Effects of heterogeneous susceptibility on epidemiological models of reinfection.

This paper studies an epidemic model with heterogeneous susceptibility which generalizes the SIS (susceptible-infected-susceptible), SIR (susceptible-infected-recovered) and SIRI (susceptible-infected-recovered-infected) models. The proposed model considers the case that some infected people are susceptible again after recovery, some infected people develop immunity after infection, and some infected people are reinfected after recovery. We perform a comprehensive theoretical analysis of the model, showing that under appropriate initial conditions, delayed outbreak phenomenon occurs that can give people false impressions. Moreover, compared with the SIRI model, the proposed model exists the delayed outbreak phenomenon under more probable conditions. Finally, we present a numerical example to illustrate the effectiveness of the theoretical results.

Introduction

As the COVID-19 continues to spread in various countries, it has a huge impact on society and economy. Because this virus is very contagious and can continue to be infected after the infection is recovered, the traditional virus transmission model cannot describe the transmission law of this virus well, and many scholars have carried out research on the transmission of COVID-19 [1-11]. The most important feature of COVID-19 is its rapid spread, which places very high requirements on epidemic prevention measures.

The spread of the virus is generally can be described by the compartment model, and the basic models include SIS and SIR [12-15]. The SIS model describes the situation that all recovered individuals are not immune and cannot avoid becoming susceptible again. Compared with the SIS model, the SIR model describes the situation that all recovered individuals gain permanent immunity. For many viruses, such as COVID-19 [16, 17] and influenza [18], many people have antibodies after infection, but the antibodies only last for a certain period of time. Reinfection models can be used to describe the situation that some population obtain partial immunity [19-21].

Due to the repeated infection of the virus, many new phenomena will appear; for example, the basic reproduction coefficient cannot be used to judge the pandemic of the virus or the resurgent epidemic phenomenon [20, 22–24]. By analyzing a Markovian SIRI model, the authors in [22] found that whether the virus is a pandemic depends on the initial number of people infected. For a SIRI model which describes the situation that it is impossible for a person to become susceptible again after first exposure to infection, the authors found a bistable phenomenon when the reinfection rate was high [23]. For some appropriate initial conditions, the proposed SIRI model existed the resurgent epidemic phenomenon [23]. The reference [24] extended the results of reference [23] to the network case and obtained parameter areas for different dynamic phenomena.

In above references about SIRI model, assuming that everyone cannot become susceptible again after first exposure to infection. This assumption does not take into account the situation in which some people become susceptible and then become infected again. Moreover, generally speaking, the susceptibility of different individuals after recovery from infection varies from person to person [25, 26]. Although there were some references about heterogeneous susceptibility [25-29], to the best of our knowledge, there is no research on the impact of heterogeneous susceptibility to SIRI model with partial immunity.

In this paper, we study a SSIRI model with heterogeneous susceptibility which generalizes the SIS, SIR and SIRI models. In the proposed model, some infected people are susceptible again after recovery, some infected people develop immunity after infection, and some infected people are reinfected after recovery. Compared with the SIRI model in [23], we consider the case that some infected people are susceptible again after recovery and heterogeneous susceptibility due to partial immunity. We analyze the stability of equilibria and the dynamic process of virus spread. Under appropriate initial conditions, the delayed outbreak phenomenon will occur, giving people an illusion. Compared with the SIRI model [23], the proposed model exists the delayed outbreak phenomenon under more probable conditions.

This paper is organized as follows. In Sect. 2, we introduce the SSIRI model. In Sect. 3, we give a comprehensive theoretical analysis of the model (1) and classify the disease transmission dynamics. In Sect. 4, a numerical example about COVID-19 is proposed to illustrate the effectiveness of the theoretical results. Section 5 summarizes our conclusions and describes future work.

Model description

Consider the population can be divided into three classes: susceptible (S), infected (I), or recovered (R). Suppose that the total population remains the same over time, and the recovered population can be divided into three parts, that is, a part will become susceptible again, a part will develop immunity, and a part will reinfect again. The density of susceptible part can be partitioned into two parts due to repeated infection, that is, and . Moreover, the infected people acquires susceptibility and after recovering from infection, with probabilities and , respectively . Suppose that the recovered individuals become infected through contact with already infected individuals at rate , and the recovery rate is . Fig. 1 depicts the flow diagram of the model, and we call it SSIRI model. Based on the above assumptions, we get the following equation:

Fig. 1

The flow diagram for model

We consider the following initial conditionand .

From model (1), one can easily see that for all . LetHence, is the invariant set for the model (1).

Model analysis

This section will give a comprehensive theoretical analysis of the model (1). The model (1) will show various behaviors under appropriate conditions. We first prove the stability of equilibria in model (1) and then we will present a detail dynamic process analysis the model (1).

