and the optimal control is solved. At last, the theoretical results are illustrated with numerical simulations.


Introduction
In conventional mathematical models of epidemic, the population is usually divided into several compartments, which may include S (susceptible), E (infected but not fell ill and not infective, and also called exposed), I (fell ill and infective), R (recovered and immune). A susceptible can be infected by an infective individual and be fitted into either E if it is not infective yet or I if it is infective immediately. An exposed individual can be fitted into I when it is infective and fell ill. An infective individual can be fitted into R if it recovers and it is immune for the disease, or be fitted into S if it recovers and it is not immune. A recovered individual can also be fitted into S when it lost immunity for the disease. For some diseases, there is incomplete recovery, and when incomplete recovery occurs, an infective individual can be fitted into E again.
Moreover, relapse may occur, that is, a recovered individual can be fitted back into E for reactivation of the bacteria. Herpes [1] and tuberculosis [2,3] are of this type.
Researches on incomplete recovery and relapse have been done for several decades.
Tudor established an SIRI model for herpes and analysed the local stability of the equilibria [1]. Moreira and Wang extended the model and analyzed the global dynamical behaviour [4]. Feng et. al formulated one-strain and two-strain SLIT models of tuberculosis with incomplete treatment and relapse, and analyzed the global dynamical behaviour [5]. As an example, the basic reproduction number of the onestrain model was calculated by van den Driessche and Watmough [6]. According to herpes, van den Driessche and Zou analyzed a general relapse phenomenon by an infectious disease model (SIRI model) of integro-differential equations and gave the global asymptotical behavior of the model [7]. According to tuberculosis, van den Driessche et. al proposed a general mathematical model (SEIRI model) with latent and relapse and gave the global asymptotical behavior [8]. Based on the model of van den Driessche et. al [8], Liu et. al introduced nonlinear incidence of infection and analyzed the local asymptotical stability of the disease-free equilibrium, existence of the endemic equilibrium and the persistence of infection [9]. Georgescu and Zhang introduced nonlinear incidence of infection, recovery and relapse to an SIRI model and analyzed the global behavior [10]. Chen et. al proposed an SIRIS model with relapse and nonlinear incidence and analyzed the global behavior [11].
The researches above have a common assumption: homogeneous host and homogeneous contact. Taking the heterogeneous host into account, Feng et. al [12], Wang and Shu [13] studied multi-group epidemic models with latency and relapse, respectively. Pastor and Vespignani introduced contact heterogeneity into a traditional SIS model by scale-free networks for the first time [14]. In the past two decades, there have been a lot of significant works of epidemic models on networks [15]. But as far as we know, results of incomplete recovery and relapse on networks are scarce. Huo et. al introduced an SIRS model with relapse on a scale-free network, analyzed the global behavior, studied proportional immunization and targeted immunization [16].
Lahrouz et. al studied the global dynamics of an SIRI model with relapse on networks [17].
The purpose of studying infectious disease models is to control the spread of infectious diseases. As an infection control precaution, vaccination has been introduced into epidemic models for a long time [18,19]. Vaccination is a method widely used in the world which can prevent infection by improving the immunity of individuals.
The advantage is that it can reduce infection and the pressure on the health system.
For an individual, vaccination will not affect the normal life and work. Moreover the cost of vaccination is much lower than the cost of treatment after infection. However, vaccination and treatment take different ways. The target of treatment is the infected individuals. The target of vaccination is uncertain. The number of individuals which need vaccination is also uncertain. So an optimal vaccination strategy is required to achieve these goals: the least vaccination number, the least cost and effective control of disease spreading.
The method of solving the optimal control above is provided by the optimal control theory [20,21]. Using the optimal control theory, optimal vaccination control strategy has been studied for a long time [22][23][24]. Chen and Sun established and solved an optimal control problem of vaccination about SIRS model on networks [25].
Following [25], they added medication to the optimal vaccination control and solved the new problem [26].
On the basis of previous works, with the introduction of latency period, an SEIRS model with incomplete recovery and relapse on networks is established in Sect. 2.
The global dynamics and the optimal vaccination control are studied. In particular, in Sect. 3, the global dynamics is studied. In Sect. 4, the optimal vaccination control is solved. In Sect. 5, numerical simulations is performed for sensitivity analysis and to support the theoretical results. In Sect. 6, the paper ends with discussion.

