key: cord-0055998-6e7la1xc
authors: Abdulmajid, Shafiu; Hassan, Adamu Shitu
title: Analysis of time delayed Rabies model in human and dog populations with controls
date: 2021-02-05
journal: Afr
DOI: 10.1007/s13370-021-00882-w
sha: 50aaa83d38c811d431097601847e35ceb54f2e37
doc_id: 55998
cord_uid: 6e7la1xc

Rabies is a fatal zoonotic disease caused by a virus through bites or saliva of an infected animal. Dogs are the main reservoir of rabies and responsible for most cases in humans worldwide. In this article, a delay differential equations model for assessing the effects of controls and time delay as incubation period on the transmission dynamics of rabies in human and dog populations is formulated and analyzed. Analysis from the model show that there is a locally and globally asymptotic stable disease-free equilibrium whenever a certain epidemiological threshold, the control reproduction number [Formula: see text] , is less than unity. Furthermore, the model has a unique endemic equilibrium when [Formula: see text] exceed unity which is also locally and globally asymptotically stable for all delays. Time delay is found to have influence on the endemicity of rabies. Vaccination of humans and dogs coupled with annual crop of puppies are imposed to curtail the spread of rabies in the populations. Sensitivity analysis on the number of infected humans and dogs revealed that increasing dog vaccination rate and decreasing annual birth of puppies are more effective in human populations. However in dog populations, the vaccination and birth control of puppies, have equal effective measures for rabies control. Numerical experiments are conducted to illustrate the theoretical results and control strategies.

model simulations agreed with the human rabies data reported by the Chinese Ministry of Health and predicted that the number of human rabies is decreasing but may reach another peak around 2030. Their study also demonstrated that reducing dog birth rate and increasing dog immunization coverage rate are the most effective methods for controlling rabies in China. In 2015, based on the model of Zhang et al. [34] , Chen et al. in [7] , proposed a multi-patch model to describe the transmission dynamics of rabies between dogs and humans in different provinces. Their study investigated how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China. They found that immigration of dogs can make the disease endemic even if it dies out in each isolated patch (province). Chapwanya et al. in [6] , formulated yet another S E I rabies model in human and dog populations with additional vaccinated compartment in humans. They proposed the discrete counterpart using the nonstandard finite difference scheme.

Introduction of time delay in mathematical modelling has shown significant impacts on dynamics of the system and disease burden. Research has shown the existence of time delay between infection to infectiousness [16] . Time delay can cause equilibria of models to change from stable to unstable or conditionally stable, thereby generating periodic solutions with delay as a bifurcating parameter [9, 14, 35, 36] . Furthermore, models with time delay are shown to decrease disease burden and more suitable for modelling severe acute respiratory syndrome (SARS) of 2003 than models without delay [25] . From the foregoing, an important reality (time delay), ought to be considered in model formulations looking at the wide range of the incubation period and the complexity involved before symptoms appeared. Motivated by the studies [6, 25, 34] , in this article, we formulate and analyze a rabies model with infections from dogs to dogs, and dogs to humans by incorporating time delay as incubation period to form a system of delay differential equations. Furthermore, in order to reduce the menace of rabies in the populations of human and dog, we propose three control strategies; vaccination of humans and dogs, coupled with annual birth control of new born puppies. The main objectives are to study the effects of control strategies and time delay in both the model dynamical properties and the endemicity of rabies virus in the populations. The rest of the article is organized as follows. In Sect. 2, we present the model formulation with basic properties results, followed by existence and stability of equilibria in Sect. 3 . Numerical simulations are presented in Sect. 4 and conclusion in Sect. 5.

The model consists of two populations: humans and dogs leaving in the same environment. At any time t, the human population is sub-divided into three sub-populations of susceptible humans (S h (t)), infected humans (I h (t)) and vaccinated humans (V h (t)). Hence, the total population of humans denoted by N h (t), is given by

Similarly, at any time t, the dog population is sub-divided into three sub-populations of susceptible dogs (S d (t)), infected dogs (I d (t)), and vaccinated dogs (V d (t)) so that the total population of dogs is

The susceptible human population is increased by the per capita growth rate (μ h K h ), where μ h is birth/death rate (which is assumed to occur in all human compartments), while K h is humans annual birth population. It is decreased by administering vaccination (at a 

Here, τ > 0, is the time lag (delay) that accounts for the time between infection and infected stage while e −μ d τ is the probability that rabid dogs survived the natural death over the period [0, τ ], and β hd is transmission rate of rabies from dogs to humans. The infected human class is increased by sufficient contact between susceptible humans and infected dogs (at a rate β hd S h I d (t − τ )e −μ d τ ), and decreased by natural and rabies induced-death at rates μ h and α h , respectively. The vaccinated human population is increased by vaccine dose administered to susceptibles (at rate v) and is decreased due to natural death (at rate μ h ).

