Analysis of a non-integer order mathematical model for double strains of dengue and COVID-19 co-circulation using an efficient finite-difference method

An efficient finite difference approach is adopted to analyze the solution of a novel fractional-order mathematical model to control the co-circulation of double strains of dengue and COVID-19. The model is primarily built on a non-integer Caputo fractional derivative. The famous fixed-point theorem developed by Banach is employed to ensure that the solution of the formulated model exists and is ultimately unique. The model is examined for stability around the infection-free equilibrium point analysis, and it was observed that it is stable (asymptotically) when the maximum reproduction number is strictly below unity. Furthermore, global stability analysis of the disease-present equilibrium is conducted via the direct Lyapunov method. The non-standard finite difference (NSFD) approach is adopted to solve the formulated model. Furthermore, numerical experiments on the model reveal that the trajectories of the infected compartments converge to the disease-present equilibrium when the basic reproduction number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_0$$\end{document}R0) is greater than one and disease-free equilibrium when the basic reproduction number is less than one respectively. This convergence is independent of the fractional orders and assumed initial conditions. The paper equally emphasized the outcome of altering the fractional orders, infection and recovery rates on the disease patterns. Similarly, we also remarked the importance of some key control measures to curtail the co-spread of double strains of dengue and COVID-19.

severe respiratory sickness will definitely require serious medical attention 3 .At the onset of the infection, efforts were made to reduce the spread of COVID-19 such as the use of face masks and self-isolation which culminated to lockdowns in many countries thereby crippling the economy.Further intervention was made to reduce the burden of this disease by developing a vaccine which has given hope to a possible end to the pandemic 6 (Fig. 1).
On the other hand, dengue is an acute febrile (feverish) disease induced by dengue virus (DENV) and flavivirus.Dengue virus, which spreads through Aedes mosquitoes, thrives in tropical and sub-tropical regions.Dengue disease is now prevalent in over 100 countries with Asia accounting for about 70% of the disease concentration globally 8 .About 390 million individuals are predicted to be infected by dengue virus each year of which 96 million may show clinical manifestations 9 .Similarly, the WHO has reported 5.2 million cases of dengue in 2019 8 .The exact figures of dengue occurrence are mis-reported because most of the cases are mild and lacking symptoms 8 .Symptomatic cases manifest in the form of high fever, joint pain, rash, nausea etc 8, 10 .Few of these symptomatic individuals may progress to a complicated fever known as "dengue hemorrhagic fever and "dengue shock syndrome".Presently, there are four distinct strains of dengue virus and they include DENV-I, DENV-II, DENV-III and DENV-IV 1,11,12 .It is also pertinent to note that infection with one strain of dengue virus may not provide permanent or cross immunity over the other strains 11 (Fig. 2).
Cases of co-infection of COVID-19 and dengue have been reported by the authors [14][15][16][17] .A co-infection of coronavirus and dengue virus invading a population can be life-threatening.This is due to the fact that co-morbidity cases in co-infection is more dangerous when compared to a sole viral infection 14 .Similarly, high mortality has been associated with individuals co-infected with dengue and coronavirus disease 15 .It has been reported that dengueinfected persons who are co-infected with coronavirus may suffer heightened sickness and hospitalization 2 .Despite the difference in pathophysiologies of the two diseases, the viruses can have impacts inside the body when compared thus, leading to indistinguishable clinical manifestations in a situation of co-infection which contributes to the complication 14 .As a result of corresponding symptoms between the two diseases, the chance of mis-diagnosis of the two infections is always high 16,18 .It is pertinent to note that increased glucose levels may manifest in individuals co-infected with dengue and coronavirus and this often leads to breeding of coronavirus 2 (Fig. 3).www.nature.com/scientificreports/more preference over other existing methods of solution such as perturbation/decomposition schemes 40 .It is hoped that, this study will provide new paths for further research studies in epidemiological modelling.

Preliminaries
Definition 1.1 2,57 The Caputo fractional derivative of order ℑ ∈ (0, 1) can be defined as where n is a natural number satisfying ℑ ∈ (n − 1, n) such that Ŵ(•) is a gamma function given by If 0 < ℑ < 1 , then the above Caputo derivative becomes Definition 1.2 2,50 The Riemann-Liouville fractional integral of a function z of order ℑ ∈ R + is defined as Definition 1.3 2 The Laplace transform, L , of the Caputo derivative is defined as Theorem 1.1 58 Let (M, � • �) be a Banach space and T : M → M a contraction mapping with constant κ ∈ [0, 1) .
Then T has a unique fixed point in M, that is, there exists a unique point x ⋆ ∈ M such that T(x ⋆ ) = x ⋆ .Further- more, for arbitrary x 0 ∈ M , the sequence {x n } defined by x n+1 = T(x n ) , n = 0, 1, 2, • • • converges strongly to x ⋆ .