Stability of equilibria

This section will analyze the equilibria of model (1), and prove the stability of equilibria. For the model (1), since , one can obtainLet be the equilibrium of the model (3). If , then we call it infection-free equilibrium (IFE). When , we call it the endemic equilibrium (EE). It is easy to see that the model (3) exists a set of infection-free equilibriums which forms a manifold. The IFE set . The endemic equilibrium (EE) satisfies and

Theorem 1

Consider the model (3). If , then the IFE set is global asymptotically stable.

Proof

Let be a Lyapunov candidate. Since , then one can obtain from system (3) thatHence, when , for all . If , then for all , and IFE set is global asymptotically stable. The proof of this theorem is complete.

Theorem 2

Consider the model (3). The model (3) has a unique EE if and only ifWhen the EE exists, then the EE is local asymptotically stable.

By the analysis of above, the model (3) has a unique EE if and only if in (4) satisfies .

In order to prove the local stability of the EE, we compute the Jacobian of model (3) at EEwhereSinceall eigenvalues of Jacobian are negative, and the EE is local asymptotically stable. The proof of this theorem is complete.

Remark 1

Note that the IFE is not an isolated equilibrium, but a manifold. We just obtain the sufficient condition such that the IFE set is global asymptotically stable. The EE does not necessarily exist. Moreover, we only get the locally stable condition for EE.

As pointed out in above theorem that the model (3) has a unique EE if and only if the inequality (6) holds. If we setthen the inequality (6) can be presented as

Dynamic process analysis

In this section, we will analyze the dynamic process behavior of model (1). One can calculate the basic reproduction number by the next generation matrix methodBased on (9), we can obtain from (1) thatHence, it is easy to obtain the following result.

Lemma 1

Consider the model (1). Then

is greater than 1 if and only if is greater than 0.

equals 1 if and only if equals 0.

is less than 1 if and only if is less than 0. Therefore, I(t) initially increases when is greater than 1 and decreases when is less than 1.

LetDefineSince is the equilibrium of the model (1), if and , then . It is easy to see that , . We can obtain the following equations from (12) and model (1)whereIn this paper, we assume that , that is . Next we will introduce the lemma about I(t) as time goes to infinity.

Lemma 2

Consider the model (1). If equals 0, then I(t) tends to 0 as time tends to infinity. If is greater than 0, then I(t) either tends to 0 or as time tends to infinity.

From (13), it is obvious that and y(t) are both positive and decreasing function. Hence, , exist. One can know from the equation about I(t) in (13) that exists. From (13), it holds that either and or . Since ,Then when and . Thus, we can obtain that either or . When , if , then one can obtain from the third equation in (13) that I(t) is unbounded. Hence, . When , then or . The proof of this lemma is complete.

We definewhere , . One can obtain from (13) thatHence, it is easy to see thatDefineIt follows that . Then we can getFrom (19) and (20), it is easy to obtain thatBy a simple computation, we can getThus, we can obtain from Lemma 1 thatNext we will analysis the value of at in different case. Since the conclusion and proof of is similar to , thus without loss of generality, we consider the cases and , respectively. It is easy to obtain the following lemma.

Lemma 3

(1) If then(2) If , then

From (20) we can compute thatIt is easy to getDefineThen we can get the following lemma.

Lemma 4

(1) If , thenfor . (2) If then

If , then

If , then , for all .

(1) If , from (23), we can getNote that , thus we can obtain the conclusion.

(2) If , we can rewrite the equation (23) asLet , has a solution of nonzero , it is defined by (24). Since where ; thus, we get when and when . If , then , and . The proof of this lemma is complete.

The following proposition shows the dynamic behavior of I(t).

Proposition 1

Assume that . (1) When , the infected population density I(t) has a unique maximum value.

(2) When , the infected population density I(t) decreases monotonically.

(1) From Lemma 3, we can get when and when . From Lemma 4, it is easy to obtain when . Hence, is monotonically decreasing on , and has a unique zero on (0, 1). Hence, the infected population density I has a unique maximum value on .

(2) Since , then we get on , thus does not have an extremum on (0, 1). Hence, the infected population density I is decreases monotonically on . The proof of this proposition is complete.

The following theorem presents the asymptotic behavior of I(t) when .

Theorem 3

Assume that . If is greater than 0, then the infected population density I(t) tends to as time tends to infinity. If equals 0, then the infected population density I(t) tends to 0 as time tends to infinity.

From Lemma 2 and Proposition 1, and the fact that , it is easy to obtain that when . Based on Lemma 2, when . The proof of this theorem is complete.