Similarly, susceptible dog population is increased by per capita growth (at a rate μ d K d ), where μ d is birth/death rate (which is assumed to occur in other dog compartment), while K d is dogs annual birth rate of newborn puppies. It is decreased by natural death (at rate μ d ), vaccination (at rate kS d ) and by infection when there is sufficient contact between susceptible dogs and infected dogs (at rate β dd S d I d (t − τ )e −μ d τ ). The infected dogs population increase by infection after sufficient contact between susceptible dogs and infected dogs (at a rate β dd S d I d (t − τ )e −μ d τ ), and is decreased by the rate at which the infected dogs die naturally (at rate μ d ), and due to the rabies virus (at rate α d ).

In the formulation of this model, we make the following assumptions:

1. there is no transmission of rabies virus between susceptible and infected humans; 2. there is no transmission between rabid humans and susceptible dogs; 3. there are vaccinations in human and dog populations with long time immunity so that vaccinated populations doesn't revert to susceptibles.

The human and dog populations dynamics in a uniform environment, that may be regarded as a single humans habitat site and single dogs breeding site, is represented by the following system of delay differential equations as illustrated in Fig. (1) . The model variables and parameters are described in Table (1) . 

with initial data,

where ( 1 (t), 2 (t), 3 (t), 4 (t), 5 (t), 6 (t)), 

+ be the humans and dogs valued function such that

). The model system (2.1)-(2.6) can be written as

From the assumed properties of the model, we can establish the following.

exists and is unique for all time.

Proof Since f is continuous and Lipschitzian, it follows from Theorem (2.2) in [13] that the delay differential equation system (2.1)-(2.6) has a unique solution given the initial data (2.7).

is positive for all time t > 0 and bounded in , given the initial data in (2.7) .

Proof Suppose all the initial data in (2.7) are positive. Consider Eq.

Therefore, S d (t) is also positive. Now, with the positivity of S h (t) and S d (t) above, we can deduce from Eqs. (2.3) and (2.6) that

Therefore,

For the other variables (I h (t) and I d (t)), we use the method of contradiction using the approach in [11] as follows. Noting that I h (θ ) and I d (θ ) are positive for any θ ∈ [−τ, 0]. Suppose that by contradiction, the variables are negative, then there exists a timet > 0 such that

which contradicts our earlier assumption. Therefore,

which also contradicts our earlier assumption. Hence,

This proves the first part of Theorem 2.

To show boundedness of solution, from the model equations in (2.1)-(2.6), adding the human subpopulations, we have

(2.10)

Using standard comparison theorem [29] , one can show that

In particular,

Therefore, the feasible solution for the human population in the model (2.1)-(2.6) is in the region,

Similarly, from the model Eqs. (2.1)-(2.6), the total subpopulations of dog at any time t is given as

(2.14)

The solution of (2.13) can be obtained using standard comparison theorem [29] as

Therefore, the feasible solution of dog population in the model is in the region

Therefore, the model (2.1)-(2.6) is mathematically well-posed and epidemiologically meaningful.

The existence and asymptotic stability properties of the model will be explored in this section. At equilibrium,

At equilibrium, we set the right hand sides of Eqs. (2.1)-(2.6) to zero. Thus,

gives the disease-free equilibrium by using appropriate substitutions in other equations, to get

(3.8) The basic control reproduction number denoted by R v is defined as the expected number of secondary cases produced by introducing one infected in a completely susceptible population in the presence of intervention. According to the approaches in [37, 38] , the next infection

Thus, the spectral radius of M 0 gives the basic control reproduction number for the model as

.

Remark 3 It is worth noting here that, all the parameters of R v are defined explicitly in terms of dog's parameters including the time delay τ . Thus for any meaningful control strategy, more attention should be directed towards these parameters as suggested in [6] .

Similarly, substituting S * d in (3.4) and simplifying,

, which holds only if R v > 1. Continuing with these back substitutions, we get unique endemic equilibrium define as

,

when R v > 1.

To establish the local asymptotic stability of the equilibria, we linearize model (2.1)-(2.6) about an arbitrary equilibrium, to have

is the Jacobian with respect to Y (t − τ ) for i, j = 1, 2, 3, 4, 5, 6, evaluated at any arbitrary equilibrium point so that

Next, we seek a solution for the system (3.11) of the form

where C is a constant matrix and λ an eigenvalue. Substituting (3.12) into Eq. (3.11), rearranging and simplifying, for non-trivial solution of (3.11), gives the transcendental equation

where I is a 6 × 6 identity matrix.