Model formulation
Let N H (t) be the total number of persons.Then, N H (t) is partitioned into the following mutually exclusive com- partments: Uninfected (susceptible) persons, (S H (t)) , infected persons with dengue strain one, (I 1D (t)) , infected persons with dengue strain two, (I 2D (t)) , infected persons with Coronavirus, (I C (t)) , co-infected persons with dengue strain one and coronavirus, (I 1DC (t)) , co-infected persons with dengue strain two and coronavirus, (I 2DC (t)) , persons who have recovered from both strains of dengue, (R D (t)) and persons who have recovered from coronavirus (R C (t)) such that; Furthermore, Let N V (t) be the total number of vectors (mosquitoes).Then, N V (t) is partitioned into the fol- lowing mutually exclusive compartments: Uninfected (susceptible) vector, (S V (t)) , infected vectors with dengue strain one, (I 1V (t)) and infected vectors with dengue strain two, (I 2V (t)) such that; Uninfected persons, S H are recruited into the population at the rate ω H .The population is depleted as a result of infection with dengue strain one, dengue strain two and coronavirus respectively.This is due to effective contacts with an infected vector(with dengue strain one) at the rate 1V , infected vector (with dengue strain two) at the rate of 2V and infected person with coronavirus at the rate of C .with, , where β 1V , β 2V , β C denote effective contact rates between susceptible humans and infectious vector with dengue strain one, dengue strain two and infectious humans with coronavirus, respectively.Individuals who have recovered from dengue may lose their dengueacquired immunity at the rate θ D and becomes susceptible.Also, Individuals who have recovered from corona- virus may lose their coronavirus-acquired immunity at the rate θ C and subsequently become susceptible.Natural mortality is assumed to be the same for all persons, at the rate µ H Similarly, uninfected vectors acquire dengue strain one or strain two as a result of effective contact with infected persons with dengue strain one or two, respectively, at the rate . Vector removal is assumed at the rate µ V .Other param- eters are detailed in Table 1.
The following assumptions are made in the formulation of the model: (1) (2) Vol.:(0123456789) iii.Co-infected persons may either recover from dengue virus or coronavirus but not from both diseases,simultaneously. iv.Individuals can only be infected with one strain of dengue but not both strains at the same time.v. Transmission rate for an infected and co-infected person are assumed the same.Thus, the model of Caputo fractional order, ℑ ∈ (0, 1) is given by:

Basic properties of the model
In order to examine the mathematical and biological well-posedness of the model (4), we shall establish the nonnegativity of solution.Furthermore, we shall prove the existence and uniqueness of the solution to the model.

Invariant regions
The dynamics of Caputo-Fractional model ( 4) is explored in the feasible region + is positively invariant subject to the nonnegative initial conditions with respect to model (4).

Proof Summing the component equations of the human population of model (4) yields
The above equation can also be rewritten in the form of the inequality below: Applying Laplace and inverse Laplace transforms respectively to the inequality and simplifying, we obtain where, the "Mittag-Leffler function" is defined as Thus, Considering that, E ℑ (−µ H t ℑ ) → 0 as t → ∞ , we have that the total population of humans, (4) Vol

Existence and uniqueness of solution
In this subsection, we shall prove the existence of a unique solution to model (4) from Banach fixed point theorem.Let ) T represent a continuous vector given as follows; Thus, model (2.1) can be re-written as is said to be Lipschitz with respect to the second argument, if the fol- lowing inequality holds where K > 0 is the Lipschitz constant Theorem 3.2 Suppose Eq. ( 9) is satisfied and , then, there exists a unique solution Proof Applying the Caputo fractional integral on both sides of (8) , we obtain 11 where g = [0, l] and with ) endowed with . is a Banach space.It suffices to show that the operator ̺ : C g, R 11 → C g, R 11 is a contraction mapping.Now, since the operator ζ satisfies Lipschitz condition, we have that By taking the supremum over t ∈ g = [0, l] we see that Thus, ̺ is a contraction and by the Banach contraction mapping principle, ̺ has a unique fixed point which is a solution to the initial value problem (8) and thus the solution to the model (4).