From (24) it is easy to see that the condition for the existence of if and only if and . From (20), if , then we can getWe define is a solution of above equation at (0, 1), that isDefineIn the following, we analyze the extremum of .

Lemma 5

exists and is a minimum value if the following conditions hold. (1) , and . (2) , .

.

and .

, and .

(1) Based on Lemmas 3 and 4, if and , then we get and . Since , thus . From (25), if , we can getandSince , we get and if , if . Then it is easy to see is a minimum value.

(2) If and , we get on , on and . (a) Since , thus . It is easy to see whatever or , exists and is a minimum value. (b) Since , . If , notes at , then it is easy to obtain . Hence, exists and is a minimum value. (c) Since , . If and , then it is easy to obtain that there is a solution on (0, 1) for . Meanwhile, since on and on , we can get is a minimum value. The proof of this lemma is complete.

In the sequel, we consider the case that . Firstly, we analyze the situation .

Proposition 2

Assume that and .

Suppose that .

If , then the infected population density I(t) has a unique maximum value.

If , then the infected population density I(t) decreases monotonically.

If , then the infected population density I(t) is monotonically increasing when , but decreasing when .

Suppose that .

Suppose that .

If holds, then the infected population density I(t) decreases monotonically.

If holds, then the infected population density I(t) decreases monotonically when , and I(t) has a unique minimum value when .

If , then I(t) is increases monotonically.

(1) Based on Lemmas 3 and 4, if , then and . Therefore, is monotonically decreasing. (a) If , then . Hence, on , and I is monotonically decreasing for all . (b) If then , then there is a unique solution of on (0, 1). Hence, I has a unique maximum value for all .

(2) If , then and . Thus, the infected population density I increases monotonically when and I decreases monotonically when .

(3) If , then and . Therefore, increases monotonically. (a) If , then . (i) Suppose that holds. Notes that I is a positive function, therefore the infected population density I decreases monotonically for all . (ii) Suppose that holds. If , it is easy to see that the infected population density I decreases monotonically for all similar to the proof (i). If , then the infected population density I has a unique minimum value for all . (b) If , then . Hence, on , and the infected population density I increases monotonically for all . The proof of this proposition is complete.

For , based on Proposition 2, we have the following theorem.

Theorem 4

Assume that and .

Suppose that . If , then the infected population density I(t) tends to as time tends to infinity. If equals 0, then the infected population density I(t) tends to 0 as time tends to infinity.

Suppose that .

If , then the infected population density I(t) tends to as time tends to infinity.

If , then the infected population density I(t) tends to (0) as time tends to infinity when ().

Suppose that .

Suppose that .

If , then the infected population density I(t) tends to 0 as time tends to infinity.

If , then the infected population density I(t) tends to 0 () as time tends to infinity when ().

If , then the infected population density I(t) tends to as time tends to infinity.

Note that , then . Hence, when . From the analysis of Proposition 2, it is easy to obtain the conclusion. The proof of this theorem is complete.

In the following, we will analyze the other situation.

Proposition 3

Assume that and .

Suppose that .

If , then the infected population density I(t) is decreases monotonically.

If , then the infected population density I(t) decreases monotonically when and has a unique minimum value when .

Suppose that .

Suppose that .

If , then the infected population density I(t) decreases monotonically.

If , then the infected population density I(t) decreases monotonically when and has a unique minimum value when .

Suppose that holds. Then the infected population density I(t) increases monotonically.

Suppose that .

If or , then the infected population density I(t) increases monotonically.

If and , then the infected population density I(t) has a unique maximum value when , and has a unique minimum and maximum value when .

If and , based on Lemmas 3 and 4, then , on and on . Thus, increases monotonically on and decreases monotonically on .

This proof is similar to Proposition 2 (3) (a), here we omit.

If , then . (a) Suppose that holds. Then has a unique minimum value, and . Note that I is a positive function. Thus, (i) if , then I decreases monotonically for all and (ii) if , then I has a unique minimum value on . (b) Suppose that holds. Then on (0, 1), and I increases monotonically for all .

If , then . (a) Suppose that or holds, then on (0, 1), and I increases monotonically for all . (b) Suppose that and . Note that , then when . If , I has a unique maximum value. If , then we get that I has a minimum and maximum value for all . The proof of this proposition is complete.

For and , based on Proposition 3, we have the following theorem about the asymptotic behavior of I(t).

Theorem 5

Assume that and .

If , then the infected population density I(t) tends to 0 () as time tends to infinity when ().

Suppose that .

If , then the infected population density I(t) tends to 0 () as time tends to infinity when ().

If , then the infected population density I(t) tends to as time tends to infinity.

Suppose that .

If or , then the infected population density I(t) tends to as time tends to infinity.

If and , then the infected population density I(t) tends to 0 () as time tends to infinity when ().