Theorem 4 If R v < 1, the disease-free equilibrium E 0 is absolutely stable for all delay τ ≥ 0 and unstable if R v > 1.

Proof Substituting the disease-free equilibrium E 0 in Eq. (3.13), we have the transcendental equation

where

It can be seen that (3.14) has five negative roots

. Therefore the stability of E 0 can now be determined by the distribution of roots for g 1 (λ) = 0. Now,

If R v > 1, then g 1 (0) < 0, and g 1 (+∞) = +∞, hence g 1 (λ) has at least one positive root, therefore E 0 is unstable. This proved the last part of Theorem 4. When τ = 0, from (3.14),

Hence, the root of (3.15) has negative real part. When τ > 0, according to Corollary 2.4 [24] , for a stability switch (a root with positive (negative) real part to cross the imaginary axis) to necessarily occurs, there must be a root λ = ±iy 1 for some y 1 ∈ R + . We assume that λ = iy 1 , is a root of Eq. (3.15). Substituting λ = iy 1 in (3.15), we get

Separating real and imaginary parts, squaring and adding we have

If R v < 1, there is no such y 1 ∈ R + , which shows that (3.15) has no purely imaginary root. This implies that all roots of g 1 (λ) must have negative real parts (no stability switch). Therefore, the disease-free equilibrium E 0 , is absolutely stable when R v < 1 for all delay τ ≥ 0.

In order to ensure total eradication of rabies infection in the populations irrespective of the initial population started with, we now prove the global stability of E 0 .

Proof Here, we use the method of Lyapunov function in conjunction with Lasalle's Invariance Principle as follows. Let 

Therefore, L(U ) is a Lyapunov function. According to Theorem 2.3.1 in [27] as applied in [11] , if R v < 1, there exists the only disease free equilibrium E 0 which is globally asymptotically stable (GAS) in for all delay τ ≥ 0. Hence, proved. (3.16) , G 2 (λ) has five negative roots given as (λ 1 ) and λ 5 = −μ d ). Therefore the stability of E 1 can be determined by the distribution of roots in g 2 (λ) = 0.

When τ = 0, we have g 2 (0) ≥ 0 and lim λ→∞ g 2 (λ) = +∞.

Hence there is no positive root of g 2 (λ) = 0 when τ = 0. Therefore E 1 is stable.

When τ > 0, we assume there is a root λ = iy 2 for any y 2 ∈ R + . Substituting λ = iy 2 in g 2 (λ) = 0, we have g 2 (iy) =iy 2 + (μ d + α d ) − β dd S * * d e −τ μ d (cos y 2 τ − i sin y 2 τ ) = 0. Separating real and imaginary parts, squaring and adding the two parts, we get

Therefore if R v > 1, there is no y 2 ∈ R + such that λ = iy 2 can be a root for g 2 (λ) = 0. From the general theory of transcendental equations, (see [3, 19] ), we conclude that when R v > 1, then endemic equilibrium E 1 is locally asymptotically stable for all delay τ ≥ 0.

Next we look at the global attractiveness for E 1 . 

(3.20)

The Lyapunov function we will consider for the global stability of the endemic equilibrium point E 1 is of the same form as those used in [12, 14, 22, 23] . Thus we let the Lyaponuv function defined as

21) where A and B are constants to be determined. Differentiating (3.21) with the respect to time along the solution of (3.17)-(3.20), we havė

At the endemic equilibrium it can be seen from (3.17)-(3.20) that interior of R 6 + limits to E 1 . Therefore, E 1 is globally asymptotically stable in whenever R v > 1 for all delay τ ≥ 0.

In this section, we present numerical simulations that support the theoretical results obtained in the previous sections. We use the parameter values in Table 2 that are mostly from published literature associated with rabies to illustrate our theoretical results. Moreover, the numerical experiments will be used to show the sensitivity of certain parameters as it affects the endemicity and control of the virus.

Figures 2 illustrate the asymptotic stabilities for equilibria (E 0 and E 1 ) in human population using parameter values in Table 2 . In Figs. 2a, b, the local and global asymptotic stabilities for disease free equilibrium E 0 are displayed using time series and 3D phase portrait respectively. Similarly, Fig. 2c , d, show the corresponding local and global asymptotic stabilities for endemic equilibrium E 1 with parameter values from Table 2 , except for β hd = 0.5, β dd = 6.58 × 10 −3 , K d = 6000 so that R v = 16.97 > 1.