Stability analysis of the model The basic reproduction number of the model
By setting the right-hand sides of model ( 4) to zero we obtain the disease-free equilibrium (DFE) as The stability of the DFE is analysed by applying the next generation matrix on model (4).The respective transfer matrices are given below; with, Hence, the basic reproduction number of model ( 4) is given as follows, and R 0C are the respective associated repro- duction numbers for Dengue strain one, Dengue strain two and COVID-19 with The above basic reproduction numbers, can be interpreted epidemiologically, as the average number of secondary infections caused by an infected individual with COVID-19 or dengue (strain one or strain two) in entirely susceptible population 60 .

Local stability of the disease-free equilibrium (DFE) of the model
Theorem 4.1 The model's DFE ( D 0 ) is locally asymptotically stable whenever R 0 < 1 and unstable when R 0 > 1 Proof Analysis of the system around the infection-free equilibrium is done with the help of the Jacobian matrix of system (8) evaluated at the DFE, which is given by: with the eigenvalues given as follows; The remaining eigenvalues can be found from the following characteristics polynomial equations = 0 From Routh-Hurwitz criterion, the above three equations will have roots of negative real parts provided that the associated reproduction numbers R 0C , R 01D and R 02D are less than one.Thus, the DFE, D 0 is locally asymptotically stable whenever the reproduction number, where ℑ ∈ (0, 1) and Vol.:(0123456789) We shall establish the global asymptotic stability of the DFE of the model for a special case.That is, in the absence of co-infection, re-infection and acquired immunity loss to COVID-19 and dengue To achieve the global stability, we shall apply the direct Lya- punov method 61 .
Theorem 4.2 Suppose there is no co-infection, re-infection and acquired immunity loss to COVID-19 and dengue in the model then, the DFE ( D 0 ) of the model is global asymptotically stable (GAS) in ℧ whenever R 0 < 1.
Proof Consider a modified version of the model when both diseases are present in the population but, no coinfection of the two diseases, re-infection and acquired immunity loss to COVID-19 and dengue.Now, we consider the following Lyapunov function, from the approach 1,34,62,63 .
with Caputo fractional derivative, Thus, Furthermore, the variables and param- eters of the model are non-negative.Hence, L 1 is an appropriate Lyapunov function on ℧ .Thus, by Lassel's invariance principle 61 , (I 1D , I 2D , I C , I 1DC , I 2DC , I 1V , I 2V ) → (0, 0, 0, 0, 0, 0, 0) as t → ∞.By substituting Therefore, every solution to model (4) having initial condi- tions in ℧, and with Epidemiologically, in the absence of the co-infection, re-infection and acquired immunity loss to COVID-19 and dengue, both infections can be eradicated when R 0 < 1, irrespective of the initial quantities of the sub-populations.

Global asymptotic stability of the disease-present equilibrium (DPE) of the model (a special case)
Theorem 4.3 Suppose there is no co-infection, re-infection and acquired immunity loss to COVID-19 and dengue in the model (4) then, the DPE ( D 0 ) of the model is global asymptotically stable (GAS) in Proof Consider a special case when there is no co-infection, re-infection with the same or different infection in the model and acquired immunity loss to COVID-19 and dengue.That is, Following the approach 1,34,63,64 , we consider the potential Lyapu- nov function constructed below.
with Lyapunov Caputo derivative, Substituting ( 10) into (11) we have that, From model (10) at steady state, we obtain Substituting equation( 13) into (12), we obtain which can be written as; ) On further simplification we have that, Furthermore, using the fact that geometric mean is less than arithmetic mean, we achieve the following inequalities Thus, L 2 is a Lyapunov function on ℧ such that L 2 ≤ 0 whenever R 0 > 1.Hence, the DPE is globally asymp- totically stable whenever R 0 > 1.
That is, in the absence of co-infection, re-infection and acquired immunity loss, every solution of model ( 4), with initial conditions in ℧ , approaches unique disease-present equilibrium as t approaches ∞ , whenever R 0 > 1.
In relation to epidemiology, Theorem 4.3 implies that in the absence of co-infection, re-infection and immunity loss, the double infections of COVID-19 and dengue (both strains) will persist over the time whenever R 0 > 1.