From the analysis of Proposition 3, it is easy to obtain the conclusion. Note that is a minimum value of , thus when . The proof of this theorem is complete.

Summarizing the above results, Fig. 2 depicts the parameter area considered in Propositions 1, 2 and 3 at different cases. Note that b is fixed in Fig. 2.

Fig. 2

a Parameter areas at the case that . b Parameter areas at the case that

Precisely, when , the parameter area can be classified into four areas: areas (A1), (A2), (A3) and (A4). The parameter area (A1) represents endemic case, including the following two partsIn the area (A1), one has . I(t) increases monotonically in area (A11) and I(t) has a peak in area (A12).

The parameter area (A2) represents epidemic case and is defined as follows:It is shown that I(t) has a unique peak and . Here holds and no EE exists.

The parameter area (A3) represents the demise of the disease and contains the following two parts:In the area (A3), and I(t) decreases monotonically. Note that in the area (A32) the EE exists and holds. From Theorem 4, since , the disease is extinct.

The disease delayed outbreak phenomenon occurs in area (A4). In this area, I has a unique minimum value and I eventually tends to the EE. Note that in the area (A4) the EE exists and holds. The area (A4) is presented asWhen , parameter area can also be classified into four areas: areas (B1), (B2), (B3) and (B4). The area (B1) is defined as follows:In the area (B1), . I(t) increases monotonically in the areas (B11) and (B12). I(t) has a peak in the area (B13).

In area (B2), I(t) has a unique peak and . Here and no EE exists; it is corresponding to the epidemic case. The parameter area (B2) is defined as follows.Similar to the area (A3), the parameter area (B3) depicts the extinct of disease and contains the following three parts:Similar to the area (A4), in the parameter area (B4) the disease delayed outbreak phenomenon occurs. It is defined as follows.In this area, I(t) has a unique minimum value and eventually tends to the EE. Note that I(t) has a minimum and maximum value in the area (B42).

Since and , it is easy to get from (11) thatHence, the areas (A1),(A2),(A3), (A4) and areas (B1), (B2), (B3), (B4) can be transformed in areas about parameters . When the parameter is fixed, Fig. 3 depicts the areas about . , , , and . The red and green areas present areas (A1) and (A2), respectively. The blue and yellow areas present areas (A3) and (A4), respectively.

Fig. 3

Classification of disease transmission dynamics in parameter plane when the parameter is fixed

Simulation

In this section, based on UK COVID-19 data and the proposed model in this paper, we will discuss how the number of infections varies with different parameter values. Various phenomena illustrate the validity of the obtained results. We divide the British population into two parts. One part consists of young children (0-14 years old) and seniors ( years old). The another part consists of people aged 15-65. The COVID-19 data come from the Office for National Statistics [30]. According to the Office for National Statistics, there are 3.14 million infected. We assume 0.868 million people recovered among infected people. The part accounts for of the total population and the part accounts for of the total population. As pointed out in [31], there is a basic value for infectious disease (the probability of transmission). When an infected person comes into contact with a susceptible person, the probability of who will be infected is . In this paper, we consider the initial population density distribution =[2.09e07, 3.79e07, 2.28e06, 8.68e05]/6.66e07. We assume for part, and for part. In addition, we assume that the relapse rate , where .

We consider the situation that . Fig. 4 displays the densities of infected people in different conditions. In Fig. 4a, we choose , and , the parameters lie in the area (B1). Hence, the conditions of Theorem 3 hold, and the endemic case occurs. In Fig. 4b, we choose , and , the parameters lie in the area (B2). As shown in above, the disease has a unique peak and . We choose , and in Fig. 4c, the parameters lie in the area (B3). It is shown that the disease is extinct. We choose , and in Fig. 4d, the parameters lie in the area (B4). In this case, the conditions of Theorem 5(3)(b) hold and . It is shown that disease has a unique minimum and maximum value, and . As shown in Fig. 4d, the disease will outbreak after a certain time delay.

Fig. 4

The densities of infected people in different conditions

Conclusion

In this paper, we investigated an epidemiological model with heterogeneous susceptibility that recapitulates SIS, SIR and SIRI models. We analyzed the stability and dynamic process of proposed model. With the change of parameter values and initial conditions, the dynamic process of the model may show four situations, that is, endemic, extinction, epidemic and delayed outbreak cases. We obtained the parameter value area corresponding to each situation. For the delayed outbreak situation, the epidemic curve initially increases and decreases after a certain time delay, but eventually increases and tends to EE. The delayed outbreak phenomenon may occur even though the basic reproduction number is less than one. Finally, we used the UK COVID-19 data to illustrate the effectiveness of the obtained results. Future research directions include multi-virus competition scenarios or multi-group network models.