Similarly, in Fig. 3 , the asymptotic stabilities for equilibria (E 0 and E 1 ) in dog population are displayed. In Fig. 3a , b, the local and global asymptotic stabilities for disease free equilibrium E 0 are shown with time series and 3D phase portrait respectively while Fig. 2c, d, show the corresponding local and global asymptotic stabilities for endemic equilibrium E 1 in human subpopulations with parameter values as in Fig. 2c , d above. In both Figs. 2, 3a, b, the value of R v is 0.67.

In order to explore the influence of control strategies on the numbers of infected humans and dogs, we varied the parameters of vaccination and annual crop of puppies as shown in Fig. 4a-d. From Fig. 4a , it can be seen that increasing the rate of human vaccination from v = 0.01 to v = 1.98, the number of infected humans with rabies, after reaching highest peak of 80 people, decline to about 30 infected humans. It can further be observed in each case, the peak and number of infected cases are reducing as the vaccine rate increases. Although human vaccine reduces the infectivity of the virus, however, as shown from Fig. 4a , the rate of human vaccine for rabies must be very high for the basic reproduction number, R v to be Table 2 , except for (c) and (d), with β hd = 0.5, β dd = 6.58 × 10 −3 , K d = 6000 so that R v = 16.97 > 1. In (a) time series of local stability for DFE, (b) 3D phase portrait for global asymptotic stability for DFE, (c) local asymptotic stability for EE (d) 3D phase portrait for global stability for EE less than one. This suggest combine control strategy, especially from the dog population. In line with this and as remarked, we consider the sensitivity of vaccination in susceptible dog's population. As observed from Figs. 4b, c, as k is increased from 0.02 to 0.2, the number of infected humans and dogs decreased to zero with value of R v < 1. The implication of this control is that, the increase in dog vaccination rate can eradicate the disease in both human and dog populations. Another effective control strategy for rabies is to reduce the dog population by culling [20] . However, there are controversies associated with this control strategy ranging from critics by pet owners, animal activists to social factors [5] . For these reasons, we employ the method of reducing the birth rate of new born puppies annually by using immunocontraception as suggested in [5] . Thus, in Fig. 4d , as K d is reduced from 6500 to 4500, the number of infected dogs reduces to almost zero, hence bringing the value of R v to below 1. In this case again, the disease can be eradicated in the population of the two organisms.

Lastly, we consider the influence of time delay (τ ) as incubation period of rabies in dogs. As shown from Fig. 5a , b, as τ is increased from 0.33 to 2, there are rapid decrease in both Table 2 , except for (c) and (d), with β hd = 0.5, β dd = 6.58×10 −3 , K d = 6000 so that R v = 16.97 > 1. In (a) time series of local stability for DFE, (b) phase portrait for global asymptotic stability for DFE, (c) local asymptotic stability for EE (d) phase portrait for global stability for EE the number and peaks in graphs of infected humans and dogs respectively. This suggest that if the incubation period can be elongated by any possible technique, the endemicity of rabies can be reduced.

In this article, we formulate and analyze a rabies model in co-population of humans and dogs by incorporating three control strategies (vaccination of dogs and humans coupled with annual birth of puppies) and time delay as incubation period to form a system of delay differential equations. This serves as an extension to several models proposed in [5, 6, 34] with respect to the control strategies and model dynamics. Basic properties of the model as par the theories of delay differential equations are established and the model is well-posed mathematically and biologically. The main findings of the study are summarized as follows:

(i) Two equilibria are identified viz: disease free and endemic equilibria. Basic control reproduction number R v , is obtained and all parameters are defined in terms of dog population. Disease-free and endemic equilibria are shown to be both locally and globally asymptotically stable whenever R v is less than/greater than unity and unstable otherwise, respectively, for any delay value. Thus in the former case, rabies can be eradicated if R v can be reduced and maintained below one. (ii) Human vaccination for rabies is found to reduce the infectivity in humans, however, combining this with dog vaccination can eradicate the disease in humans. (iii) Similarly, administering vaccination on susceptible dogs can eradicate rabies in dog population in about 10 years. (iv) Decreasing the crop of new puppies annually using immunocontraception can also eliminate rabies in dog's population within 10 years. (iv) Increasing time delay as incubation period is shown to decrease the infectivity of rabies in both human and dog populations. 

K d =6500 K d =6000 K d =5500 K d =5000 K d =4500 (d)

Disclosure statement The authors hereby declare that there is no conflicts of interest whatsoever.

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 (3.27) Substituting, the derivatives from (3.17)- (3.20) and further with the constants from (3.24)- (3.27) , in Eq. (3.23), we havė(3.28)Expanding and simplifying (3.28), we geṫ From the last two terms in (3.30) , using the fact that geometric mean is less than or equal to arithmetic mean, it impliesV (t) ≤ 0.