Numerical scheme of the Caputo fractional model
Then, with where E ℑ,ℑ+1 is the Mittag-Leffler function and h N and N ∈ Z + .Now, we present a Non-Standard Finite Difference (NSFD) scheme for the Caputo fractional model following the approach of 40 .For t > 0 and 0 < ℑ < 1 , the Caputo fractional derivative is expressed as Upon discretizing dϕ(s) ds on [t i , t i+1 ] we have that where h is the mesh size and To determine the denominator function (ψ(h)) , we compare equations ( 14) and (22) Hence, Next, we apply the NSFD scheme on model ( 4) and we obtain the following difference equations Vol.:(0123456789)  4) are set as follows:S H (0) = 3, 600, 000, 000, I 1V (0) = 5, 000 and I 2V (0) = 5, 000 .Fitting of the model to real data was conducted using the MATLAB fmincon optimization algorithm.We relied on the data of Amazonas state, Brazil for COVID-19 66 and dengue 67 curve fittings.Based on the data, we cumulated COVID-19 and dengue active cases for a period of twelve weeks (between February and April, 2021).Within this period, it was observed that, there was a rise in combined incidences of COVID-19 and arboviruses.The estimated parameters and others from the literature, are detailed in Table ().Furthermore, model fitting to the real data are demonstrated in Fig. 4a and b.From the figures, it is clearly shown that, our model has a good fit to the real data.

Comparison of NSFD and ODE45 solver
The numerical simulation for comparing the solutions of NSFD and ODE45 is performed at different values of the fractional-order, ℑ .This is done for each of the disease compartment as shown in Figs. 5 and 6.It is observed that, the solutions of the NSFD scheme are sufficiently close to solutions of the ODE45 solver whenever the fractional-order is close to one.In other words, the NSFD scheme is dynamically-conformable with ODE45 when the fractional-order ( ℑ = 0.98 ) approaches one.This is expected since ODE45 can be seen as a numerical solution to fractional-order one.On the contrary, when the NSFD scheme is simulated against fractional values (23) www.nature.com/scientificreports/( ℑ = 0.85, 0.75 ), it is observed that the corresponding curves differ from the curves obtained from ODE45.It is further observed that, the NSFD scheme is superior to the ODE45 solver since it shows the correct dynamic behavior of the model at different fractional values.Epidemiologically, a faster decay in the evolution of infections is observed at lower fractional values.Reverse is the case when the fractional value is higher.

Impact of fractional order on the dynamics of each compartment
The numerical simulations and solution curves at varying fractional-orders (ℑ : 0.95, 0.85, 0.75, 0.65) are pre- sented in Figs.7 and 8 when R 0 < 1. Figure 8 depicts the solution curves for the human components at different fractional-orders when R 0 < 1.From the perspective of epidemiology, the population of susceptible individuals decays fast and stabilizes at about 3,350,000 over a period of 200 days.This happens when the fractional-order is 0.95 (close to 1).Also, it can be observed from Fig. 8a that, when ℑ is as low as 0.65, the population of sus- ceptible individuals decreases slowly and stabilizes at a higher value of about 3,400,000.As depicted in Fig. 8b  and c, when ℑ = 0.95 (maximum), the populations of human-infected dengue strain one and strain two decay fast from a maximum value of about 200 and 1500 respectively, to a minimum value of about 0, over a period of 200 days.However, when ℑ = 0.65 (minimum), both populations decay slowly, from a maximum value of about 250 and 1800, to a minimum value of about 50 and 500 respectively, over the same period of time.Thus, less people get infected with dengue strain one and two as the fractional-order gets close to one and vice versa.Similar trends are observed in the populations of I C , I 1DC , I 2DC though, with less impact of fractional-orders.
There is something remarkable in the dynamics of recovered individuals.As observed in Fig. 8g, the number of individuals that recovered from dengue (strain one or strain two) decreases from about 4000 to 500 when ℑ = 0.95 and,increases from 0 to 6000 individuals, when ℑ is as low as 0.65.However, there is a direct relation- ship between the value of the fractional-order and, COVID19-recovered persons.Thus, more people recover from either dengue strain one or strain two for less fractional values when R 0 < 1 .A contrary scenario is the case for individuals recovered from COVID-19.Furthermore, it can be shown in Fig. 7 that, the populations of the vector components (S V , I 1V , I 2V ) at any given time is inversely proportional to the fractional-order.That is, the higher the fractional-order, the faster the decay and vice versa.
On the other hand, we examined the impact of fractional-order, (ℑ) on the trajectories of human and vector components when R 0 > 1.This is depicted in Figs. 9 and 10.The simulation is for a period of 200 days with ℑ ranging from 0.65 to 0.95.It can be observed that the fractional-order has a significant impact in the dynamics of both components as observed in the solution curves.From Figs. 9b,c, 10b and c, it can be observed that, the number of individuals and vectors infected with both strains of dengue, decreases from its peak to the lowest number, as ℑ increases from 0.65 to 0.95.The same observation is made for the populations of susceptible humans, suscep- tible vectors and dengue-recovered individuals as shown in Figs.9a, 10a and 9f respectively.Reverse situation is observed for the populations of COVID19-infected individuals, co-infected individuals and individuals recovered from COVID-19.This is presented in Figs.9d, e and g.Thus, with respect to epidemiology, the burden of dengue and COVID-19 infections can be better managed with a good knowledge of fractional-order.(e) I 1DC (t) for NSFD and ODE45.(f) I 2DC (t) for NSFD and ODE45.