Model formulation
We propose and study an SEIRS model with incomplete recovery and relapse on scale-free networks: (2.1) In the process of disease transmission, there are four states: S k (susceptible), E k (exposed), I k (infected), R k (removed). Following [14], P k (k = 1, 2, · · · , n) is the degree distribution, ⟨k⟩ = ∑ k kP k is the average degree, and θ is the link probability pointing to an infected individual: Assume that b = d, that is, the population of nodes with the same degree is constant and when one individual is removed from a node with degree k, another individual with degree k will be added. So S k (t) + E k (t) + I k (t) + R k (t) = 1, and model (2.1) can be rewritten as: The transmission flow is shown in Fig. 1 is a positively invariant set for model (2.3).
It is easy to find that E 0 = (1, 0, 0, 0, 1, 0, 0, 0, · · · , 1, 0, 0, 0) is a disease-free equilibrium of model (2.3). Next we will study the basic reproduction number R 0 by the method of van den Driessche and Watmough [6], Thus we have Remark 3.1. If the birth, death and incomplete recovery are not considered, , which reduces to the result in Liu and Zhang [29].

Optimal vaccination control
In order to save the vaccination control cost and ensure the achievement of the vaccination control goal, we will use the optimal control theory to establish and solve a time-varying vaccination control problem. Define the time-varying vaccination T > 0 is the terminal time. Then the optimal vaccination control problem of model (2.3) can be written as where V k (t) is the density of people with immunity after vaccination at node k, w is the rate of loss of immunity. The objective function is and L(v(t)) is the Lagrangian function: where B k is the ratio of the cost of immunization to the cost of treatment. Our goal is to find an time-varying vaccination control (optimal control) v * such that Next, use the same method as [27,28], we obtain the existence of the optimal vaccination control of Eq. (4.1): Theorem 4.1. There exists an optimal vaccination control v * (t).

Numerical simulations
In this section, numerical simulations will be presented to show the correlation and sensitivity of the basic reproduction number to the parameters, to illustrate the global behavior of model 2.3 in Sect. 3, and to show the optimal control problem (4.1) in Sect. 4. Use 4 steps Runge Kutta algorithm, the differential equations are solved by Matlab. The dynamical process is considered on powerlaw networks with P k = ck −γ and ∑ n k=1 P k = 1, ⟨k⟩ = ∑ n k=1 kP k = 1.3668, ⟨k 2 ⟩ = ∑ n k=1 k 2 P k = 5.6510. For simulation, γ = −3, n = 500, T = 30, other parameters are listed in Table 1. Fig. 2 shows the sensitivity of the basic reproduction number R 0 to γ 1 , γ 2 , δ 1 and δ 2 . R 0 decreases with the increase of γ 1 , γ 2 , δ 1 , and increases with the increase of δ 2 .
The incomplete recovery is related to γ 1 and the relapse is related to δ 2 . Only relapse can lead to the increase of R 0 . Fig. 3 shows that the correlations of γ 1 , γ 2 , δ 1 , δ 2 and R 0 are significant, and only the correlation of δ 2 and R 0 is positive. Therefore, the correlations may be influenced by the parameter values. Following the definition of the sensitivity index [33,34], if x is a parameter, the sensitivity index of R 0 with respect to x can be written as   Fig. 7 shows similar results as those of Fig. 6. The difference is that the total cost increases with the increase of B k . And the difference in cost also increases with the increase of B k . In addition, different value of B k leads to different optimal control. To see the optimal control degree in details, control degree of each node is shown in Fig. 8. Consistent with Fig. 6 and Fig. 7, Fig. 8 shows that control degree decreases as times go on. When B k = 0.05, for each node, the control degree is different. The control degree of nodes with small degree and nodes with big degree are both strong. When B k = 0.1, the result is similar to that of the case of B k = 0.05, but when n > 200 the result is complicated.

Conclusion and Discussion
In this paper, an SEIRS model with incomplete recovery on scale-free networks is proposed. We analyzed the global behavior of the model. R 0 is obtained. In addition, when R 0 ≤ 1, the disease-free equilibrium E 0 is globally stable; when R 0 > 1, the The numerical simulation shows that optimal control can restrain the infected density as well as the maximal control but the cost is less. And the optimal control degree varies as the ratio of vaccination cost to treatment cost varies.
In our model, appropriate vaccination can reduce the infected density to a very low level but can not clear up the infection. The reason may be that incomplete recovery and relapse may lead to complicated situation. In our model, the study is based on the assumption that incomplete recovery and relapse are independent. In fact, incomplete recovery can lead to relapse. There is no detailed information about the relationship between incomplete recovery and relapse in existing studies. Thus the relevant research is expected.
Herpes and tuberculosis are infectious diseases spread all over the world. Due to incomplete recovery and relapse, they cannot be cleared up despite of the high degree of attention. The efforts of medical workers are expected to remove incomplete recovery and relapse. And we hope that our research will have practical value in controlling the spread of similar diseases. Figure 1 The disease transmission ow Global stability of the model Figure 6 Sensitivity of control strategies