Numerical experiment of the reproduction number
In this section, we present the numerical experiment of the associated reproduction numbers, as a response function to the disease transmission and recovery rates.This is illustrated in Fig. 11.It can be observed from the respective Fig. 11a-f that, the value of the associated reproduction numbers depends on the disease transmission and recovery rates.That is, an increase in transmission rates of dengue strain one, strain two and COVID-19, will result to an increase in the respective reproduction numbers.Conversely, an increase in recovery rates of dengue strain one, strain two and COVID-19, will result to a decrease in the corresponding reproduction numbers and vice versa.Epidemiologically, the co-infection or single infections of dengue (both strain one and strain two) and COVID-19 can be abated if the transmission rates are adequately low.More so, the burden of dengue and COVID-19 infections can be reduced when the recovery rates are adequately high.This may be achieved through enhanced recovery strategies and interventions.manner the trajectories approach the DPE over time.It is important to note that, as fractional-order, ℑ gets near 1, the trajectories sufficiently gets close to the disease-present equilibrium and vice versa.Thus, it may be said epidemiologically that dengue and COVID-19 infections will persist within the population if R 0 > 1 and ℑ substantially close to 1.

Conclusion
In this paper, we have designed and analyzed a fractional-order mathematical model for the co-circulation of double strains of dengue and COVID-19 endowed with Caputo derivative.We established the existence and uniqueness of the solution for the given model through some fixed point theorem.The model's solution was analyzed with the non-standard difference scheme.We also highlighted the impact of different Caputo fractional implication is that, while effort is being made to lower the reproduction number, serious effort should be made also by the government and other related health organizations in understanding the impact of fractional-orders in the dynamics and control of dengue and COVID-19 infections.This is to ensure that the fractional-order is high as possible in order to reduce the viral load and increase the number of R D (dengue-recovered individuals).For the particular case of COVID-19-recovered individuals, efforts should be made to keep fractional-order as low possible to enhance the number of individuals recovering from COVID-19.Furthermore, in a situation where the reproduction is greater than unity, it is expected that the co-circulation of the infections will persist.However, a fractional-order as low as possible can substantially reduce the populations of COVID-19 related components while, a fractional-order as high as possible can significantly reduce the number of individuals infected with both strains of dengue virus.
In any case, understanding the phenomenon of fractional-order as applied to disease dynamics will enhance the struggle of health officials in controlling dengue and COVID-19 infections.(ii) It was shown that the disease reproduction numbers are functions of corresponding transmission and recovering rates as depicted in Fig. 11.It follows that an increasing transmission rate results to an increasing reproduction number and a decreasing recovery rate returns a decreasing reproduction number.Thus, to reduce the number of secondary infections caused by a typically infected person to below one, health officials should ensure that the transmission rate is reduced to barest minimum.This may be achieved by the use of treated nets, insecticides etc may reduce transmission rate of dengue virus.Similarly, the use of face-masks, lock-downs as observed during the epidemic can reduce the transmission rate of COVID-19.Furthermore, effort should be be made by health officials and governments to enhance recovery strategies and interventions.This could be possible through equipping the hospitals, sufficient supply of health workers and training etc. (iii) It was observed that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 while, the disease-present equilibrium is globally asymptotically stable when R > 1 .This is demonstrated in Theorems 4. achieved faster when the fractional-order is low.Unfortunately, the co-circulation will persist if average secondary infections produced by an infected is above one.
Our model is primary built on double strains of dengue and COVID-19 co-circulation.Furthermore, our model did not take into account the impact of vaccine for COVID-19 in the system.Future directions of our work could look at other more efficient numerical scheme for obtaining a solution, and other biological aspect of the model such as within-host dynamics.
I C (t) for NSFD and ODE45.
R D (t) for NSFD and ODE45.
R C (t) for NSFD and ODE45.
S V (t) for NSFD and ODE45.
(a) Surface plot of reproduction number of dengue strain one.Contour plot of reproduction number of dengue strain one.(c) Surface plot of reproduction number of dengue strain two.Contour plot of reproduction number of dengue strain two.(e) Surface plot of reproduction number of COVID-19.Contour plot of reproduction number of COVID-19.

Figure 11 .
Figure 11.Surface and Contour plots of respective reproduction numbers as a function of transmission and recovery rates.

Table 1 .
Description of parameters in the model (4).